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  • Home
  • What are A Levels?
  • Exam boards and how the exams work
  • Grades, marks and UCAS points
  • Choosing your subjects
  • How to use this guide
The road to exam day
  • The preparation road at a glance
  • Build your personal study plan
  • Phase 1: Fix your foundations
  • Phase 2: Learn the content
  • Phase 3: Practise and test yourself
  • Phase 4: Past papers and mark schemes
  • Phase 5: Final revision
  • Phase 6: Exam day, results and resits
Skills that raise grades
  • Study methods that actually work
  • Exam technique
  • Command words decoded
  • Motivation, sleep and stress
  • The best resources by subject
Beyond the exams
  • Studying as a private candidate
  • University applications and admissions tests
Before you start
  • Mathematics: the course map
  • Numbers, fractions and percentages
  • Algebra from scratch
  • Indices, surds and standard form
  • Graphs and straight lines
  • Geometry and trigonometry basics
Pure Mathematics 1
  • QuadraticsAS
  • FunctionsAS
  • Coordinate geometry: lines and circlesAS
  • Circular measureAS
  • TrigonometryAS
  • Series: binomial expansion and progressionsAS
  • DifferentiationAS
  • IntegrationAS
Pure Mathematics 2 and 3
  • Modulus and polynomialsAS
  • Logarithms and exponentialsAS
  • Further trigonometryAS
  • Further differentiationAS
  • Further integrationAS
  • Numerical solution of equationsAS
  • Partial fractions and the binomial seriesA2
  • Integration techniques: parts, substitution and partial fractionsA2
  • Differential equationsA2
  • VectorsA2
  • Complex numbersA2
Probability and Statistics
  • Representing and summarising dataA2
  • Permutations and combinationsA2
  • ProbabilityA2
  • Discrete random variables: binomial and geometricA2
  • The normal distributionA2
  • The Poisson distributionA2
  • Linear combinations of random variablesA2
  • Continuous random variablesA2
  • Sampling and estimationA2
  • Hypothesis testingA2
  • Correlation and regression (UK boards)
Mechanics
  • Forces and equilibriumAS
  • Kinematics: describing motionAS
  • Newton’s laws and connected particlesAS
  • Momentum and collisionsAS
  • Work, energy and powerAS
  • Projectiles and moments (UK boards)
Further Mathematics
  • Proof: how mathematicians know
  • Further Mathematics: a guided tour
Exam preparation
  • Your calculator: exam skills
  • Writing maths for marks
  • Paper 1 survival guide
  • Papers 2 and 3 survival guide
  • Paper 4 survival guide
  • Paper 5 survival guide
  • Paper 6 survival guide
  • Mixed practice: Pure 1
  • Mixed practice: Pure 2 and 3
  • Mixed practice: Mechanics
  • Mixed practice: Statistics 1
  • Mixed practice: Statistics 2
  • Mock paper: Pure 1 (A)
  • Mock paper: Pure 2 (A)
  • Mock paper: Pure 3 (A)
  • Mock paper: Mechanics (A)
  • Mock paper: Probability and Statistics 1 (A)
  • Mock paper: Probability and Statistics 2 (A)
Start here
  • Physics: the course map
  • Physics from zero
AS Level
  • Physical quantities, units and uncertaintyAS
  • KinematicsAS
  • Dynamics and momentumAS
  • Forces, density and pressureAS
  • Work, energy and powerAS
  • Deformation of solidsAS
  • WavesAS
  • SuperpositionAS
  • ElectricityAS
  • D.C. circuitsAS
  • Particle physicsAS
A Level (A2)
  • Motion in a circleA2
  • Gravitational fieldsA2
  • TemperatureA2
  • Ideal gasesA2
  • ThermodynamicsA2
  • OscillationsA2
  • Electric fieldsA2
  • CapacitanceA2
  • Magnetic fieldsA2
  • Electromagnetic inductionA2
  • Alternating currentsA2
  • Quantum physicsA2
  • Nuclear physicsA2
  • Medical physicsA2
  • Astronomy and cosmologyA2
Practical physics
  • Practical skills: measurement, tables, graphs and uncertaintyAS
  • Planning, analysis and evaluationA2
Exam preparation
  • Paper 1 survival guide: multiple choice
  • Papers 2 and 4 survival guide: structured questions
  • Mixed practice: AS Level
  • Mixed practice: A Level
  • Mock paper: Paper 1 multiple choice
  • Mock paper: Paper 2 AS structured questions
  • Mock paper: Paper 3 practical skills (written)
  • Mock paper: Paper 4 A Level structured questions
  • Mock paper: Paper 5 planning, analysis and evaluation
Start here
  • Chemistry: the course map
  • Chemistry from zero
AS physical chemistry
  • Atomic structureAS
  • Moles and stoichiometryAS
  • Chemical bondingAS
  • Shapes of molecules and intermolecular forcesAS
  • States of matterAS
  • Chemical energeticsAS
  • Redox and oxidation numbersAS
  • Chemical equilibriaAS
  • Acids and basesAS
  • Reaction kineticsAS
AS inorganic chemistry
  • Periodicity: Period 3AS
  • Group 2AS
  • Group 17: the halogensAS
  • Nitrogen and sulfurAS
AS organic chemistry and analysis
  • Introduction to organic chemistryAS
  • IsomerismAS
  • Hydrocarbons: alkanes and alkenesAS
  • HalogenoalkanesAS
  • AlcoholsAS
  • Aldehydes and ketonesAS
  • Carboxylic acids, esters and nitrilesAS
  • Organic synthesis (AS)AS
  • Infrared spectroscopy and mass spectrometryAS
A Level physical chemistry
  • Lattice energy, entropy and Gibbs free energyA2
  • Electrochemistry: electrolysis and electrode potentialsA2
  • pH, buffers, solubility product and partitionA2
  • Rate equations and catalysisA2
A Level inorganic chemistry
  • Transition elementsA2
  • Complexes: colour, isomerism and stabilityA2
A Level organic chemistry and analysis
  • Arenes: benzene and its compoundsA2
  • Phenol and the acidity of organic compoundsA2
  • Acyl chlorides and A Level carboxylic acidsA2
  • Amines, amides and amino acidsA2
  • PolymerisationA2
  • Organic synthesis (A Level)A2
  • Chromatography and NMR spectroscopyA2
Practical skills
  • Practical skills and qualitative analysisAS
  • Planning, analysis and evaluationA2
Exam practice
  • Paper 1 survival guide: multiple choice
  • Papers 2 and 4 survival guide: structured questions
  • Mixed practice: AS Level
  • Mixed practice: A Level
  • Mock paper: Paper 1 multiple choice
  • Mock paper: Paper 2 AS structured questions
  • Mock paper: Paper 3 practical skills (written)
  • Mock paper: Paper 4 A Level structured questions
  • Mock paper: Paper 5 planning, analysis and evaluation
Start here
  • Biology: the course map
  • Biology from zero
Cells and molecules
  • Microscopy and magnificationAS
  • Eukaryotic cells and organellesAS
  • Prokaryotes and virusesAS
  • CarbohydratesAS
  • Lipids and waterAS
  • ProteinsAS
  • Testing for biological moleculesAS
  • EnzymesAS
  • Enzyme kinetics, inhibitors and immobilised enzymesAS
  • Membrane structure and cell signallingAS
  • Movement into and out of cellsAS
Genetics at AS
  • Chromosomes and the mitotic cell cycleAS
  • Nucleic acids and DNA replicationAS
  • Protein synthesis and gene mutationAS
Transport and exchange
  • Plant tissues and water transportAS
  • Translocation in phloemAS
  • Blood vessels, blood and tissue fluidAS
  • Transport of oxygen and carbon dioxideAS
  • The heart and the cardiac cycleAS
  • The gas exchange systemAS
Infection and immunity
  • Infectious diseasesAS
  • Antibiotics and resistanceAS
  • The immune responseAS
  • Antibodies and vaccinationAS
AS skills
  • Maths skills for biologyAS
  • Practical skills: investigations (Paper 3)AS
  • Practical skills: microscopy and drawing (Paper 3)AS
Energy
  • Energy, ATP and respiratory quotientA2
  • Aerobic respirationA2
  • Anaerobic respiration and investigating respirationA2
  • PhotosynthesisA2
  • Limiting factors in photosynthesisA2
Control and coordination
  • Homeostasis and the kidneyA2
  • Blood glucose, cell signalling and biosensorsA2
  • Neurones and action potentialsA2
  • Synapses and muscle contractionA2
  • Control and coordination in plantsA2
Inheritance and evolution
  • Meiosis and genetic variationA2
  • Monohybrid inheritanceA2
  • Dihybrid crosses and the chi-squared testA2
  • Genes, phenotype and gene controlA2
  • Variation and the t-testA2
  • Natural and artificial selectionA2
  • Evolution and speciationA2
Ecology and genetic technology
  • ClassificationA2
  • Biodiversity and samplingA2
  • ConservationA2
  • Tools of genetic technologyA2
  • Genetic technology in medicine and agricultureA2
A Level skills
  • Statistics for biologistsA2
  • Planning investigations (Paper 5)A2
  • Analysis, conclusions and evaluation (Paper 5)A2
Exam readiness
  • Paper 1: multiple choice
  • Papers 2 and 4: structured questions
  • Papers 3 and 5: the practical skills papers
  • Mixed practice: AS
  • Mixed practice: A Level
  • Mock Paper 1
  • Mock Paper 2
  • Mock Paper 3
  • Mock Paper 4
  • Mock Paper 5
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Zero to A*

Learn A Level Mathematics, the sciences, Engineering, Business, Economics and Accounting from the very first idea, with worked examples marked the way examiners mark them. Then follow a step-by-step road all the way to exam day.

Start from zero Build my study plan
  1. 0
  2. Fix your foundationsThe GCSE skills every A Level assumes.
  3. Learn the contentOne chapter at a time.
  4. PractiseUntil you can do it without notes.
  5. Past papersTimed, marked, with an error log.
  6. Final revisionThe last eight weeks, day by day.
  7. Exam dayThe night before, the paper, results.
  8. A*
Pick up where you left off: …

Each phase of the road has its own page saying exactly what to do, for how long, and how to know you are ready to move on. The overview of the road puts all six on one page.

Choose a subject

Every course is taught in full, from absolute basics to the hardest A Level ideas. The number beside each subject is its Cambridge International syllabus code. Each chapter also notes where Pearson Edexcel, AQA and OCR differ.

    How this guide works

    Every chapter is written for someone meeting the topic for the first time. It starts with the idea in plain English, builds up step by step, and finishes at real exam standard. Look out for these parts:

    AaKey term cards hold the exact definitions examiners expect. They are also collected in the glossary.
    M1Worked examples show every step. The marks sit in the margin, as on a marked script: M for method, A for an accurate answer, B for an independent fact.
    ✗Common mistakes are the errors examiners see every year, and how to avoid them.
    !Exam technique notes explain what the question really wants and where the marks are.
    ≠Board differences say where Edexcel, AQA or OCR do things differently from Cambridge.
    ✓Try it yourself questions have full answers, and quick checks mark you instantly. Tick Mark as learned at the end of a chapter to track your progress.

    Tools you can use any time

    • Build your study plan: enter your exam date and subjects to get a week-by-week timeline.
    • Practice quizzes: mixed questions from every chapter, with instant feedback.
    • Formula sheets, the periodic table, a graph plotter, calculators and the glossary.
    NoteAbout this guide

    Facts about syllabuses, exam papers and university admissions were checked in October 2026 against the official Cambridge International syllabuses and university admissions pages. Exam boards update their syllabuses every few years, so always confirm the details for your exam year in the official syllabus for your board. Your progress is saved in this browser only.

    Zero to A* by UncertaintyNihilityOmega. Free to use, share and change with credit: content under CC BY 4.0, code under the MIT License.

    Start Here/Welcome

    What are A Levels?

    Start hereStudy time20 minutes

    A Levels are two-year qualifications in single subjects, usually taken at 16–18 after GCSE, IGCSE or O Level, and used for university entry around the world. This chapter explains how they work, how they compare with other systems, and what studying them is really like.

    Why this matters

    Before you open a textbook, it helps to know the rules of the game. Students lose time and money because nobody explained the basics: which exam board they are entered for, whether their first-year exams count, or which subjects their chosen degree requires. The next twenty minutes will save you months later.

    A Levels in plain English

    A Level is short for Advanced Level. An A Level is a qualification in one subject, such as Mathematics, Chemistry or Business. Most students study three of them at the same time, over two years, between the ages of 16 and 18. At the end you receive one grade for each subject, from A* (the highest) down to E.

    A Level

    An Advanced Level qualification in a single subject, normally studied over two years after GCSE, IGCSE or O Level and graded A* to E. It is the main qualification for university entry in England, Wales and Northern Ireland, and schools in many other countries offer it too.

    A Levels follow the exams most students take at about 16: GCSE (General Certificate of Secondary Education) in England, Wales and Northern Ireland, IGCSE (International GCSE) in international schools, or O Level (Ordinary Level), still offered by Cambridge in some countries.

    In England, the two A Level years are Year 12 and Year 13, together called the sixth form. In Turkey they match the ages of Grades 11 and 12 of lise. Scotland has its own system (Highers and Advanced Highers), so A Levels are less common there.

    Two halves: AS and A2

    An A Level course splits naturally into two halves of about one year each.

    AS Level

    Advanced Subsidiary Level: a qualification covering roughly the first half of an A Level, normally studied in one year. Cambridge International grades it a to e (lower-case letters); UK boards use A to E. There is no A* at AS Level.

    The second half is usually called A2, and this guide labels first-year content “AS” and second-year content “A2”.

    Whether your AS result counts depends on the exam board:

    Exam board Does the AS count towards the full A Level?
    Cambridge International Yes, on the “staged” route: AS first, then the result is carried forward. You can also sit everything at the end.
    Pearson Edexcel International (IAL) Yes: the first-year units make up half the qualification.
    AQA, OCR, Pearson Edexcel and Eduqas in England No. The AS is a separate qualification; all A Level exams come at the end of Year 13.
    WJEC (Wales) and CCEA (Northern Ireland) Yes. The AS is worth 40% of the A Level.
    Year 12 · AS contentYear 13 · A2 contentAS examsMay–JuneA Level examsMay–JuneSONDJFMAMJJASONDJFMAMJJAStart in SeptemberAS results (Aug)Apply to university (Oct–Jan)Final results (Aug)
    A typical two-year course. Year 12 exams count towards the A Level on Cambridge’s staged route and at Edexcel International. With the boards in England, every exam that counts comes at the end of Year 13.

    How many subjects, and which grades?

    Most students take three A Levels. Students who choose Further Mathematics usually take four, because Further Maths is studied alongside Mathematics, not instead of it. Some schools ask everyone to start four subjects and drop one at the end of the first year, keeping it as an AS.

    Qualification Pass grades, highest first
    A Level (all boards) A*, A, B, C, D, E
    AS Level (Cambridge International) a, b, c, d, e
    AS Level (UK boards) A, B, C, D, E

    A result below E is recorded as U (ungraded), which is not a pass. Universities usually make offers in grades, such as “AAB”, or in UCAS tariff points, which turn grades into numbers. Grades, marks and UCAS points explains both.

    How much work is an A Level?

    Guided learning hours

    The time a student spends being taught, or doing study directed and supervised by a teacher, for a qualification. Cambridge International plans for about 360 guided learning hours for each A Level and about 180 for each AS Level. Private study comes on top.

    Two school years have about 36 teaching weeks each, so 360 hours works out at about 5 hours of teaching per subject per week. Add homework, revision and practice, and a normal A Level week looks like this:

    Activity Hours per subject per week For three subjects
    Lessons (or, if you study alone, working through chapters) 4–5 12–15
    Homework and reviewing notes 2–3 6–9
    Practice questions, and later past papers 1–2 3–6
    Total 7–10 21–30

    Three A Levels are close to a full-time job. In school, that means lessons in each subject on most days, homework in every subject, and a short test or practical most weeks. If you study alone, you are teacher and student at once: lesson time becomes reading, working through examples and checking answers.

    TipA self-study week that works

    Give each subject at least three separate sessions a week rather than one long one: short, regular sessions beat a weekend marathon because you forget less in between. Keep one session a week for older topics. Build your personal study plan turns your exam date into a week-by-week timetable.

    Why depth matters

    At IGCSE you probably studied eight to ten subjects. At A Level you study three, and each goes much deeper: A Level Chemistry, for example, explains why reactions happen and asks you to predict ones you have never seen.

    Because of this depth, examiners test understanding more than memory. Only part of the marks are for recalling facts; the rest reward applying ideas to unfamiliar situations, handling data, and analysing or evaluating.

    Assessment objectives (AOs)

    The skills an exam is designed to reward, listed with their weightings in every syllabus. Typical AOs are knowledge and understanding, application, and analysis and evaluation; science subjects also have one for experimental skills.

    Think of it like this

    IGCSE checks that you can follow a recipe. A Level checks whether you can cook a good meal from whatever is in the fridge. You still need to know the recipes, but the marks go to people who understand why each step works.

    Most students find the first term the hardest: the pace is faster, the ideas are more abstract, and you are expected to work more independently. This is normal, and it is why every chapter in this guide starts from zero and builds up in small steps. The same depth is why universities value A Levels: two years of serious work on a few hard subjects is excellent preparation for a degree.

    Who uses A Levels?

    A Levels are the standard entry qualification for UK universities, and universities in many other countries accept them too. Each university sets its own requirements, so always check the entry requirements for international students on each course page.

    Several organisations, called exam boards, write and mark A Levels. Schools in the UK use boards such as AQA, OCR, Pearson Edexcel, WJEC/Eduqas and CCEA. International schools mostly use Cambridge International (often shortened to CAIE) or Pearson Edexcel International (IAL), and OxfordAQA also offers international A Levels. UK universities accept A Levels from all the major boards. This guide follows Cambridge International and points out where the others differ.

    How A Levels compare with other systems

    A Levels IB Diploma US Advanced Placement (AP) National systems (e.g. Turkey)
    Length 2 years 2 years Usually 1 year per course Final years of school, then an entrance exam
    Subjects Usually 3, in depth 6 (3 Higher Level, 3 Standard Level) plus a core Any number of single-subject exams A broad school curriculum
    Extra parts Optional Extended Project (EPQ) at UK boards Theory of Knowledge, Extended Essay, CAS None Varies by country
    Grading A* to E per subject 1 to 7 per subject, 45 points maximum 1 to 5 per exam School grades plus an exam score
    Main use University entry in the UK and worldwide University entry worldwide US college admission and credit Entry to that country’s universities

    None of these is “better”. The IB Diploma suits students who want breadth (CAS stands for creativity, activity, service). AP exams are taken alongside an American high school diploma, and UK universities usually ask for several AP scores as well as the diploma. National systems, such as Turkey’s lise diploma plus the YKS entrance exam run by ÖSYM, are designed mainly for entry to that country’s own universities. A Levels suit students who are ready to specialise.

    NoteStudying in Turkey?

    A Levels do not replace the YKS for most places at Turkish universities. Students in Turkey usually take A Levels to apply abroad, and many UK universities ask applicants from Turkey for A Levels, the IB, AP results or a foundation year rather than the national diploma alone. To keep both routes open, check the current rules with ÖSYM and with each university.

    Can anyone take A Levels?

    Yes. There is no age limit, and you do not need to be at a school. Adults, home-schooled students and school students adding a subject their school does not teach can all enter as a private candidate.

    Private candidate

    A student who enters for exams through an approved exam centre without being taught there. Private candidates sit the same papers and receive the same grades and certificates as school students.

    Private candidates register with an exam centre that accepts external students, pay the fees and sit the exams there. In Turkey, the British Council (in Istanbul and Ankara) and some approved schools act as centres. Sciences need care: Cambridge’s science A Levels include a laboratory exam that your centre must be able to run, while Edexcel International tests practical skills in written papers, one reason many private candidates choose it. Studying as a private candidate explains the process.

    Where to go from here

    1. Decide roughly what, and where, you want to study next.
    2. Check the subjects that degree needs: Choosing your subjects.
    3. Pick a board and exam series: Exam boards and how the exams work.
    4. Learn how grades and offers work: Grades, marks and UCAS points.
    5. Make a plan and start: How to use this guide.
    Quick check
    Summary
    • An A Level is a two-year qualification in one subject, usually taken at 16–18 after GCSE, IGCSE or O Level.
    • Most students take three A Levels, or four if they add Further Mathematics.
    • A Level grades are A* to E. AS grades are a to e (Cambridge) or A to E (UK boards), with no A* at AS.
    • The AS counts towards the A Level at Cambridge (if carried forward), Edexcel International, WJEC and CCEA, but not with the boards in England.
    • Each A Level is designed for about 360 guided learning hours (about 5 hours of teaching a week), plus your own study.
    • A Levels reward understanding and application, not just memory, and universities worldwide accept them.
    • Anyone can take A Levels at any age, including as a private candidate.

    Next: Exam boards and how the exams work.

    PreviousHomeNextExam boards and how the exams work
    Start Here/Welcome

    Exam boards and how the exams work

    Start hereStudy time25 minutes

    Who sets A Level exams, how Cambridge International, Pearson Edexcel International and the boards in England organise them, when exams and results happen, and how papers, mark schemes and grade boundaries work.

    Why this matters

    Two students can both be “doing A Level Physics” and sit completely different exams: different papers, different months, different resit rules. Knowing your board and its rules tells you which syllabus to follow, which past papers to practise and when your deadlines fall.

    What an exam board does

    An exam board writes the syllabus, sets the papers, marks them and awards the grades. All boards teach broadly the same subjects at the same level, but the details differ: the exact topics, the number and style of papers, the exam months and the resit rules.

    Exam board

    An organisation that writes the syllabus for a qualification, sets and marks the exams, and awards grades and certificates. Also called an awarding body.

    Syllabus

    The official document listing exactly what can be examined in a subject, how the exams are structured and what each assessment objective is worth. Cambridge calls it a syllabus; Pearson and the UK boards call it a specification.

    The main boards

    Board Qualification Where it is used
    Cambridge International (CAIE) Cambridge International AS & A Level International schools in many countries; private candidates at approved centres
    Pearson Edexcel International International Advanced Level (IAL) International schools; widely used by private candidates, including at British Council centres
    AQA AS and A Level Schools in England
    OCR AS and A Level Schools in England
    Pearson Edexcel (UK) AS and A Level Schools in England, and some British schools abroad
    WJEC / Eduqas AS and A Level WJEC in Wales; Eduqas is the same organisation’s brand in England

    Northern Ireland has its own board, CCEA, and OxfordAQA offers international A Levels in some countries. Cambridge International and OCR both belong to Cambridge University Press & Assessment, but they set different qualifications. AQA, OCR, Pearson Edexcel and Eduqas follow the same rules for A Levels in England, set by the regulator Ofqual, so their A Levels share one structure.

    Cambridge International: AS first, or everything at the end

    A Cambridge A Level is made of several papers, which Cambridge calls components. The AS papers make up the AS Level and also half of the full A Level; the A Level adds more papers on second-year content. You can take them in three ways:

    1. AS Level only. Sit the AS papers and receive an AS qualification.
    2. Staged route. Sit the AS papers in one series and the remaining A Level papers in a later one. Your AS result is carried forward and combined with them.
    3. Linear route. Sit every paper, AS and A Level, in the same series at the end of the course.

    An AS result can be carried forward to complete the A Level within 13 months. In practice, an AS taken in June can be completed in the following June series.

    Common mistakeThe 13-month rule

    If you sit AS in June 2026, you must complete the A Level by the June 2027 series for the AS to count. If you wait longer, you would have to sit all the papers, AS ones included, in a single series.

    A real example is Cambridge International AS & A Level Physics (9702). Chemistry (9701) and Biology (9700) follow the same pattern.

    Paper Type Time Marks Share of A Level
    1 Multiple choice (AS) 1 h 15 min 40 15.5%
    2 AS structured questions 1 h 15 min 60 23%
    3 Advanced practical skills (laboratory exam) 2 h 40 11.5%
    4 A Level structured questions 2 h 100 38.5%
    5 Planning, analysis and evaluation 1 h 15 min 30 11.5%

    Papers 1–3 form the AS (50% of the A Level); Papers 4 and 5 are the second-year papers.

    Cambridge runs two main exam series each year: June (exams from roughly late April to mid-June) and November (October and November). A third, March, series is available in India only. June results come out in August (the June 2026 A Level results were released on 11 August 2026), and November results in January.

    Exam series

    A period in which a board runs a full set of exams, named after its main month, such as the June series or the November series.

    Pearson Edexcel International: units taken in stages

    An IAL is modular: it is split into units, each examined by one paper. IAL sciences have six units, three for the International AS and three more for the full IAL. IAL Mathematics also has six: four Pure units and two applied units (mechanics, statistics or decision mathematics).

    Modular qualification

    A qualification split into separately examined units that can be taken at different times, with the results combined at the end. Edexcel International A Levels are modular; A Levels from the boards in England are linear.

    • There are three sittings a year: January, May/June and October/November. Most units are offered in every sitting.
    • You can spread units over several sittings, for example Pure 1 in January and Pure 2 in June.
    • You can resit a unit to improve it, and the better result counts.
    • Each unit’s raw mark is converted to a uniform mark so that units from different sittings can be added fairly (how this works).
    • When you have all the units you need, your centre claims the qualification (called “cashing in”) and your overall grade is calculated.
    • In the sciences, practical skills are tested in written papers (Units 3 and 6), not in a laboratory exam.

    The boards in England: linear A Levels

    A Levels from AQA, OCR, Pearson Edexcel and Eduqas are linear:

    • Every exam that counts is sat in May and June at the end of Year 13.
    • The AS is a separate qualification whose marks never count towards the A Level, so many schools no longer enter students for it.
    • There is one exam series a year, in the summer. To improve a grade, you resit all the exam papers the following summer; coursework marks can usually be carried forward.
    • In Biology, Chemistry and Physics you also receive a Practical Endorsement, reported separately as Pass or Not classified, based on at least 12 practical activities assessed by your teacher. Many universities expect a Pass.
    • Some subjects include coursework, officially called non-exam assessment (NEA): typically 20% in History, Geography, English Literature and Computer Science, and more in subjects such as Design and Technology and Art.

    The three systems side by side

    Cambridge International Edexcel International (IAL) Boards in England
    Structure Papers; AS papers form half the A Level Units; AS units form half the IAL Linear: all papers at the end
    Does the AS count? Yes, if carried forward within 13 months Yes No
    Exam series June and November (March in India only) January, May/June, October/November Summer only
    Improving a grade Retake in a later series Resit single units; the better result counts Resit all papers the next summer
    Science practicals Laboratory exam plus a planning paper Written practical-skills papers Practical Endorsement plus practical questions in written papers
    Results August (June series), January (November series) Roughly two months after each sitting Mid-August
    Board differences

    WJEC A Levels in Wales and CCEA A Levels in Northern Ireland are unitised: the AS counts for 40% of the A Level. Scotland uses its own qualifications (Highers and Advanced Highers). Rules change from time to time, so always check the current syllabus for your board.

    What the papers look like

    Paper type What you do Common in
    Multiple choice Answer 30–40 questions, each with four options Cambridge sciences, Economics, Accounting
    Structured questions Answer a question in parts (a), (b), (c): short answers, calculations, explanations Every subject
    Extended response Write a long answer, marked by “levels of response” Business, Economics, English, History
    Data response or case study Answer questions on unseen data, a text or a business case Business, Economics, Geography
    Practical exam Carry out experiments in a laboratory under exam conditions Cambridge sciences
    Planning and analysis Plan an experiment and analyse given results, on paper Cambridge and Edexcel International sciences
    Coursework (NEA) Complete a project over several weeks, marked by teachers and checked by the board Some subjects at the boards in England

    Mark schemes, examiner reports and grade boundaries

    After each series, boards publish three documents that are gold for revision:

    • Mark schemes show exactly what earned each mark. In Maths they use M (method), A (accuracy) and B (independent) marks; in essay subjects they describe levels of response.
    • Examiner reports explain, question by question, what candidates did well and where they lost marks.
    • Grade boundaries (Cambridge calls them grade thresholds) give the minimum mark for each grade. They are set after marking, for each paper in each series. Grades, marks and UCAS points explains why they move.

    Cambridge puts a selection of past papers, mark schemes and examiner reports on its public website, and schools get more through its School Support Hub. Pearson, AQA and OCR publish theirs on their qualification pages, although the most recent papers are sometimes released to teachers first.

    Exam techniqueUse mark schemes from day one

    Do not wait for “past-paper season”. After every topic, try two or three past-paper questions on it and mark them yourself with the official scheme. You will quickly learn which words and steps examiners need to see. Phase 4: Past papers and mark schemes shows a full method.

    Entries and deadlines

    You cannot simply turn up on exam day. Your school, or your exam centre if you are a private candidate, must enter you for each paper months before the series. For a June series, the deadline usually falls early in the same year, and many centres set their own earlier deadline. Late entries, if accepted at all, cost extra. Fees are set by each centre and vary, so ask for the full price per subject, including any centre charges.

    Common mistakeConfirm four things before you start studying

    Check with your centre: the board, the syllabus code (for example 9702 for Cambridge Physics), the exam series, and the entry deadline. A year spent studying the wrong syllabus is an expensive mistake.

    Quick check
    Summary
    • An exam board writes the syllabus, sets and marks the papers and awards grades. Always study the syllabus with the correct code for your exam year.
    • Cambridge: papers called components; AS papers form half the A Level; take AS only, AS first (carried forward within 13 months), or everything at the end.
    • Cambridge series: June (results in August) and November (results in January); March in India only.
    • Edexcel International: modular units, sittings in January, May/June and October/November, single-unit resits, and written practical-skills papers.
    • Boards in England: linear, all exams in May–June of Year 13, AS does not count, Practical Endorsement in sciences, NEA in some subjects.
    • Mark schemes, examiner reports and grade boundaries are published after each series: use them throughout the course.
    • Entries close months before the exams, so confirm your board, syllabus code, series and deadline early.

    Next: Grades, marks and UCAS points.

    PreviousWhat are A Levels?NextGrades, marks and UCAS points
    Start Here/Welcome

    Grades, marks and UCAS points

    Start hereStudy time25 minutes

    How raw marks become grades, why grade boundaries move, how the A* is awarded, and how UCAS tariff points, university offers and predicted grades work.

    Why this matters

    Your grades open university doors, so it pays to know exactly how they are calculated. Understanding grade boundaries also changes how you study: it shows why every single mark matters, and why a hard paper is not a disaster.

    The grades

    Qualification Pass grades, highest first
    A Level A*, A, B, C, D, E
    AS Level a, b, c, d, e at Cambridge; A, B, C, D, E at UK boards

    A result below the lowest pass grade is reported as ungraded (U), and no qualification is awarded in that subject. Your certificate shows a grade for each subject, not a percentage.

    From raw marks to a grade

    Each paper is marked and you receive a raw mark. Your raw marks are combined, using each paper’s weighting, and the total is compared with the grade boundaries for that series.

    Raw mark

    The number of marks you actually score on a paper, before any conversion; for example, 52 out of 75.

    Grade boundary

    The minimum mark needed for a particular grade on a paper or qualification in a given exam series. Boundaries are set after marking and can change from series to series. Cambridge calls them grade thresholds.

    Why boundaries move

    No two papers are equally difficult. If this year’s paper is harder than last year’s, the same student scores fewer marks. To keep an A worth the same every year, senior examiners study the papers, samples of marked scripts and statistics, and then set the boundaries. A harder paper gets lower boundaries; an easier paper gets higher ones.

    Mark needed (out of 100) Series 1: harder paper Series 2: easier paper
    Grade A 68 76
    Grade C 47 55

    These numbers are invented to show the idea; each board publishes the real boundaries for every series. Three lessons follow:

    1. There is no fixed percentage for an A. “70% is an A” is a myth.
    2. A hard paper is hard for everyone. If a paper feels brutal, keep collecting marks: the boundaries will reflect it.
    3. Every mark counts. Neighbouring grades are often separated by only a handful of marks on each paper, so one extra method mark here and there can lift a whole grade.
    TipEstimating your grade from a past paper

    Mark the paper with the official mark scheme, then compare your total with the published boundaries for that same series, not this year’s or an average. Do this for papers from several different years to get a fair picture.

    How the A* is awarded

    • Cambridge International awards A* for the full A Level only, based on your total across all the papers. There is no A* at AS.
    • Edexcel International requires an A overall plus an especially high score on the A2 (second-year) units, so an A* reflects strength in the hardest material.
    • The boards in England set the A* as the top boundary on your total mark, like any other grade.

    Uniform marks at Edexcel International

    In a modular system, you might sit one unit in a January with a hard paper and another in a June with an easy one. Simply adding the raw marks would be unfair. Pearson solves this with uniform marks.

    Uniform mark

    A raw unit mark converted onto a fixed scale, so that results from papers of different difficulty, taken in different sittings, can be added fairly. The grade boundaries on the uniform scale stay the same every series.

    The conversion uses each sitting’s grade boundaries: a raw mark exactly on the A boundary always becomes the same uniform mark, however hard the paper was. Your uniform marks from all the units are then added, and the total decides your overall grade. Pearson’s results show the uniform mark for each unit, which tells you how close you were to the next grade.

    Think of it like this

    Uniform marks work like currency exchange. Raw marks from different sittings are like dollars, euros and lira: you cannot add them directly. Converting everything into one currency first lets you add them fairly.

    UCAS tariff points

    UCAS (the Universities and Colleges Admissions Service) handles applications to UK universities. Its tariff turns grades into points so that different qualifications can be compared.

    UCAS tariff points

    A points system that converts qualifications and grades into numbers so that different qualifications can be compared. Some universities state their entry requirements in tariff points.

    Grade A Level AS Level Extended Project (EPQ)
    A* 56 no A* 28
    A (AS: a) 48 20 24
    B (AS: b) 40 16 20
    C (AS: c) 32 12 16
    D (AS: d) 24 10 12
    E (AS: e) 16 6 8

    An EPQ is worth exactly half an A Level at every grade. An AS that you later complete as a full A Level in the same subject does not count separately: only the A Level counts.

    Worked exampleAdding up tariff points

    A student has A Levels in Mathematics (A*), Physics (A) and Chemistry (B), an AS in Further Mathematics (grade b) and an EPQ (grade A). How many UCAS tariff points does she have?

    A Levels: 56+48+40=144

    AS Further Mathematics, grade b: 16

    EPQ, grade A: 24

    Total: 144+16+24=184 points.

    Try your own combination in the calculator: choose a grade for each qualification and watch the total change.

    Common mistakePoints are not everything

    AAA and AAB are both worth 144 points, but a course asking for "AAA including Mathematics" accepts neither. Most selective universities make offers in grades and name the subjects. Use points only when a course actually asks for them.

    What offers look like

    You apply through UCAS to up to five courses (four at most in medicine, dentistry or veterinary science). Each university replies with an offer or a rejection. Typical offers read like this:

    • “AAA”: three A grades, sometimes excluding certain subjects.
    • “AAA, including A in Mathematics and A in Physics”: grades plus named subjects, as for many engineering courses.
    • “ABB including Biology and Chemistry”: a common pattern for life sciences.
    • “112 UCAS points, including a C in Business”: a tariff offer, which BBC would meet (40+40+32=112).
    • “AAB, or ABB with an A in the EPQ”: some universities lower an offer for a strong EPQ.
    Conditional offer

    A university place that becomes yours only if you meet stated conditions, usually particular grades. An unconditional offer has no grade conditions, usually because you already have your results.

    From your offers you choose one firm choice and, if you wish, one insurance choice with lower grades as a backup. On results day, the university confirms your place if you meet the conditions. If you miss both, Clearing matches students without a place to courses that still have vacancies. Some courses also set admissions tests or interviews; see University applications and admissions tests.

    Key factsUCAS deadlines for 2027 entry
    • 15 October 2026: Oxford, Cambridge, and most courses in medicine, dentistry and veterinary science.
    • 13 January 2027: deadline for equal consideration for most other courses.

    Predicted grades

    Universities usually decide before your final exams, so they rely on predictions.

    Predicted grades

    The grades your teachers expect you to achieve, entered on your UCAS application. Universities make most offers on the basis of predicted grades, because final results arrive after most decisions are made.

    Teachers predict from evidence: classwork, mock exams, AS results if you have them, and the progress you have shown. A prediction below a course’s typical offer usually means no offer, so the evidence you give your teachers matters. Private candidates still need predicted grades and a reference; a tutor, a former teacher or a school can provide them. Studying as a private candidate explains how.

    Exam techniqueEarn your predictions

    Do your mocks properly, under timed conditions, and keep a record of your past-paper scores. A strong AS result, on Cambridge’s staged route or at Edexcel International, is the most convincing evidence of all.

    Quick check
    Summary
    • A Level grades run A* to E, AS grades a to e (Cambridge) or A to E (UK boards); U is not a pass.
    • Raw marks are compared with grade boundaries set after marking. Harder papers get lower boundaries, so there is no fixed percentage for an A.
    • Cambridge awards A* for the full A Level only. Edexcel International requires an A overall plus a very high score on the A2 units.
    • Edexcel International converts raw unit marks into uniform marks so that units from different sittings add up fairly.
    • UCAS tariff: A* 56, A 48, B 40, C 32, D 24, E 16; AS a 20, b 16, c 12, d 10, e 6; EPQ A* 28 down to E 8.
    • Selective universities mostly make grade offers that name subjects; some use tariff points.
    • Offers are based on predicted grades, so mocks and past-paper scores matter. For 2027 entry, UCAS deadlines are 15 October 2026 and 13 January 2027.

    Next: Choosing your subjects.

    PreviousExam boards and how the exams workNextChoosing your subjects
    Start Here/Welcome

    Choosing your subjects

    Start hereStudy time30 minutes

    How to choose A Level subjects that fit the degree or career you want: typical requirements for popular courses, how many subjects to take, strong combinations, entry requirements, honest workload notes and the mistakes to avoid.

    Why this matters

    Your A Level subjects decide which degrees you can apply for. At most universities you cannot study medicine without Chemistry, or engineering without Mathematics. An evening of research now protects options you may want in two years’ time.

    Work backwards from your goal

    The safest way to choose is to start at the end:

    1. Write down two or three degree courses you might want to study, even if you are unsure.
    2. Look up each course’s subject requirements on the websites of a few universities you like.
    3. Choose subjects that satisfy all of them, so you keep every option open.
    4. Fill any remaining place with a subject you enjoy and can do well in.

    University websites separate subjects that are required from those that are only preferred, and the difference matters.

    Required subject

    An A Level subject that a course insists on; without it, an application is normally rejected. Universities also list “preferred” or “recommended” subjects, which strengthen an application but are not compulsory.

    Typical requirements for popular courses

    If you want to study Usually required Recommended or useful Also note
    Medicine Chemistry, and usually Biology Maths or Physics UCAT at most schools; interviews; apply by 15 October
    Dentistry Chemistry and Biology Maths or Physics UCAT at most schools; interviews; apply by 15 October
    Veterinary science Chemistry and Biology Maths or Physics Animal work experience expected; apply by 15 October
    Engineering Maths, and usually Physics Further Maths, strongly recommended for top courses; Chemistry for chemical engineering ESAT at some universities (e.g. Cambridge, Imperial, Oxford)
    Computer science Maths Further Maths; Physics or Computer Science The A Level of the same name is rarely required; TMUA at some universities
    Mathematics or physics Maths (and Physics for physics) Further Maths, often expected at top universities Admissions tests at some universities
    Economics Maths at most selective universities Further Maths for the most competitive courses The A Level of the same name is rarely required; TMUA at some universities
    Business, management, accounting Usually none Maths, especially for finance and accounting A Level Business is not required
    Law None in the UK Essay subjects such as History, English Literature or a language LNAT at several universities
    Architecture Often none Art or Design; Maths or Physics A portfolio of creative work is usually required
    Psychology Often none Biology or Maths; some courses want one science A psychology degree contains a lot of statistics
    Natural sciences Two or more of Maths, Physics, Chemistry and Biology Maths for any physical science Requirements depend on which sciences you plan to take

    These are typical UK requirements. Every university sets its own, they change from year to year, and universities in other countries have different rules, so confirm the details on each course page for your year of entry. University applications and admissions tests explains UCAT, LNAT, TMUA and ESAT.

    How many subjects?

    Three A Levels is the standard, and university offers are almost always based on three. A fourth makes sense mainly when it is Further Mathematics, or when you are strong enough to take it without your other grades suffering. A fourth A Level rarely earns you a lower offer, but a weaker grade in one of your main three can cost you a place.

    TipThe “3 + 1” option

    If you cannot decide between four subjects, start all four. At Cambridge or Edexcel International you can sit the AS in one of them at the end of the first year and stop there, keeping a useful extra qualification. An AS at the boards in England is also possible, but it is a separate exam and many schools no longer offer it.

    Strong combinations and keeping doors open

    Combination Keeps these doors open
    Maths, Physics, Chemistry Engineering, physics, chemistry, natural sciences, computer science
    Maths, Further Maths, Physics Engineering, mathematics, physics, computer science: the strongest choice for the most competitive courses
    Biology, Chemistry, plus Maths or Physics Medicine, dentistry, veterinary science, pharmacy, biomedical sciences, and still some physical sciences
    Biology, Chemistry, Psychology Medicine at universities that accept any third subject, psychology, life sciences
    Maths, Economics, plus History or Further Maths Economics, finance, management, politics
    English Literature, History, plus a language or Politics Law, history, politics, languages, journalism
    Maths, Physics, Art Architecture, design engineering

    If you are not sure yet what you want to study, choose subjects that keep the most doors open:

    • Mathematics is the subject most often required or preferred: engineering, physics, computer science, economics and many sciences ask for it, and it helps with the statistics in psychology and biology.
    • Chemistry opens medicine, dentistry, veterinary science, pharmacy, biochemistry and chemical engineering.
    • One essay subject keeps law, the humanities and the social sciences possible.
    • It is easier to move from sciences to a humanities degree than the other way round, because most humanities degrees do not require particular subjects.

    Entry requirements: are you ready?

    Exam boards set no formal entry requirements, but schools do, and so does common sense: A Level builds directly on IGCSE. Typical school requirements look like this:

    A Level subject Typical GCSE/IGCSE grade needed
    Mathematics 7 (A), sometimes 6 (B)
    Further Mathematics 8 or 9 (A*), and a confident start in A Level Maths
    Physics, Chemistry, Biology 6 or 7 (B or A) in that science, and at least 6 in Maths
    Essay subjects (History, English Literature) 6 (B) in English and a related subject
    New subjects (Economics, Psychology, Law) Good grades in English and Maths; no earlier study needed

    If you are teaching yourself and never took the IGCSE, read what the syllabus recommends as prior learning, then fill gaps with the foundation chapters: Algebra from scratch, Physics from zero, Chemistry from zero and Biology from zero. Phase 1: Fix your foundations gives a checklist.

    Honest workload notes

    No A Level is easy at A* standard, but subjects are hard in different ways.

    • Mathematics demands fluent algebra and a lot of practice. Answers are right or wrong, so it needs less extended writing, which helps if English is your second language.
    • Further Mathematics roughly doubles your maths load. Take it only if you are strong at Maths and enjoy it; it also makes ordinary Maths feel easier.
    • Physics needs confident Maths and clear explanations of ideas, not just formulas.
    • Chemistry combines a large amount of content with calculations, and later topics build directly on earlier ones, so falling behind early is costly.
    • Biology has a large amount of content and precise vocabulary, with many longer written answers.
    • Essay subjects (History, English Literature, Economics, Business) need wide reading and plenty of writing practice. If English is not your first language, start practising timed writing early; it is entirely possible to reach the top grades.
    • Private candidates should check that their centre can run every part of the course, such as a science laboratory exam or coursework, before choosing.

    Common mistakes

    Common mistakeChoosing for the wrong reasons
    • Picking a subject because your friends take it or because you like one teacher.
    • Dropping Mathematics without checking: it is required or preferred for engineering, computer science, physics, economics and more.
    • Assuming you need the A Level of the same name: law, economics, psychology, business and computer science degrees rarely require it.
    • Taking Further Maths when your Maths is not yet secure.
    • Taking four or five subjects “to impress” and getting lower grades in all of them.
    • Choosing two very similar subjects, such as Business and Economics, without checking: some selective universities prefer them not to be two of your three.
    Exam techniqueGrades matter as much as subjects

    A university looks at both what you studied and how well you did. A subject you enjoy and can score an A in is usually a better choice than a “prestigious” one in which you would get a C, provided the required subjects for your course are covered.

    Quick check
    Summary
    • Work backwards: list possible degrees, check their required subjects, then choose subjects that satisfy all of them.
    • Medicine, dentistry and veterinary science need Chemistry and usually Biology; engineering needs Maths and usually Physics, with Further Maths strongly recommended.
    • Computer science and economics usually need Maths but rarely the A Level of the same name; law usually needs no particular subject.
    • Three A Levels is standard; a fourth makes most sense as Further Mathematics.
    • Maths plus one or two sciences keeps the most options open.
    • Schools usually expect a 6 or 7 (B or A) at GCSE/IGCSE in a subject, and higher for Further Maths.
    • Requirements differ between universities and change over time: always check the course page for your entry year.

    Next: How to use this guide.

    PreviousGrades, marks and UCAS pointsNextHow to use this guide
    Start Here/Welcome

    How to use this guide

    Start hereStudy time15 minutes

    How Zero to A* is organised, what each type of box means, how to search, track your progress and use the tools, and a simple learning loop that turns every chapter into exam marks.

    Why this matters

    A guide only helps if you use it actively. Reading a chapter once feels productive, but most of it is forgotten within days. Working through it in the right order, and coming back to it later, is what turns reading into marks.

    How the guide is organised

    Open the menu (the ☰ button on a phone, or the sidebar on a larger screen) to see every subject: the six main subjects (Mathematics, Physics, Chemistry, Biology, Engineering and Business), Economics and Accounting, short guides to other subjects, the preparation road, the toolbox and practice quizzes. Each subject is split into groups of chapters in a sensible order of study, usually starting with a course map.

    The top of each chapter shows a level label, a short summary and, for subject chapters, the exams that test it and a realistic study time for a beginner. The level labels mean:

    Label What it means
    Foundations GCSE/IGCSE skills that A Level assumes. Start here if you are rusty.
    AS Level First-year content.
    A Level (A2) Second-year content, needed for the full A Level.
    Guide, Reference, Summary Course maps, the preparation road, formula sheets and short subject summaries.
    Interactive Tools such as the graph plotter and calculators.
    Beyond A Level Extension material, such as the tour of Further Mathematics.

    Most chapters begin with Before you start links to the chapters you need first, and end with a Next link. On a wide screen, the On this page panel lists the sections and gives an estimated reading time.

    The boxes, and how to use them

    Box What it holds How to use it
    Key term An exact definition in examiner-quality wording Learn it word for word: definitions are easy marks
    Worked example A full solution, with marks shown in exam-style ones Cover the solution and try it first
    Common mistake An error examiners see every year Check your own work against it
    Exam technique What a question wants and where the marks are Apply it when you practise past papers
    Board differences Where Edexcel International, AQA or OCR differ from Cambridge Read the parts for your board
    Tip, Think of it like this Shortcuts and everyday analogies Use them to understand, then learn the precise version
    Key facts, Formula Facts and formulas to memorise Turn them into flashcards
    Practical skills Methods, apparatus, safety and errors in experiments Essential for practical and planning papers
    Try it yourself Practice questions, each with a hidden full answer Attempt first; only then press Show answer
    Quick check Multiple-choice questions marked instantly Read the explanation every time, even when you are right
    Summary The facts to remember from the chapter Test yourself on them a week later

    Here is what three of them look like.

    Mark scheme

    The document examiners use to award marks. It lists the accepted answers and shows exactly where each mark is earned, including method (M), accuracy (A) and independent (B) marks in Mathematics.

    Worked exampleA rectangle is (x+3) cm long and x cm wide. Its area is 40 cm2. Find x.3 marks

    Form the equation: x(x+3)=40, so x2+3x−40=0. B1

    Factorise: (x+8)(x−5)=0. M1

    So x=5; reject x=−8, because a length cannot be negative. A1

    Check: a rectangle 8 cm long and 5 cm wide has area 40 cm2.

    The mark labels follow the system used in Cambridge and Edexcel mathematics mark schemes:

    • M (method): for a correct method applied to the problem, even if you then make a slip.
    • A (accuracy): for a correct answer or step. You can only earn it if you have earned the method mark it depends on.
    • B (independent): for a correct result or statement that does not depend on a method.

    So a correct answer with no working may not earn full marks, while a wrong answer with a sound method can still earn some. You will meet problems like this one in Quadratics.

    Common mistakeReading is not the same as doing

    Following a worked solution feels easy because every step is already there. The real test is whether you can produce it yourself with the page covered. Always try first, then compare.

    Track your progress

    At the end of every chapter there is a Mark as learned button. Press it when you can do the practice questions without help. Learned chapters get a tick in the menu, the progress bar fills up, and the home page shows how many chapters you have learned in each subject, with a link to pick up where you left off.

    Common mistakeProgress is saved in this browser only

    Your ticks, theme and text size are stored on this device, in this browser. They do not move between your phone and your computer, and clearing your browsing data or using a private window removes them. Choose one main device for studying.

    Search, glossary, tools and display

    • Search: press / from anywhere (or tap the search box) and type a topic, key term or formula name. Use the arrow keys and Enter to open a result, or Esc to close the list.
    • Glossary: every key-term card in the guide, in one alphabetical list that you can filter by subject: Glossary.
    • Tools: the graph plotter, the periodic table, calculators for UCAS points, formula masses, break-even and more, and formula sheets such as the maths formula sheet.
    • Practice: Practice quizzes mix questions from every chapter and tell you which chapters to revise.
    • Planning: Build your personal study plan turns your exam date into a weekly timetable.
    • Display: the buttons at the top right switch between light and dark mode and cycle the text size from normal to large to extra large.

    The learning loop

    Use the same six steps for every chapter.

    STEP 1Read the chapterSTEP 2Worked examplesSTEP 3Try it yourselfSTEP 4Quick checkSTEP 5Mark as learnedSTEP 6Revisit in a weekrepeatThe learning loop, for every chapter
    Work through each chapter in this order. If the quick check goes badly, go back to the worked examples before moving on. The dashed arrow is the review a week later that brings the chapter back into your memory.
    1. Read the chapter with pen and paper. Write each key term in your own words, then compare it with the card.
    2. Study the worked examples. Cover each solution, attempt it, then compare, noting where the marks are.
    3. Try it yourself with the chapter closed. Open Show answer only after a real attempt, and mark your work honestly.
    4. Take the quick check and read every explanation.
    5. Mark as learned only when you can do the practice questions without help.
    6. Revisit after a week. Redo a few questions and the quick check from memory. Each time you pull an idea back out of memory after a gap, you keep it for longer.
    TipLittle and often

    A main-subject chapter usually takes several sessions, as its study time shows. End each session by writing down three things you learned, and start the next one by recalling them without notes.

    Use the guide with your syllabus, a textbook and past papers

    Resource Use it for When
    This guide Learning each topic from zero, worked examples, practice Throughout the course
    Official syllabus The final word on what can be examined At the start, then as a checklist
    A textbook written for your syllabus Extra explanations and more questions Alongside each chapter
    Past papers, mark schemes, examiner reports Real exam questions and how they are marked Topic questions from the start; full timed papers in the final months

    Download the syllabus for your board and exam year from the board’s website and tick off each learning outcome as you finish it. Facts in this guide were checked in October 2026, but if anything here disagrees with your official syllabus, the syllabus wins. For recommended books and websites, see The best resources by subject; for a past-paper method, see Phase 4: Past papers and mark schemes.

    Try it yourselfCheck you know your way around
    1. In a mark scheme, what is the difference between an M mark and an A mark?

      Show answer

      An M mark is for a correct method applied to the problem. An A mark is for an accurate answer or step, and it can only be awarded if the method mark it depends on has been earned.

    2. You have just read a chapter and its worked examples. What should you do next, and why?

      Show answer

      Try the practice questions with the chapter closed, and only then check the answers. Recognising a solution is much easier than producing one, and producing answers from memory is what builds lasting understanding.

    Quick check
    Summary
    • Each subject is split into chapters labelled Foundations, AS Level or A Level (A2), with the exams that test them and a realistic study time.
    • Key-term cards hold exact definitions; worked examples show marks: M for method, A for accuracy (dependent on M), B for independent results.
    • Common-mistake, exam-technique and board-difference boxes show where marks are lost and how the boards differ.
    • Try it yourself hides full answers; quick checks mark you instantly and explain why.
    • Press / to search. The glossary, tools, formula sheets, practice quizzes and study planner are in the menu.
    • Progress, theme and text size are saved in this browser only.
    • The learning loop: read, study the worked examples, try without looking, quick check, mark as learned, revisit after a week.
    • Use the guide alongside the official syllabus, a textbook and past papers.

    Next: The preparation road at a glance, or choose a subject from the home page.

    PreviousChoosing your subjectsNextThe preparation road at a glance
    Preparation Roadmap/The road to exam day

    The preparation road at a glance

    GuideExamsAll subjects · all boardsStudy time20 minutes to read

    Six phases take you from your first day of study to results day. This page shows what each phase is for, how long it takes on three different timelines, the milestones you must not miss, and how to know when you are ready to move on.

    Why this matters

    Two years, or even one, is a long time, and long projects fail in a predictable way: weeks drift past, the hard topics get pushed back, and past papers start far too late. A road with clear phases turns one big, frightening goal into small weekly jobs. It also tells you what to do next on any given day, which is half the battle when you study alone.

    The six phases

    Every successful exam preparation, whatever the subject or board, passes through the same six phases. They are not strict boxes. Learning and practice run side by side, and past-paper work starts before the last topic is finished. But the order matters: you cannot practise what you have not learned, and timed papers only help once you know most of the content.

    main focusalongside other work1Fix your foundations2Learn the content3Practise and test4Past papers5Final revision6Exam dayChoose boardFind a centreEntries closeMocksFirst paperResultsMilestones
    The six phases from your first study day to your first exam. Solid bars show the main focus; pale bars show work that runs alongside it. The phases overlap: you practise each topic as soon as you have learned it. "Find a centre" applies to private candidates.
    Phase What you do Why it matters How long
    1. Fix your foundations Check and repair the GCSE/IGCSE skills each subject assumes: algebra, units, graphs, essay writing Every A Level topic is built on them, and small gaps slow down everything later 2–6 weeks
    2. Learn the content Work through the syllabus topic by topic: read actively, make notes, follow worked examples You cannot practise or revise what you never understood About half of your total time
    3. Practise and test yourself Retrieval practice, spaced review, flashcards, topic questions, an error log Understanding fades within days unless you test it; testing makes it stick Alongside phase 2, then 3–6 weeks of its own
    4. Past papers Real papers under timed conditions, marked honestly with the official mark scheme Teaches the exam’s style, its timing and the wording that earns marks 8–12 weeks
    5. Final revision Condense notes, target weak topics, sit full mock exams Turns what you know into marks you can rely on under pressure The last 6–8 weeks
    6. Exam day Prepare the logistics, manage your time in the paper, then deal with results, reviews and resits Avoids losing marks to nerves, admin or a missing plan B The exam series and results day

    Three tracks: how long each phase takes

    How long you spend in each phase depends on how much time you have. The three most common situations are below. Find yours, then use the study planner to turn it into dated weeks.

    2-year school route 12-month intensive 5–6-month resit or crash plan
    Who it suits Students in Years 12–13 at a school or college Private candidates studying full-time, often straight after IGCSE Students resitting, or who already know much of the content
    Typical dates September of Year 12 → May/June of Year 13 June → June, or October → October/November of the next year June → October/November, or December → May/June
    1. Foundations 3–4 weeks (the summer before, plus September) 3–4 weeks 1 week: a diagnostic test only
    2. Learn September of Year 12 to about February of Year 13 26–30 weeks 4–8 weeks, weak topics only
    3. Practise Every week, plus about 4 weeks of consolidation Every week, plus about 3 weeks Every week: the main activity
    4. Past papers Topic questions from Year 12; full papers January–April of Year 13 8–10 weeks 8–10 weeks
    5. Final revision 8 weeks (April–May of Year 13) 6–8 weeks 3–4 weeks
    6. Exam day May/June; results in August Your chosen series Your chosen series
    Study outside lessons 10–15 hours a week 30–36 hours a week 8–12 hours a week per subject
    Board differencesStaged, linear and modular

    Cambridge International lets you take the AS papers at the end of Year 12 and the rest of the A Level a year later (staged assessment). An AS result can be carried forward for up to 13 months. If your school works this way, you run phases 4–6 twice: a short version for AS in Year 12, then the full version in Year 13. AQA and OCR A Levels are linear: every paper is sat in May/June at the end of the course. Edexcel International A Levels (IAL) are modular, with exams in January, May/June and October, so you can sit some units early and spread the load.

    Milestones you must not miss

    When Milestone Details
    Before you start Choose your subjects and board, and download the syllabus for your exam year Choosing your subjects, Exam boards
    First month (private candidates) Find an exam centre; ask about entry deadlines, fees and practical papers In Turkey: the British Council (Istanbul, Ankara) and Pearson-approved schools. See Studying as a private candidate
    Every 6–8 weeks A timed test on everything learned so far Record the score in your progress tracker
    Autumn of your final year University applications UCAS deadlines for 2027 entry: 15 October 2026 (Oxford, Cambridge and most medicine, dentistry and veterinary courses), 13 January 2027 (most other courses). See University applications
    Several months before the exams Exam entries close Schools enter you; private candidates enter through their centre. Late entries cost extra
    2–4 months before the first paper Mock exams A full set of papers in exam conditions
    Start of phase 5 Past-paper target At least 5 full papers per component done and marked; 8–12 by exam day
    August (for June exams) Results day Cambridge results for the June 2026 series came out on 11 August 2026

    How many hours a week?

    Cambridge designs each A Level around 360 guided learning hours of teaching (about 180 for an AS Level). That is class time with a teacher; practice and revision come on top. As a rough guide:

    • At school: lessons plus 3–5 hours of your own study per subject per week, so 10–15 hours a week for three subjects. Raise this to 20 or more in the final 8 weeks.
    • Private candidate, full-time: 10–12 hours per subject per week, so 30–36 hours for three subjects. Treat it like a job: a fixed start time, fixed breaks, and one day a week off.
    • Resitting one or two subjects: 8–12 hours per subject per week.
    TipHours are not the goal

    Two hours of testing yourself beat five hours of rereading. Track your hours to make sure the time goes in, but judge your progress by your scores on questions you have not seen before.

    Are you ready to move on?

    Use these checks at the end of each phase. If you fail one, do not panic: spend another week or two on that phase, then check again.

    Phase You are ready to move on when…
    1. Foundations You score at least 8 out of 10 on each foundation self-check without looking anything up
    2. Learn For every syllabus learning outcome, you can explain the idea aloud and answer a basic question without notes
    3. Practise You score 70% or more on mixed topic questions a week after learning them, and the same mistakes have stopped appearing in your error log
    4. Past papers Your last three full papers, timed and honestly marked, reach your target grade
    5. Final revision Every topic in your tracker is green, and you can write out the key formulas and definitions from memory
    Common mistakeThe commonest way to fall behind: living in phase 2

    Many students spend months rereading notes and making them prettier, then reach the exam having answered almost no questions. Rereading feels productive because everything looks familiar, but recognising an idea is not the same as being able to use it. From the first week, end every study session with questions you answer without looking.

    Exam techniqueStart from the syllabus, finish with the mark scheme

    Two documents define what you need: the syllabus (what can be asked) and the mark schemes (what earns marks). Download the syllabus for your exam year before you study anything. Read your first mark scheme early, during phase 2, so you know what examiners look for from the start.

    TipAlready part-way through?

    You do not have to start at phase 1. If your exams are four months away, do one past paper per subject this week, mark it, and study the results. Then jump to the phase that fixes your biggest problem, and use the 5–6-month track for timing.

    Where to go next

    Start by building your personal study plan, then follow the phases in order, beginning with Phase 1: Fix your foundations. Along the way, read Study methods that actually work, Exam technique and Motivation, sleep and stress.

    Summary
    • Six phases: fix foundations, learn, practise, past papers, final revision, exam day. They overlap, but the order matters.
    • Pick your track: the 2-year school route, the 12-month intensive, or the 5–6-month resit or crash plan.
    • Download the syllabus for your exam year first, and read a mark scheme early.
    • Private candidates: find an exam centre and check its entry deadlines in the first month.
    • Cambridge designs an A Level around 360 guided learning hours of teaching; your own practice comes on top.
    • Judge progress by scores on unseen questions, not by hours or pages of notes.
    • Move on from a phase only when you pass its “ready” check.

    Next: Build your personal study plan.

    PreviousHow to use this guideNextBuild your personal study plan
    Preparation Roadmap/The road to exam day

    Build your personal study plan

    InteractiveExamsAll subjects · all boardsStudy time30 minutes

    Turn the six phases into a dated, week-by-week plan for your own exam date and subjects. Then learn how to read the plan, adapt it when life gets in the way, and fill in a weekly timetable, with three complete example plans.

    A good plan answers two questions: is my goal realistic? and what should I do this week? Enter your exam date and your subjects below. The planner counts the weeks you have and spreads them across the six phases, so you can see when each phase should start and finish.

    TipWhich date should I enter?

    Use the date of your first paper, not the last, because final revision has to be finished by then. If the timetable is not published yet, take the earliest date in your board’s window for that exam series. Planning for a week too early is harmless; planning for a week too late is not.

    Build or check a plan by hand

    The planner does the arithmetic, but you should understand it so that you can check the result, or redo it with a pen and a calendar.

    1. Count the whole weeks from next Monday to your first paper.
    2. Keep about one week in ten free as a buffer for illness, holidays and family events.
    3. Working back from the exam, mark the last 6–8 weeks as final revision (phase 5) and the 4–6 weeks before that as full past papers (phase 4).
    4. Mark the first 2–4 weeks as foundations (phase 1).
    5. The weeks left in the middle are for learning (phase 2), with practice (phase 3) built into every week. Share them between your subjects’ topics.
    6. Add the fixed dates: entry deadline, mocks, holidays and UCAS deadlines.

    Then do a reality check. Each chapter of this guide shows a study time at the top (for example, 6–8 hours for Quadratics). Add about half as much again for practice, multiply by the number of topics, and compare the total with the hours your learning weeks give you. If it does not fit, you have three honest choices: study more hours a week, take fewer subjects in this series, or move to a later series. Never solve the problem by cutting phases 4 and 5, because that is where grades are made.

    How to read your plan

    • Phase dates are targets, not walls. Phases overlap. Start past-paper questions on each topic as soon as you finish it, long before phase 4 officially begins.
    • The learning weeks are the tightest. If your plan gives you fewer than about 10 weeks per subject to learn the content, you are on an intensive track, and your weekly hours matter more than anything else.
    • Phase 5 has a minimum. Even a short plan needs at least 3–4 weeks of final revision.

    Adapting the plan

    Every plan goes wrong at some point. What matters is how you respond.

    Situation What to do
    1–2 weeks behind Use a buffer week, or add 3–5 hours a week for a month. Do not push every later phase back
    3 or more weeks behind Decide now: more hours, one subject moved to the next series, or an AS Level only this time (a Cambridge AS Level is a qualification in its own right)
    Ahead of schedule Do more mixed practice and past-paper questions; do not race ahead to new content
    A topic takes twice as long as planned Normal for hard topics. Borrow time from a buffer week, never from practice
    Ill or away for a week Restart with a weekly review; do not try to “catch up” by working through the night

    Run the planner again once a month: it shows whether you are on track while there is still time to act.

    Common mistakeA plan with no spare time breaks in week three

    If every hour of every week is filled, the first cold or family wedding pushes everything back, and the plan starts to feel like a list of failures. Leave buffer weeks, keep one day a week free, and plan only the next week in detail.

    Three example plans

    A. School student: Year 12 from September 2026, exams in June 2028

    Term Main phases Targets and milestones
    Autumn 2026 (Year 12) 1 in September, then 2 and 3 Foundation self-checks in the first 3 weeks; weekly reviews from October; first unit tests in November
    Spring 2027 2 and 3, plus past-paper questions by topic Most AS content finished by Easter
    Summer 2027 AS exams in May/June if your school uses staged assessment; otherwise school exams Start the A Level content in June; in the holidays, research universities and draft your personal statement
    Autumn 2027 (Year 13) 2 and 3 for the A Level content UCAS: mid-October deadline for Oxford, Cambridge and medicine, January for most courses; admissions tests
    Spring 2028 Content finished by February; mocks; then 4 From February, 1–2 timed papers per subject per week; at least 5 per component by Easter
    Summer 2028 5 (the 8 weeks before your first paper), then 6 Exams in May/June; results in August 2028

    B. Private candidate: October 2026 to June 2027, Mathematics, Physics and Chemistry

    A full-time candidate studying about 36 hours a week for Cambridge 9709, 9702 and 9701 in the May/June 2027 series. Week 1 starts on Monday 5 October 2026, and the plan assumes a first paper in early May.

    Weeks Dates Phase Mathematics 9709 Physics 9702 Chemistry 9701
    1–2 5–18 Oct 1 Algebra, powers, graphs Units, standard form, rearranging Formulas, equations, basic moles
    3–8 19 Oct – 29 Nov 2 + 3 Pure 1 AS topics 1–6 AS topics 1–8
    9–12 30 Nov – 27 Dec 2 + 3 Pure 3, first half AS topics 7–11 AS topics 9–16
    13 28 Dec – 3 Jan Buffer Rest and catch up Rest and catch up Rest and catch up
    14–17 4–31 Jan 2027 2 + 3 Pure 3, second half A Level topics 12–18 AS 17–22, A Level 23–26
    18–21 1–28 Feb 2 + 3 Probability & Statistics 1, Mechanics A Level topics 19–25 A Level topics 27–37
    22–25 1–28 Mar 4 Two timed papers a week Timed papers; Paper 5 planning practice Timed papers; Paper 5 planning practice
    26–30 29 Mar – 2 May 5 Final revision; full mocks in weeks 26 and 29 As Maths As Maths
    31 on From 3 May 6 Exams Exams Exams
    Common mistakeThis plan is intensive

    Thirty weeks of 36 hours gives about 360 hours per subject: the same as Cambridge’s guided learning hours, but without a teacher. It suits students with strong IGCSE grades in all three subjects who can study full-time. If you are more than two weeks behind in week 8, move your weakest subject to the October/November 2027 series. Cambridge science candidates also sit a practical exam (Paper 3), so check early that your centre can run it, and book some laboratory practice before March.

    C. Resit student: sitting again in November

    Emre sat A Level Chemistry in June 2027 and knows that Paper 4 went badly. He needs an A for a university application next year, so he starts preparing for the October/November 2027 series the week after his last June paper. Waiting for results day in August would cost him eight weeks. He plans 10–12 hours a week.

    Weeks When Phase Focus
    1 Late June 1: diagnosis One paper of each component done cold and marked; every topic sorted into red, amber or green; ask the centre for the November entry deadline
    2–7 July to mid-August 2 + 3 Relearn two red topics a week; daily flashcards; topic questions after each one
    Results day August – Confirm the grade; ask the centre about a copy of the marked script; make sure the November entry is in
    8–13 Mid-August to late September 4 One or two timed papers a week, marked the same day; every lost mark redone a week later
    14–17 October 5 Amber topics, condensed notes, two full mocks
    18 on October/November 6 The exams; results in January

    A weekly timetable template

    Copy this pattern into a notebook or calendar and adapt it to your subjects. Each day mixes new learning with practice on earlier topics, and every week ends with a review.

    Day Private candidate, full-time (about 36 hours) School student (about 9 hours, plus homework)
    Monday 09:00–12:00 Maths, new topic · 13:30–16:30 Physics, new topic · 20 min flashcards 1 hour: redo today’s lesson examples without looking · 20 min flashcards
    Tuesday Chemistry, new topic · Maths, questions on Monday’s topic · flashcards 1 hour Chemistry questions · 20 min flashcards
    Wednesday Physics, questions · Chemistry, questions · flashcards 1 hour Physics questions · 20 min flashcards
    Thursday Maths, new topic · Physics, new topic · flashcards 1 hour Maths questions · 20 min error log
    Friday Chemistry, new topic · mixed questions from all three · flashcards Free evening
    Saturday 09:00–12:00 one timed test or past-paper section · 30 min marking it and updating the error log 2–3 hours: one timed section, marked; catch up
    Sunday 45 min weekly review and plan; the rest of the day free 45 min weekly review and plan; homework
    Exam techniquePlan around the exam timetable

    When the timetable is published, mark every paper on your plan. Note which paper comes first in each subject and any days with two papers. Then arrange your last two weeks in the same order, so the paper you sit first gets the first final mock.

    Summary
    • Enter the date of your first paper; if the timetable is not out yet, use the earliest date of the exam series window.
    • Work back from the exam: final revision, then past papers, then learning. Keep about one week in ten as a buffer.
    • Reality-check the learning weeks: each chapter’s study time plus half again for practice, times the number of topics.
    • If you fall 3 or more weeks behind, decide early: more hours, a later series for one subject, or AS only.
    • Never shorten phases 4 and 5 to rescue phase 2.
    • Review every Sunday and re-run the planner once a month.

    Next: Phase 1: Fix your foundations.

    PreviousThe preparation road at a glanceNextPhase 1: Fix your foundations
    Preparation Roadmap/The road to exam day

    Phase 1: Fix your foundations

    GuideExamsAll subjects · all boardsStudy time2–6 weeks (this phase)

    Every A Level assumes skills from GCSE or IGCSE. This phase checks those skills with a short self-test and repairs the gaps before they slow you down. It also covers the academic English that second-language learners need.

    Why this matters

    Most students who struggle at A Level do not struggle with the new ideas. They struggle because an old skill is shaky: rearranging a formula, handling a negative number, writing a clear paragraph. Every new topic then takes twice as long, and the student concludes “I’m not good at this subject” when the real problem is a two-hour gap from three years ago. A few weeks spent on foundations now saves months later.

    What each A Level assumes

    Cambridge writes its AS and A Level syllabuses for students who have studied the subject at IGCSE or GCSE (or an equivalent course). You do not need a top grade, but the core skills must be quick and reliable, because at A Level they are tools inside bigger problems, not the problem itself.

    Mathematics, and the maths inside the sciences

    Skill You should be able to… Repair it here
    Number Work with fractions, negative numbers, percentages and ratios without a calculator Numbers, fractions and percentages
    Algebra Expand and factorise; solve linear and simultaneous equations; rearrange formulas Algebra from scratch
    Powers Use the laws of indices, simplify surds, work in standard form Powers, roots, surds and standard form
    Graphs Find a gradient, use y=mx+c, plot points and read values from a graph Coordinates and straight lines
    Geometry Use Pythagoras’ theorem, sine, cosine and tangent, angle facts, areas and volumes Geometry and basic trigonometry

    Physics, Chemistry and Biology

    The sciences use five maths skills constantly, so check these even if Mathematics is not one of your subjects.

    • Units and prefixes: converting mm to m, kJ to J, cm3 to dm3, and knowing that 1 dm3=1000 cm3.
    • Rearranging: making any letter the subject of a formula such as pV=nRT or v2=u2+2as.
    • Standard form: writing 0.000 045 6 as 4.56×10−5 and entering it correctly on a calculator.
    • Significant figures: giving an answer to the same number of significant figures as the data, usually 2 or 3.
    • Graphs: choosing scales, drawing a line of best fit, and finding a gradient from a large triangle.

    Each science also has its own starting knowledge: atoms, ions and equations in Chemistry; forces, energy and circuits in Physics; cells, organs and basic genetics in Biology. Check and repair these with Physics from zero, Chemistry from zero and Biology from zero.

    Business, Economics and other essay subjects

    Essay subjects assume you can read a long case study quickly, pick out the useful facts and numbers, calculate a percentage change, and write a clear paragraph that makes one point and explains it. No prior Business or Economics course is needed, but if these skills feel slow, start with Business from zero and Writing top-band business answers.

    The ten-minute self-check

    Do these without a calculator or notes. Give yourself one mark for each fully correct answer. Score 8 or more and you can start phase 2; otherwise, repair the skills you missed first, using the links above.

    Try it yourselfFoundation self-check (Mathematics and sciences)
    1. Work out 23+34.

      Show answer

      Use a common denominator of 12: 812+912=1712, which is 1512.

    2. Expand and simplify (2x−3)(x+4).

      Show answer

      2x2+8x−3x−12=2x2+5x−12.

    3. Factorise x2−7x+12.

      Show answer

      Two numbers that multiply to 12 and add to −7 are −3 and −4, so (x−3)(x−4).

    4. Make u the subject of v2=u2+2as.

      Show answer

      u2=v2−2as, so u=±v2−2as.

    5. Solve the simultaneous equations 2x+y=7 and x−y=2.

      Show answer

      Adding them gives 3x=9, so x=3. Then y=7−2(3)=1.

    6. Simplify x5×x−2x2.

      Show answer

      Add the powers when multiplying and subtract when dividing: x5−2−2=x1=x.

    7. Write 0.000 045 6 in standard form, and give 0.004 562 to 2 significant figures.

      Show answer

      4.56×10−5 and 0.0046.

    8. Convert 250 mm to metres, 3.5 kW to watts, and 250 cm3 to dm3.

      Show answer

      0.25 m, 3500 W and 0.25 dm3 (because 1 dm3=1000 cm3).

    9. Find the gradient and the y-intercept of the line 2y−6x=8.

      Show answer

      Rearrange to y=3x+4: the gradient is 3 and the line crosses the y-axis at (0,4).

    10. A price rises from $40 to $46. Find the percentage increase.

      Show answer

      The increase is $6, and 640×100=15%.

    For essay subjects, the check is a list of “I can” statements. Be honest: tick one only if you could do it today, under time pressure.

    • I can read a one-page article in English in under five minutes and summarise it in three sentences.
    • I can write a paragraph that makes one point, explains why, and gives an example.
    • I can calculate a percentage change and say what it means in words.
    • I can read a simple table or chart and describe the trend with numbers.
    • I can define ten basic terms in my subject (for Business: profit, revenue, costs, market, stakeholder and so on) in one sentence each.

    Academic English for second-language learners

    Every paper is set and marked in English, and many students who know the science well lose marks to language. The good news: exam English is a small, learnable language of its own.

    • Learn the command words. “Describe” (say what happens) and “explain” (say why it happens) earn different marks. See Command words decoded.
    • Keep a vocabulary notebook. Four columns: the English term, the word in your own language, the definition in English, and an example sentence. Learn definitions as fixed phrases, exactly as the syllabus words them.
    • Write short, complete sentences. Examiners reward correct science and clear reasoning, not elegant style. One idea per sentence is safer than one long sentence with three ideas.
    • Read in English every day. Fifteen minutes of your textbook or a good subject website builds reading speed, which matters in long case-study and data questions.
    • Use the glossary in this guide: every key term from every chapter is collected there.
    English term Turkish Definition in English
    gradient eğim the change in y divided by the change in x
    dynamic equilibrium dinamik denge forward and reverse reactions happen at equal rates, so concentrations stay constant
    demand talep the quantity buyers are willing and able to buy at each price
    evaluate değerlendirmek judge the strengths and weaknesses, then reach a conclusion
    Common mistake“I got an A at IGCSE, so I can skip this”

    Skills fade, especially over a long summer, and A Level needs them to be fast as well as correct. Do the self-check anyway: it takes ten minutes, and it either confirms you are ready or finds a gap while it is still cheap to fix.

    Exam techniqueFoundation skills carry marks in every paper

    A correct method with a slip in rearranging, a wrong power of ten or a missing unit can still lose the final accuracy mark. In Cambridge Mathematics, a non-exact answer not given to 3 significant figures (unless the question asks otherwise) can also cost marks. Examiners usually accept small spelling mistakes, but not when the misspelt word could be another technical term, such as ethane and ethene, or mitosis and meiosis.

    How long should phase 1 take?

    Your self-check result Plan
    8–10, and quick About 2 weeks: repair anything you missed and read the “from zero” chapter for each science, then start phase 2
    6–7 3–4 weeks: work through the foundation chapters for every skill you missed, then redo the self-check
    5 or fewer, or no IGCSE in the subject 5–6 weeks: work through all the foundation chapters for that subject in order
    Any score, if English is your second language Add 15 minutes of subject reading in English every day, all the way to the exam
    TipFix the core now, the rest just in time

    You do not need to repair everything before you start learning. Fix the core skills now (algebra, rearranging, units, paragraph writing), then repair smaller gaps as topics need them. Every chapter in this guide begins with a “Before you start” line that links to exactly what it assumes.

    Summary
    • Most A Level difficulty comes from shaky GCSE/IGCSE skills, not from the new ideas.
    • Maths skills to check: number, algebra, powers and standard form, graphs, geometry and trigonometry.
    • Science skills to check: units and prefixes, rearranging, standard form, significant figures, graphs.
    • Essay subjects need fast reading, clear paragraphs and confident use of data.
    • Score 8 out of 10 on the self-check before moving on; repair what you missed with the foundation chapters.
    • Second-language learners: learn the command words, keep a four-column vocabulary notebook and read in English daily.
    • Phase 1 takes 2–6 weeks, depending on your score.

    Next: Phase 2: Learn the content.

    PreviousBuild your personal study planNextPhase 2: Learn the content
    Preparation Roadmap/The road to exam day

    Phase 2: Learn the content

    GuideExamsAll subjects · all boardsStudy timeAbout half of your preparation time

    How to work through the syllabus so that the ideas stay: studying each chapter actively, turning the syllabus into a checklist, making notes that help, choosing a textbook, keeping a steady pace, and getting unstuck.

    Why this matters

    Phase 2 is the longest part of the road, and the easiest to get wrong. Reading a chapter is not the same as learning it. You have learned a topic when you can explain it and use it without looking, a week later. Everything in this phase is designed to get you to that point with as little wasted time as possible.

    How to study one chapter actively

    Passive study (reading, highlighting, copying) feels comfortable because the material looks familiar while it is in front of you. Active study makes your brain produce the ideas itself, which is harder in the moment and far more effective. Use this routine for every chapter in this guide or your textbook.

    1. Preview (5 minutes). Read the summary, the headings and the key-term cards. Ask yourself what the topic is for.
    2. Study one section at a time. At the end of each section, look away and explain it aloud or on paper in your own words.
    3. Work through each worked example with the solution covered. Attempt each step before you reveal it.
    4. Answer the practice questions without notes, then check them against the full answers and mark honestly.
    5. Write your notes from memory, then compare them with the chapter and fill the gaps in a different colour.
    6. Finish with the quick check, and make flashcards from the key terms and formulas.

    Work in focused blocks of 25–50 minutes with a short break between them, and keep your phone in another room. The timer below helps.

    Turn the syllabus into a checklist

    The syllabus for your exam year is the official list of everything that can be examined. Download it free from your board’s website. In the subject content section, each topic is broken into numbered learning outcomes, short statements of what you must be able to do. Copy them into a tracker, in a notebook or a spreadsheet, and colour each one as you go.

    Learning outcome (Quadratics, Cambridge Paper 1, in our words) Learned Practised Exam-ready
    Complete the square and use it to find the vertex Yes Yes Green
    Use the discriminant to decide how many real roots there are Yes Yes Amber
    Solve quadratic inequalities Yes No Red
    Solve one linear and one quadratic equation simultaneously No No Red
    • Red: not learned yet, or you cannot do it even with notes.
    • Amber: you can do it with notes, or slowly.
    • Green: you can do it without notes, under time pressure, at least a week after learning it.

    The tracker becomes the backbone of your whole preparation. In phase 5 it tells you exactly which topics need work.

    Making notes that help

    Good notes are short, written partly from memory, and built for testing yourself later. Three formats work well. Use whichever suits the topic.

    Method How it works Best for
    Cornell notes A page split into a cue column, a notes column and a summary strip First notes on a topic: definitions, explanations, methods
    Concept map The main idea in the centre, with branches and labelled links Connections: organic reactions, the cardiovascular system, macroeconomic policy
    One-page summary One sheet per topic: key terms, formulas, one diagram, one worked example, common mistakes The end of a topic, and final revision
    What isb² − 4ac?3 cases?Tangenttest?disc. = b² − 4ac> 0: two roots= 0: one root< 0: no rootsset line = curve,then use b² − 4acThe sign of b² − 4ac gives thenumber of roots without solving.Cue columnquestions and keywords, added afterthe lessonHow to revisecover the right side,answer each cuealoud, then checkNotes columnideas, diagrams andexamples, in shortphrasesSummary3–4 sentences in yourown words, written last
    A Cornell page has three parts. You write the notes column while you study, add cue questions afterwards, and finish with a short summary. Covering the notes and answering the cues turns your notes into a self-test.
    Common mistakeNotes are not the goal

    Beautiful notes with colours and perfect handwriting take hours and teach little. If you are copying sentences from the textbook, stop. Close the book, write what you remember, then check. Notes are a tool for testing yourself, not a product.

    Choosing a textbook

    One good textbook per subject, this guide and the past papers are enough. Collecting more resources feels productive, but it is usually a way of avoiding practice.

    • Cambridge: look for books that carry Cambridge’s endorsement mark on the cover. Cambridge lists endorsed resources on its website, from publishers such as Cambridge University Press and Hodder Education.
    • Edexcel IAL: Pearson publishes its own student books for the IAL specifications.
    • AQA and OCR: each board approves or endorses textbooks from several publishers for its own specifications.
    • Check the syllabus years. A book written for an older syllabus can miss new topics or include ones that were removed. Compare its contents with the current syllabus before you buy, especially second-hand.

    Pacing: how fast should you go?

    Your plan from the study planner gives you a number of learning weeks. Divide the topics by the weeks to get your weekly target. For Cambridge, the number of topics in the syllabus is:

    Subject (Cambridge) Topics or sections Topics per week with 20 learning weeks
    Mathematics 9709 (Pure 1, Pure 3, Mechanics, Probability & Statistics 1) 8 + 9 + 5 + 5 = 27 about 1.4
    Physics 9702 25 (1–11 AS, 12–25 A Level) about 1.3
    Chemistry 9701 37 (1–22 AS, 23–37 A Level) about 1.9
    Biology 9700 19 (1–11 AS, 12–19 A Level) about 1

    Topics differ in size, so treat these as averages. Study all your subjects every week rather than one subject per month: switching between subjects is tiring at first, but it gives each topic time to settle and keeps every subject fresh. If a topic takes more than one and a half times its budget, mark it amber, move on, and come back to it in your buffer week.

    Flashcards as you go

    Make flashcards on the day you finish each chapter, while you still know what matters. Aim for 5–15 cards per chapter:

    • one fact per card, written as a question: “What is the discriminant of ax2+bx+c?”
    • definitions word for word on the back, exactly as the syllabus uses them;
    • “why” questions as well as “what” questions: “Why does the rate of reaction increase with temperature?”

    Paper cards in a box or a free flashcard app both work. How and when to review them is in Phase 3.

    The weekly review

    Once a week, for 30–45 minutes, step back from new content.

    1. Test yourself for 15 minutes, without notes, on this week’s topics and one older topic.
    2. Update the colours in your tracker.
    3. Read your error log: is any mistake appearing again and again?
    4. Compare where you are with your plan, and write next week’s topics into your timetable.

    When you are stuck

    Being stuck is a normal part of learning, not a sign that you cannot do the subject. Work down this list.

    1. Slow down. Reread the section, then copy out the worked example line by line, asking why each step follows from the one before.
    2. Go back one step. Most stuck points are really an earlier gap. Follow the “Before you start” link at the top of the chapter.
    3. Find a different explanation: your textbook, a video, or a friend who explains it in other words.
    4. Ask a precise question. “I don’t understand moles” is hard to answer; “I don’t see why we divide by 2 in line 3” is easy. Teachers, study groups and online forums all respond better to precise questions.
    5. Set a time limit. After 30 minutes stuck, mark the topic red, move on, and return in two days. Sleep often does the work for you.
    TipExplain it to a younger student

    If you can explain a topic simply to someone two years younger, without notes, you understand it. If you get stuck halfway, you have found exactly the part to study again.

    Exam techniqueLearn definitions word for word

    Definitions are quick marks, and mark schemes look for precise key words. “Acceleration is the rate of change of velocity” scores; “acceleration is how fast something speeds up” does not. Put every definition from the syllabus and from this guide’s key-term cards onto flashcards, and practise writing them out exactly.

    Summary
    • Learning means being able to explain and use an idea without looking, a week later.
    • Study each chapter actively: preview, explain each section, attempt worked examples before reading them, answer questions without notes.
    • Turn the syllabus learning outcomes into a red/amber/green tracker.
    • Use Cornell notes, concept maps or one-page summaries; write notes from memory.
    • Use one endorsed textbook per subject that matches your syllabus years.
    • Pace yourself: divide topics by learning weeks, and study every subject every week.
    • Make flashcards on the day you finish a chapter, and review your week every weekend.

    Next: Phase 3: Practise and test yourself.

    PreviousPhase 1: Fix your foundationsNextPhase 3: Practise and test yourself
    Preparation Roadmap/The road to exam day

    Phase 3: Practise and test yourself

    GuideExamsAll subjects · all boardsStudy timeAlongside phase 2, then 3–6 weeks

    Practice turns understanding into marks. This phase uses retrieval practice, spaced repetition, flashcards and mixed questions, with an error log that makes sure each mistake happens only once.

    Why this matters

    You can understand a topic perfectly on Monday and be unable to answer a question on it by Friday. Forgetting is normal, and it happens to everyone. The cure is not more reading: it is testing yourself, spread out over time. This phase runs alongside phase 2 from the first week, and becomes your main activity once all the content is learned.

    Retrieval practice: test, don’t reread

    Retrieval practice

    Learning by recalling information from memory, for example by answering questions or writing down everything you remember, instead of reading the material again.

    Every time you pull an idea out of memory, the memory becomes stronger and easier to find next time. Rereading does not do this, because the answer is already in front of you. A large 2013 review of ten common study techniques rated practice testing and spaced practice as the two most useful, and rereading and highlighting among the least useful.

    Ways to practise retrieval:

    • Blank page: write everything you remember about a topic on a blank sheet, then check it against your notes and add what you missed in a different colour.
    • Flashcards, answered aloud or on paper before you turn the card over.
    • Quizzes: the quick check at the end of each chapter, and the mixed practice quizzes.
    • Exam-style questions, from your textbook, this guide and past papers.

    Spaced repetition: when to review

    Spaced repetition

    Reviewing material at increasing intervals (for example after 1, 3, 7, 14 and 30 days), so that each review happens just as the memory starts to fade.

    07142128350%50%100%days after you first learn the topichow much you can recallreviews on days 1, 3, 7, 14, 30no review
    A schematic picture of forgetting. Without review (dashed), most of a new topic is gone within a week. Each review (dot) brings recall back up, and the memory then fades more slowly, so the gaps between reviews can grow.

    Review a topic about 1, 3, 7, 14 and 30 days after you first learn it, then once a month until the exam. For example, for a topic learned on Monday 12 October 2026:

    Review Day Date What to do (5–15 minutes)
    1st 1 Tuesday 13 October Blank page, then check
    2nd 3 Thursday 15 October Flashcards for the topic
    3rd 7 Monday 19 October 3–4 exam-style questions
    4th 14 Monday 26 October Quiz or mixed questions
    5th 30 Wednesday 11 November A past-paper question, timed

    Every review is a test, never a reread. If a review goes badly, restart the cycle for that topic from day 1. Write the review dates in your diary on the day you learn the topic, so you never have to remember them.

    Flashcards that work

    The Leitner system organises paper flashcards in five boxes. Box 1 is reviewed every day, box 2 every 3 days, box 3 every week, box 4 every 2 weeks and box 5 every month. A card you answer correctly moves up one box; a card you get wrong goes back to box 1. Flashcard apps such as Anki do the same scheduling automatically.

    • Answer before you flip. Say or write the full answer first. A vague “I knew that” counts as wrong.
    • Be strict. Half right is wrong for a definition, because the mark scheme wants the key words.
    • Mix card types: definitions, formulas, “why” questions, and blank diagrams to label.

    Interleaving: mix your topics

    Interleaving

    Practising a mixture of different topics or question types in one session, so that you must decide which method each question needs, instead of practising one method many times in a row.

    Blocked practice Interleaved practice
    What it looks like 20 questions on completing the square 12 questions mixing quadratics, functions, series and trigonometry
    How it feels Smooth and fast Slower and harder
    What it trains Repeating a method you already know is needed Choosing the right method, which is what the exam demands
    When to use it The first practice on a new topic Everything after that

    Exam questions never tell you which chapter they come from. Once you have learned three or four topics, make most of your practice interleaved.

    Topic questions

    After each topic, climb this ladder: textbook exercises, then the practice questions in this guide, then past-paper questions on that topic. Collections of Cambridge questions sorted by topic (“topical past papers”) are widely available, and they let you practise a topic in exam style long before you sit full papers. Aim for at least 10–15 exam-style questions on each topic, more for big topics, before you call it green.

    The error log

    The error log is the most valuable document you will make. Every mark you lose goes into it, with the reason and a rule to stop it happening again.

    Date Question Topic Marks lost What went wrong Type Rule for next time Redo on
    12 Oct Maths practice Q4 Quadratics 2 Used −b=−4 when b=−4 Careless slip Write a, b, c with their signs before substituting 19 Oct
    14 Oct Physics topic test Q2 Kinematics 2 Did not convert km h−1 to m s−1 Method Convert every quantity to SI units first 21 Oct
    15 Oct Chemistry worksheet Q5 Moles 3 Ignored the 1 : 2 ratio in the equation Knowledge Always use the mole ratio from the balanced equation 22 Oct
    20 Oct Business 12-mark question Motivation 4 No judgement at the end Exam technique End with a justified conclusion that answers the question 27 Oct

    Use five types: knowledge (you did not know it), method (you knew the topic but chose or applied the wrong method), reading (you misread the question), careless slip, and exam technique (timing, missing working, no evaluation). Once a week, read the whole log and count the types. If most are knowledge errors, go back to phase 2 for those topics. If most are careless slips, slow down and check every answer. Then redo each logged question on its redo date, without looking at your old answer.

    Calculation subjects and essay subjects

    Calculations (Maths, Physics, Chemistry)
    • Do many questions: volume matters.
    • Write full working, as in the exam: method marks need visible steps.
    • Check units and significant figures at the end of every answer.
    • Redo wrong questions from a blank page, not by reading the solution.
    • Time yourself from early on: about 1.5 minutes per mark in Cambridge Maths.
    Essays (Business, Economics, long-answer Biology)
    • Plan every answer for 3–5 minutes before writing.
    • Build each paragraph: knowledge, application, analysis, evaluation.
    • Write timed answers, then mark them against the mark scheme’s levels.
    • Collect strong chains of reasoning (“this leads to… which means…”).
    • See Writing top-band business answers.

    Measuring readiness

    Test each topic with mixed questions a week or more after learning it, and record your score.

    Score Meaning What to do
    80% or more Exam-ready Mark it green; review it once a month
    60–79% Nearly there Mark it amber; work through its error-log entries and do more topic questions
    Below 60% Not secure yet Mark it red; go back to the chapter (phase 2)

    You are ready to give past papers most of your time when your average across mixed sets is 70% or more, no topic is red, and the same mistakes have stopped appearing in your error log.

    TipStart every session with retrieval

    Spend the first 10 minutes of each study session on questions or flashcards from older topics, before any new work. It warms you up, and it builds your spaced reviews into every day without extra planning.

    Common mistakeLooking at the answer too soon

    When a question gets hard, it is tempting to look at the solution after a minute. Struggle for at least 5–10 minutes first: the effort of trying is what makes the method stick. If you do look, close the solution and redo the whole question from a blank page.

    Exam techniquePractise in exam format from the start

    Write practice answers exactly as you would in the exam: full working, units, sentences rather than notes, and diagrams labelled. Habits formed in practice are the ones you will use under pressure, so make them exam habits from the first week.

    Summary
    • Retrieval practice (testing yourself) beats rereading; use blank pages, flashcards, quizzes and exam-style questions.
    • Review each topic about 1, 3, 7, 14 and 30 days after learning it, then monthly; every review is a test.
    • Use a Leitner box or a flashcard app, and be strict with yourself.
    • Interleave: once you know a few topics, mix them in every practice set.
    • Keep an error log with a type and a rule for every lost mark, and redo each question a week later.
    • Green at 80% or more on mixed questions a week after learning; move on to past papers at a 70% average with no red topics.

    Next: Phase 4: Past papers and mark schemes.

    PreviousPhase 2: Learn the contentNextPhase 4: Past papers and mark schemes
    Preparation Roadmap/The road to exam day

    Phase 4: Past papers and mark schemes

    GuideExamsAll subjects · all boardsStudy time8–12 weeks

    Where to find official past papers, mark schemes, examiner reports and grade thresholds; how many papers to do and in what order; how to sit them under timed conditions and mark them honestly; and the redo cycle that turns each lost mark into a gained one.

    Why this matters

    Past papers are the closest thing to the real exam that exists. They show you the style of the questions, how long you really need, and the exact ideas that earn marks. Students who do many past papers, mark them honestly and learn from every mistake usually outscore students who simply know more content.

    Where to find them

    Use the official sources first. Each board publishes four kinds of document for each exam series: question papers, mark schemes, examiner reports (what candidates did well and badly) and grade thresholds or grade boundaries (the raw marks needed for each grade).

    Board Where to look Notes
    Cambridge International The syllabus page on cambridgeinternational.org Cambridge publishes some past papers on its website. The full archive is on the School Support Hub, which registered schools use; private candidates can ask their exam centre
    Pearson Edexcel (IAL) Your subject’s page on qualifications.pearson.com Papers, mark schemes and examiner reports under the exam materials; grade boundaries on their own page
    AQA aqa.org.uk Past papers and mark schemes for each specification
    OCR ocr.org.uk Past papers in the assessment section of each qualification

    With every board, the newest series is sometimes available only to teachers for a while. Many other websites copy the official files. If you use one, make sure each paper comes with its matching mark scheme.

    Reading Cambridge file names

    Cambridge files have names like 9709_s25_qp_12. Once you can read them, finding the right paper takes seconds.

    Part Meaning
    9709 The syllabus code (Mathematics)
    s25 The series and year: s = May/June, w = October/November, m = February/March; 25 = 2025
    qp, ms, er, gt Question paper, mark scheme, examiner report, grade thresholds
    12 Paper 1, variant 2

    Cambridge sets several versions (variants) of most papers in each series for different time zones. That is why there are so many papers to practise with: a few recent years give you 20 or more versions of some components.

    How many, and in what order?

    Aim for 8–12 full papers per component by exam day, plus topic questions, and more for components you find hard. Use them in this order:

    1. Older papers (about 6–10 years old): cut into topic questions during phase 3.
    2. Middle years: full timed papers in phase 4.
    3. The two or three most recent series: saved for full mocks in phase 5.

    Before you start, read the “changes to the syllabus” page at the front of your syllabus. Papers written before a syllabus change can include topics that are no longer examined, and miss new ones. Skip those questions, or replace them with questions from this guide.

    Sitting a paper under timed conditions

    1. Choose the same time of day as the real exam, a quiet desk, and only the permitted equipment: pens, pencil, ruler, an allowed calculator, and the formula list or data booklet your board provides (for Cambridge Mathematics, the list called MF19).
    2. Put your phone in another room and set a timer for the real length of the paper.
    3. Read the front cover as if it were the real exam.
    4. Work to time. Cambridge Mathematics Paper 1 has 75 marks in 1 hour 50 minutes, so allow about 1.5 minutes per mark.
    5. When time runs out, draw a line. You may finish in a different colour to see what more time would give, but count only the marks above the line.

    Marking honestly

    Mark the paper the same day, or the next, with the official mark scheme. Be strict: if your answer does not clearly contain the point the mark scheme describes, do not give yourself the mark. Generous self-marking only hides the problems you need to fix.

    Mathematics mark schemes use a short code. Learn it, because it tells you exactly how marks are earned:

    Code Meaning
    M1 A method mark, for a correct method applied to the problem
    A1 An accuracy mark: only possible if the related M mark is earned
    B1 An independent mark for a correct answer or statement
    FT Follow through: credit for correct work that uses your own earlier wrong answer
    CAO Correct answer only
    OE Or equivalent: another correct form of the answer is accepted
    AWRT Answer which rounds to the value given
    AG Answer given in the question: every step must be shown

    Science mark schemes list the acceptable answers for each mark point, and essay mark schemes describe levels: what an answer must show to reach each band of marks. Then turn your raw mark into a grade with the grade thresholds for that series and variant. Thresholds change from series to series because some papers are harder than others, so always use the matching one. Aim to be comfortably above your target grade, not just on it.

    Exam techniqueWhy the method mark matters

    An A mark depends on its M mark. A correct final answer from a wrong or invisible method can score nothing, while a wrong final answer from a correct method can still score most of the marks. In “show that” questions, the answer is printed (the mark scheme labels it AG), so all the marks are for the steps. Always write your working where the examiner can follow it.

    Read the examiner report

    For every paper you sit, read the examiner report’s comments on it. Examiners describe the common errors, the misunderstandings they saw most often, and what the best answers did. Ten minutes with the report often explains a lost mark better than the mark scheme does. Copy anything that applies to you into your error log.

    Topic by topic, or full papers?

    Topic-by-topic questions Full papers
    When Phase 3 and the start of phase 4 Most of phase 4, and all of phase 5
    What it trains Depth in one topic; finding gaps quickly Timing, stamina, switching between topics, choosing methods
    The risk The topic heading tells you which method to use One paper cannot cover every topic

    Use both. Start phase 4 with one full paper per component per week, plus topic sets for your weakest areas. Build up to two or three full papers a week in the last month of the phase.

    The redo cycle

    1. Sit the paper, timed.
    2. Mark it honestly, the same day.
    3. Log every lost mark in your error log, with its type and a rule for next time.
    4. Relearn: for a knowledge gap, go back to the chapter; for a method gap, do three to five similar questions.
    5. Redo every question that lost marks, from a blank page, a week later.
    6. Track your scores for each component, with the grade thresholds next to them, so you can see the trend.
    Common mistakeDo not memorise mark-scheme wording without understanding it

    Mark schemes are written for examiners, not as model answers. Phrases learned by heart without understanding end up in the wrong context, where they earn nothing. Questions are reworded every year, and the mark is for the idea used correctly. Learn why each point earns its mark, then practise explaining it in your own words while keeping the key technical terms.

    TipKeep the newest papers for later

    Do not use the two or three most recent series before phase 5. They are the best guide to your own exam, so save them for full mocks under real conditions.

    Summary
    • Use official sources: Cambridge (public website and the School Support Hub through your centre), Pearson, AQA and OCR, with mark schemes, examiner reports and grade thresholds.
    • Read Cambridge file names: syllabus, series and year, document type, paper and variant (9709_s25_qp_12).
    • Aim for 8–12 full papers per component: old papers as topic questions, middle years timed, newest saved for mocks.
    • Sit papers in real conditions and to time: about 1.5 minutes per mark in Cambridge Maths Paper 1.
    • Mark strictly; learn M, A, B, FT and AG; compare your mark with the thresholds for that exact paper.
    • Read the examiner report for every paper, and follow the redo cycle: sit, mark, log, relearn, redo, track.
    • Understand mark-scheme points; never memorise their wording blindly.

    Next: Phase 5: Final revision.

    PreviousPhase 3: Practise and test yourselfNextPhase 5: Final revision
    Preparation Roadmap/The road to exam day

    Phase 5: Final revision

    GuideExamsAll subjects · all boardsStudy timeThe last 6–8 weeks

    A week-by-week plan for the last eight weeks before your first paper, with daily timetables, how to run your own mocks, how to condense your notes, formula practice, sleep, and the habits to avoid.

    Why this matters

    The last eight weeks decide more grades than any other part of the course. By now you know most of the content; this phase makes it reliable under pressure. Done well, it is calm and organised: you know exactly what to do each day, and each week you can see the gaps closing.

    The last eight weeks, week by week

    Count the weeks back from your first paper. If you have only six weeks, combine weeks 8 and 7, and weeks 6 and 5.

    Weeks to go Main focus Full papers per subject Also this week
    8 Audit: update your tracker, list every red and amber topic, plan the eight weeks 1, as a baseline Get the final exam timetable and mark every paper on your plan
    7 Red topics, first half: relearn, then practise 1 Condensed notes for each red topic
    6 Red topics, second half 1–2 Start daily formula and definition drills
    5 Mock week: a full set of papers in exam conditions A full set Mark everything; update the error log
    4 Fix the mock: relearn every topic that lost marks, then amber topics 2 Finish a one-page summary for every topic
    3 Timed papers from recent series 2–3 The redo cycle after every paper
    2 Final mocks with the newest papers, in the order of your exams 2–3 Shift your sleep to fit the exam start times
    1 Light review: summaries, flashcards, error-log rules; nothing new Parts of papers only Logistics: route, equipment, ID, statement of entry

    The last two weeks

    • Follow the timetable. The subject you sit first gets the first mocks and the most review time in the final days.
    • Sit papers at exam times. If your papers start in the morning, do your mocks in the morning, so your brain is used to working hard at that hour.
    • Read your rules daily. Read the “rule for next time” column of your error log every day. It is a personal list of the traps you fall into.
    • Make one sheet per paper: the formulas, definitions and mistakes that matter most for that paper. Read it the evening before.
    • Plan the gaps between papers. With three days between two papers, for example: one timed paper on day 1, weak topics on day 2, light review on day 3.

    Daily timetable templates

    On a full revision day (study leave, weekends, or every day for a private candidate):

    Time Activity
    08:30–09:00 Flashcards and error-log rules
    09:00–10:50 A timed paper (subject A)
    10:50–11:10 Break, away from your desk
    11:10–12:30 Mark the paper and log every lost mark
    12:30–13:30 Lunch, away from screens
    13:30–15:00 Relearn the topics the morning’s paper exposed
    15:00–15:20 Break and some movement
    15:20–16:50 Subject B: topic questions on a weak topic
    16:50–19:00 Exercise, free time, dinner
    19:00–20:00 Subject C: condensed notes and flashcards
    From 20:00 Free time; screens off an hour before bed

    That is about 7.5 hours of real work. More than this, day after day, usually means less is learned per hour.

    On a school day, add about two hours after lessons:

    Time Activity
    17:00–17:45 Topic questions on a red or amber topic
    18:00–18:45 One timed section of a past paper
    19:45–20:30 Mark it, update the error log, 20 minutes of flashcards

    Running your own mocks

    If your school runs mock exams, treat them as the real thing. If you study alone, run your own:

    • the same length, the same time of day and the same rules as the real exam;
    • a clear table, no notes, phone in another room, and if possible someone in the room acting as the invigilator;
    • the whole set of papers over a few days, in the same order as your real timetable;
    • marking within 24 hours, followed by the redo cycle from Phase 4.

    A disappointing mock is useful: it shows you the problems while there is still time to fix them. Treat the score as information, not as a judgement of you.

    Condensing your notes

    Condensing works because every step forces you to choose what matters and to recall it.

    1. Chapter notes from phase 2.
    2. One page per topic: write it from memory first, then check and fill the gaps.
    3. One page per paper: only the formulas, definitions and traps for that paper.
    4. A final card per paper to read on the evening before.

    Formulas, definitions and data

    Get a clean copy of what your board provides in the exam: for Cambridge, the formula list MF19 for Mathematics, the data and formulae pages printed in each Physics paper, and the Data Booklet for Chemistry. Learn what is on it, and where, so you never waste time searching. Then memorise everything else. Each day in the last three weeks, write out every formula and definition for one paper on a blank page in five minutes, check it, and correct it. Our sheets help: Maths, Physics, Chemistry, Biology and Business, economics and accounting.

    Sleep, food and exercise

    Your brain stores the day’s learning while you sleep, so sleep is part of revision, not time taken away from it. Most teenagers need 8–10 hours a night. An all-nighter before a paper makes you slower and more error-prone the next day. Two weeks before the exams, move your bedtime and wake-up time so that you are alert at the time your papers start. A daily walk or other exercise helps you sleep and manage stress, and avoiding caffeine after mid-afternoon protects your sleep. There is more in Motivation, sleep and stress.

    What not to do

    Don’t Do instead
    Start new textbooks, courses or apps Stick to your notes, this guide and past papers
    Reread your notes for hours Test yourself: papers, flashcards, blank pages
    Revise only your favourite topics Work from your red and amber list
    Sit papers without marking them Mark every paper within 24 hours
    Use the newest papers too early Save them for the last three weeks
    Stay up late to “finish” a topic Keep the same bedtime every night
    Compare hours with friends or on social media Compare your scores with your own scores last week
    TipRevise by question, not by chapter

    In the final weeks, start from questions rather than from your notes. Do a question; if you cannot, open the chapter, fix that one gap, and try a similar question. Your time goes exactly where your marks are being lost.

    Common mistakeThe last-week panic switch

    In the final week, many students abandon their plan and start rereading everything from the beginning. It feels safe but achieves little, and it leaves no time for the subject you sit first. Trust the plan you made when you were calm: light review, error-log rules, sleep.

    Exam techniqueTurn your mock into a timing plan

    After each mock, note how long each question took. Then write a timing plan for each paper: for Cambridge Mathematics Paper 1 (75 marks in 1 hour 50 minutes), about 1.5 minutes per mark, so roughly 45 minutes for the first 30 marks. Practise the plan in your final mocks so that pacing feels automatic on the day.

    Summary
    • Count back from your first paper; use the 8-week table, merging weeks if you have only six.
    • Weeks 8–6 for red topics, week 5 a full mock, week 4 fixing the mock, weeks 3–2 timed papers and final mocks, week 1 light review.
    • Sit papers at real exam times, mark within 24 hours, and read your error-log rules daily.
    • Condense: chapter notes, one page per topic, one page per paper, one card per paper.
    • Know what is on your formula list or data booklet, and memorise everything else.
    • Sleep 8–10 hours; shift your bedtime to the exam timetable two weeks ahead.
    • No new resources, no all-nighters, no last-week panic.

    Next: Phase 6: Exam day, results and resits.

    PreviousPhase 4: Past papers and mark schemesNextPhase 6: Exam day, results and resits
    Preparation Roadmap/The road to exam day

    Phase 6: Exam day, results and resits

    GuideExamsAll subjects · all boardsStudy timeRead it a week before your first paper

    Checklists for the night before and the morning of each exam; how to read the paper, manage your time and keep going when you are stuck; and what happens next: results day, enquiries about results, resits, UCAS Clearing and a plan B.

    Why this matters

    By exam day the learning is done, but marks can still be lost to things that have nothing to do with knowledge: a forgotten calculator, a misread question, fifteen minutes stuck on one part. This chapter removes those risks. It then walks you through what happens after the last paper, including what to do if results day does not go to plan.

    The night before

    • Check the details on your statement of entry, the document from your centre that lists your candidate number and each paper’s date and start time. Confirm the time and the room, and set two alarms.
    • Pack your bag:
      • photo ID (a passport or national ID card): private candidates will not be allowed in without it;
      • your statement of entry;
      • a permitted calculator with fresh batteries;
      • at least three black pens, two pencils, an eraser, a sharpener and a ruler, plus a protractor and compasses for Mathematics;
      • a clear pencil case and a clear water bottle, if your centre allows them.
    • Check your calculator is allowed. Rules differ by board and subject. For example, Cambridge Mathematics does not allow graphic calculators or calculators that can do algebra.
    • Read your one-page sheet for the paper, then stop. Do not start anything new.
    • Plan your route, with time to spare for traffic, and go to bed at your normal time.
    Common mistakePens, pencils and correction fluid

    Cambridge papers ask for black or dark blue pen, and do not allow erasable pens or correction fluid. Use pencil only for diagrams and graphs, and for multiple-choice answer sheets, which are marked by machine and need a soft pencil. If you make a mistake in pen, cross it out with a single neat line and write the correction next to it.

    The morning

    • Eat breakfast and drink some water, even if you are nervous.
    • Arrive at least 30 minutes early. Private candidates should check the arrival time their centre gives.
    • Take a last look at your one-page sheet, then put it away before you enter the room.
    • Leave your phone, smartwatch and any other device at home or hand them in as instructed. Having one with you in the exam room can count as malpractice, even if it is switched off.
    • Go to the toilet before the exam starts.

    In the exam

    Reading the paper

    1. Write your name, centre number and candidate number exactly as they appear on your statement of entry.
    2. Read the front cover: the number of questions, the total marks, and whether you must answer every question or choose some.
    3. If you have time, look through the whole paper for a minute so that nothing surprises you later.
    4. Read each question twice. Underline the command word and the key data.
    5. Look at the marks for each part and the space given for the answer. Both tell you how much the examiner expects.

    Timing: minutes per mark

    Divide the time by the total marks to get your pace. For example, Cambridge Mathematics Paper 1 has 75 marks in 1 hour 50 minutes, which is about 1.5 minutes per mark, so a 6-mark question deserves about 9 minutes.

    Cambridge paper Marks Time Minutes per mark
    Mathematics Paper 1 or 3 75 1 h 50 min about 1.5
    Mathematics Paper 2, 4, 5 or 6 50 1 h 15 min 1.5
    Physics, Chemistry or Biology Paper 1 (multiple choice) 40 1 h 15 min just under 2 per question
    Physics, Chemistry or Biology Paper 2 (AS structured) 60 1 h 15 min 1.25
    Physics, Chemistry or Biology Paper 4 (A Level structured) 100 2 h 1.2
    Business Paper 2 60 1 h 30 min 1.5
    Economics Paper 2 60 2 h 2

    To keep 10 minutes for checking in Mathematics Paper 1, work at about 1.3 minutes per mark. Edexcel IAL, AQA and OCR papers have different lengths, so work out your own pace from the front cover of a past paper.

    Exam techniqueLet the marks guide you

    The marks for each part tell you what the examiner wants. A 2-mark “explain” question usually needs two separate points, or one point and its reason. A 1-mark “state” question needs a short answer, not a paragraph. For calculations, a part worth 4 or 5 marks needs several clear steps of working, because most of those marks are method marks.

    When you are stuck

    • Move on once you have spent about twice the time the marks deserve. Circle the question number and come back at the end.
    • Write what you know. A formula, a labelled diagram or the first line of working can earn method marks.
    • Use answers given in the question. If you cannot show that x=4 in part (a), use x=4 in part (b) anyway: the later marks are still available.
    • Never leave a multiple-choice question blank. No marks are taken away for wrong answers, so eliminate what you can and choose.
    • If panic starts, put your pen down, breathe slowly for thirty seconds, and read the next question. One hard question is only a few marks.
    • Need more space? Use the extra page at the back of the booklet, and write the question number clearly.

    The last ten minutes

    Check that you answered what was asked (“exact”, “in terms of π”, “to 3 significant figures”), that every answer has a unit where one is needed, that no part is left blank, and that your multiple-choice answer sheet is complete.

    After each paper

    Leave the room and do not go through the answers with friends: it only creates worry about marks you can no longer change. Eat, rest, then spend thirty minutes planning for your next paper. If something affected your performance, such as illness, a family emergency or a disturbance in the room, tell your exams officer the same day. Your centre may be able to apply for special consideration, but only within a short time after the exam.

    Results day

    • When: Cambridge results for the May/June series come out in August (the June 2026 results were released on 11 August 2026), and results for the October/November series in January. Each Edexcel IAL series has its own results day; check Pearson’s key dates.
    • How: your centre gives you your results, and some centres also give you a login to see them online.
    • University places: if you applied through UCAS, check UCAS Hub on results day to see whether your place is confirmed. Before the exams, ask your centre whether it will send your results to UCAS.

    Read your results calmly, and decide what to do before you phone anyone.

    Enquiries about results

    If a grade is lower than you expected, your centre can ask the board to check it. Cambridge calls these enquiries about results; Pearson, AQA and OCR call them post-results services or reviews of results. The usual options are:

    • a clerical re-check that all the marks were counted and added correctly;
    • a review of marking, with a faster priority service when a university place depends on the result;
    • a copy of your marked script, which shows exactly where you lost marks. This is very useful if you plan to resit.

    Every service has a fee and a strict deadline, and it must be requested through your centre, including by private candidates. A review looks for marking errors; it is not a second marking, and marks can go down as well as up. It is usually worth it only if you believe you are close to a grade boundary.

    Resits

    Board When you can resit How it works
    Cambridge International October/November (results in January), or the next May/June You normally sit all the papers of the AS or A Level again; ask your centre whether anything can be carried forward, such as an AS result within 13 months
    Pearson Edexcel IAL January, May/June or October Modular: you can resit only the units you choose, and your best result for each unit counts
    AQA and OCR (UK A Levels) May/June only Linear: you sit all the papers again, a year later

    Resit when a university offer or a career depends on the grade, and when you know exactly what went wrong. If your university has already accepted you, the time is usually better spent preparing for the course. If you decide to resit, read the study planner example for a resit student, and start straight away.

    TipWrite your plan B before results day

    In July, while you are calm, write down what you will do if you miss your offer by one grade, and by two. On results day you will then follow a plan instead of making big decisions in a panic.

    UCAS Clearing and plan B

    If you miss the conditions of your offer, do not assume the worst. Universities sometimes accept students who just miss their grades, so check UCAS Hub first, and phone the admissions office if your status is unclear (have your UCAS Personal ID ready). If you have no place, Clearing lets you apply for university courses that still have places. You phone universities, and if one makes you an offer, you add it as your Clearing choice in UCAS Hub. Clearing is busiest on A Level results day and in the days after.

    Other good plan B routes include:

    • a course with a foundation year, which has lower entry requirements;
    • a gap year with resits, then a new application with real grades instead of predicted ones;
    • universities in Turkey or other countries that accept A Level results;
    • apprenticeships and work-based routes.

    There is more on applications in University applications and admissions tests.

    Common mistakeDo not grab the first Clearing offer

    An offer on the phone can feel like relief, but you will spend three or four years on that course. Ask about the course content, the city and the costs, compare two or three options, and talk it through with someone you trust before you accept.

    Summary
    • Night before: statement of entry, photo ID, permitted calculator, black pens, pencils, eraser and ruler; normal bedtime.
    • Morning: breakfast, arrive at least 30 minutes early, no phone or smartwatch in the room.
    • Pace yourself by minutes per mark: about 1.5 in Cambridge Maths Paper 1 (75 marks in 1 hour 50 minutes).
    • Stuck? Move on, write what you know, use given answers, never leave a multiple-choice question blank.
    • Report illness or disruption to your exams officer the same day.
    • Cambridge June results come out in August, November results in January. Enquiries about results have fees and short deadlines, and marks can go down.
    • Resits: Cambridge in November or June, Edexcel IAL in January, June or October (best unit result counts), AQA and OCR in June only.
    • Write a plan B before results day; Clearing, foundation years and gap years are all good routes.

    You have reached the end of the road. Go back to The preparation road at a glance at any time to check where you are, or read Studying as a private candidate if you are preparing without a school.

    PreviousPhase 5: Final revisionNextStudy methods that actually work
    Preparation Roadmap/Skills that raise grades

    Study methods that actually work

    GuideExamsAll boards · all subjectsStudy time45 minutes to read, then use all year

    Research shows that some popular study habits, such as rereading and highlighting, do very little. This chapter shows you the methods that do work (testing yourself, spacing and mixing your practice) and how to build them into a normal week.

    Why this matters

    Two students can study for the same hours and get very different grades. Usually the difference is method, not talent. Rereading feels productive but builds familiarity, not memory. Testing yourself feels harder, and that difficulty is what makes knowledge stay. Change how you study and every hour from now on is worth more.

    Before you start, read The preparation road at a glance so you know where these methods fit. They are the engine of Phase 2: Learn the content and Phase 3: Practise and test yourself.

    What the research says

    In 2013 the psychologist John Dunlosky and four colleagues reviewed the research on ten popular learning techniques and rated how useful each one is for real students.

    Rating Technique What it means
    High Practice testing Recalling or answering questions without looking
    High Distributed practice Spreading study of a topic over days or weeks
    Moderate Elaborative interrogation Asking yourself “why is this true?”
    Moderate Self-explanation Explaining each step of a solution to yourself
    Moderate Interleaved practice Mixing problem types in one session
    Low Summarising Writing shorter versions of a text
    Low Highlighting Marking key words while reading
    Low Keyword mnemonics Linking new words to memorable images
    Low Imagery for text Picturing what you read
    Low Rereading Reading the same notes again

    “Low” does not mean useless; it means a small benefit for the time it takes. Yet rereading and highlighting are the most popular study habits of all. They feel good because familiar words make your brain mistake “I recognise this” for “I know this”, an illusion of competence. The exam gives you a blank space, not the page, so producing knowledge from memory is what you must practise.

    Retrieval practice: test yourself first

    Retrieval practice

    Learning by recalling information from memory (for example by answering questions or writing everything you remember on a topic) instead of rereading it. Each successful recall makes the memory stronger and easier to find later.

    Retrieval works even when you get answers wrong, as long as you check and correct them straight away. Good ways to retrieve:

    • Blank-page recall (“blurting”). Close your notes and write everything you remember about a topic for 10 minutes. Then fill the gaps from your notes in a different colour. The coloured parts are your revision list.
    • Chapter questions. Do the practice questions at the end of each chapter here without looking back.
    • Flashcards (see below) and past-paper questions, which are retrieval in the exact form the exam uses. See Phase 4: Past papers and mark schemes.
    TipTurn your notes into questions

    After you finish a section, write three questions it answers in the margin, for example “What does the discriminant tell you?” Next time, cover the notes and answer the questions. Your notes become a self-test instead of something to reread.

    Spacing: little and often beats one long session

    Spaced practice

    Spreading study of the same material over several sessions separated by gaps of days or weeks, instead of doing it all at once. Also called distributed practice.

    Memory fades quickly after you first learn something, and each recall after a gap makes it fade more slowly. A useful rule of thumb is to review a new topic about 1 day, 3 days, 1 week, 2 weeks and 1 month after first learning it. The exact gaps matter less than the habit of coming back.

    The opposite is cramming: everything in one long session just before a test. It can get you through tomorrow’s test, but most of it is gone weeks later, and A Levels examine two years of content at the end.

    Common mistakeRewatching is rereading

    Watching the same video lesson twice is the video version of rereading. Watch it once, carefully, then close it and test yourself. Rewatch only the part you could not recall.

    Interleaving: mix your practice

    Interleaving

    Mixing different topics or types of problem within one practice session (for example differentiation, vectors and probability), so that you must decide which method each question needs.

    When you do twenty questions on one method in a row (blocked practice), you know the method before you read the question. In the exam, nobody tells you, and interleaving trains that choice. Use blocked practice for a brand-new topic, then switch to mixed sets once the basic questions feel easy.

    Making ideas stick: elaboration, examples and pictures

    Elaboration means connecting a new idea to what you already know by asking “why?” and “how?”. Why does a higher temperature speed up a reaction? How does that link to activation energy? Every connection is another route your memory can use.

    Concrete examples make abstract ideas easy to picture. “Price elastic demand” is hard to remember; “when one brand of crisps raises its price, many buyers switch brands” is easy. Collect one or two examples of your own for every definition.

    Dual coding means combining words with visuals. Draw blood flow through the heart, a supply and demand diagram or a flowchart of protein synthesis, and label it from memory. Words and pictures give you two routes to the same idea.

    Worked examples, then fade them

    When you are a beginner, solving problems alone can overload you, so study a worked example first. Then remove the support step by step:

    Stage What you do
    1. Study Read a full worked example. Explain each step to yourself: why this step, why now?
    2. Complete Try a similar example where the last steps are hidden. Finish it.
    3. Solo Do a similar question with no help, then check against a full solution.
    4. Mixed Do the same type of question inside a mixed set, under time pressure.

    Once you reach stage 4 easily, stop studying examples for that topic and spend your time on questions.

    The Feynman technique

    The physicist Richard Feynman was famous for explaining hard ideas in simple words. This technique finds exactly where your understanding breaks.

    1. Choose one concept, for example “moles” or “opportunity cost”.
    2. Explain it out loud or on paper as if teaching a younger student. Use plain words and an example.
    3. Notice where you hesitate, use jargon you cannot explain, or say “it just is”. Those are your gaps.
    4. Go back to your notes or the chapter and fix those gaps only.
    5. Explain it again, more simply. Finish with the exact syllabus definition.
    TipIf English is your second language

    Explain the idea first in your own language to check you understand it. Then explain it again in English, using the precise syllabus terms. Examiners reward the correct technical words, so collect them in a personal word list with a short meaning next to each.

    Spaced-repetition flashcards

    Flashcards combine retrieval and spacing. A spaced-repetition app such as Anki (free on computers, Android and the web) shows each card just before you are likely to forget it: cards you know appear rarely, and cards you miss come back soon.

    • One fact per card. “State the units of the Planck constant” is a good card; “quantum physics” is not.
    • Make your own cards. Writing them is learning, and shared decks often do not match your syllabus.
    • Use cards for exact facts: definitions, equations, units, reagents and conditions. Use diagrams and questions for processes and problem-solving.
    • Review every day, and add only about 10 to 20 new cards a day so reviews do not pile up.

    On paper, use the Leitner system: review box 1 daily, box 2 every three days and box 3 weekly. A correct card moves up a box; a wrong card goes back to box 1.

    Focus: your phone is the biggest problem

    Every time you check a message, it takes minutes to regain full concentration, and a phone on the desk distracts you even when it is silent.

    • Put the phone in another room, or use a focus mode that blocks apps while you study.
    • Choose one specific task before you start: “Questions 1 to 6 on the discriminant”, not “do maths”.
    • Study at the same place and time when you can. Routine makes starting easier.

    The Pomodoro technique makes this easy: 25 minutes of full focus, then a 5-minute break; after four rounds, a longer break of 15 to 30 minutes. Try it now with the timer below: pick one task, press start, and do not touch your phone until it rings. As your focus grows, you can stretch the blocks to 40 or 50 minutes.

    Maths and science versus essay subjects

    The same principles apply everywhere, but what you practise is different.

    Subject type Main thing to practise Best retrieval tool Typical trap
    Mathematics Many questions, in mixed sets Questions with worked answers Reading solutions before trying
    Physics and Chemistry Definitions, equations, calculations, practical skills Flashcards; past-paper questions Learning calculations but not explanations
    Biology Precise terms, processes, data analysis Diagrams from memory; long answers Vague words such as “it affects”
    Business and Economics Definitions, diagrams, evaluation Timed paragraph plans; case studies Describing the case instead of analysing it

    For essay subjects, write full answers regularly and compare them with the mark scheme’s levels; see Writing top-band business answers.

    Studying with friends

    Group study works when everyone tests and explains, and fails when it turns into chatting. Keep groups to two to four people, agree a topic and a finishing time, and use the time to quiz each other, take turns teaching a topic (the Feynman technique with a real audience), mark each other’s answers with the official mark scheme, or compare essay plans. If you study alone as a private candidate, an online study partner gives the same benefits.

    Using AI tools responsibly

    AI chat tools can be excellent tutors or terrible ones. The rule: use AI to explain and to check, never to do the practice for you. Learning happens when your own brain retrieves and works things out. If a tool does that work, you get none of the benefit, and you still meet the exam alone.

    Good uses Bad uses
    “Explain the discriminant in a different way, with an example.” “Solve these ten homework questions for me.”
    “Here is my answer and the mark scheme. Which marks did I miss?” Copying an AI answer into your notes without trying first
    “Quiz me on Group 2 trends, one question at a time.” Trusting AI calculations or facts without checking
    “Translate this sentence from the question into simpler English.” Submitting AI-written work in coursework or applications
    Common mistakeAI can be confidently wrong

    AI tools sometimes give wrong facts, wrong numbers or answers for a different syllabus, in a confident tone. Check against your syllabus, a trusted textbook or the mark scheme. In coursework or non-exam assessment, presenting AI-generated work as your own is malpractice and can lead to disqualification.

    A sample weekly schedule

    Here is a realistic week for a student taking Mathematics, Physics and Chemistry, with about two hours of independent study on weekdays. Each block is two Pomodoros (about 50 minutes). Swap in your own subjects; the structure is what matters. For a plan built around your exam dates, use Build your personal study plan.

    Day Block 1 Block 2 Short review (15 min)
    Monday Maths: learn a new topic Physics: questions on last week’s topic Flashcards
    Tuesday Chemistry: learn a new topic Maths: questions on Monday’s topic Flashcards + blank-page recall of Monday’s maths
    Wednesday Physics: learn a new topic Chemistry: questions on Tuesday’s topic Flashcards
    Thursday Maths: mixed question set Physics: questions on Wednesday’s topic Flashcards + update error log
    Friday Chemistry: mixed question set Maths: redo last week’s wrong questions Flashcards
    Saturday One timed past-paper section (rotate the subject each week) Mark it with the mark scheme Add mistakes to error log
    Sunday Rest Plan next week (10 minutes) Optional light flashcards

    Each subject returns several times (spacing), new content is tested on a later day (retrieval), Thursday and Friday mix topics (interleaving), and Saturday is a test. Sunday’s rest protects your motivation; see Motivation, sleep and stress.

    Quick check
    Summary
    • Test yourself instead of rereading: retrieval practice and spaced practice were the two techniques rated high utility in the 2013 review.
    • Review new topics after gaps of about 1 day, 3 days, 1 week, 2 weeks and 1 month. Avoid cramming.
    • Learn new methods with blocked practice, then switch to mixed (interleaved) question sets.
    • Connect ideas by asking “why?”, collect concrete examples, and draw diagrams from memory.
    • Study worked examples first, then fade the support until you can solve questions alone.
    • Use flashcards with spaced repetition (such as Anki) for facts that must be exact.
    • Put your phone in another room and work in focused blocks such as 25-minute Pomodoros.
    • Use AI to explain and check, never to do your practice or your coursework.

    Next: Exam technique shows how to turn what you know into marks.

    PreviousPhase 6: Exam day, results and resitsNextExam technique
    Preparation Roadmap/Skills that raise grades

    Exam technique

    GuideExamsAll boards · all subjectsStudy time1 hour to read, then practise with every past paper

    Knowing the content is half the job; the other half is turning it into marks. This chapter covers reading questions, timing, working, accuracy and units, and how to answer every type of question, from multiple choice to essays and practical papers.

    Why this matters

    Examiners can only mark what is on the page. Every year, examiner reports repeat the same message: many candidates lose marks they could have earned by misreading the question, leaving out units, rounding too early or running out of time. These are skills, and skills can be practised. The habits in this chapter are worth a grade or more to many students.

    Before you start, it helps to know what each instruction word means: see Command words decoded. Practise every habit here with real papers, as described in Phase 4: Past papers and mark schemes.

    Read the question, then read it again

    Most lost marks start in the first ten seconds, when a student sees a familiar topic and starts writing the answer to a question they expected, not the one printed. Use the same routine for every question:

    1. Read the whole question, including any introduction, data or diagram. Information is there because you need it.
    2. Underline the command word (describe, explain, calculate, evaluate…) and the topic.
    3. Circle any conditions: “two reasons”, “exact”, “to 3 significant figures”, “with reference to the data”, “using your answer to (a)”.
    4. Look at the marks and plan that many separate points.
    5. Answer, then reread the question to check you answered what was asked.

    Marks tell you how much to write and how long to spend

    As a rough guide, one mark needs one correct, separate point. A 3-mark “explain” question needs three linked steps of reasoning, not one long sentence. The number of answer lines is a second clue: if you have filled two lines for a 6-mark question, you have probably missed something.

    Marks also set your time budget. Divide the time by the total marks:

    Cambridge paper Time Marks Time per mark
    Mathematics 9709 Paper 1 or 3 1 h 50 min 75 about 1.5 min
    Mathematics 9709 Paper 4, 5 or 6 1 h 15 min 50 1.5 min
    Sciences Paper 1 (multiple choice) 1 h 15 min 40 questions about 1 min 50 s per question
    Sciences Paper 2 (AS structured) 1 h 15 min 60 1.25 min
    Sciences Paper 4 (A Level structured) 2 h 100 1.2 min
    Business 9609 Paper 2 1 h 30 min 60 1.5 min
    Economics 9708 Paper 2 2 h 60 2 min

    So an 8-mark maths question deserves about 12 minutes. If you have spent 20, move on and come back later.

    TipWrite checkpoints on the front page

    At the start, write two or three checkpoint times, for example “halfway marks by 10:00”. A quick glance at the clock then tells you whether you are on pace, without constant calculation.

    Show your working

    In mathematics and in science calculations, many marks are for method. In mark schemes, M marks are for a correct method, A marks are for accuracy and depend on the method mark, and B marks are independent. If your final answer is wrong but your method is visible, you can still earn many of the marks. A bare wrong answer earns nothing.

    Examiners also use error carried forward (often called “follow through”): if you make a mistake in part (a) and then use your wrong answer correctly in part (b), you can usually still earn the part (b) marks. That only works if they can see what you did.

    Use the same pattern every time: formula, substitution, answer with unit.

    Ek=12mv2=12×0.50×4.02=4.0 J
    Exam technique“Show that” questions

    When the answer is printed in the question, the marks are for the route, not the result. Every step must be visible, with no jumps. Then use the given result in the next part, even if you could not prove it: the later marks are still available.

    Significant figures, units and rounding

    • Do not round in the middle of a calculation. Keep the full value on your calculator (use the memory or ANS key) and round only the final answer.
    • Mathematics: give answers to 3 significant figures unless told otherwise, and angles in degrees to 1 decimal place. If the question says exact, leave answers as surds, fractions or in terms of π or ln⁡.
    • Sciences: give the final answer to the same number of significant figures as the least precise data, usually 2 or 3. Never fewer.
    • Units: write a unit on every final answer and check the prefix. A missing or wrong unit often costs a mark on its own.
    Common mistakeEarly rounding changes the answer

    A ball of mass 0.65 kg falls 1.8 m from rest. Take g=9.81 m s−2. Its speed is v=2×9.81×1.8=5.9427… m s−1. If you round v to 5.9 and then calculate 12mv2, you get 11.3 J. Keeping the full value gives 11.5 J, which matches mgh=0.65×9.81×1.8. One early rounding lost the accuracy mark.

    Multiple choice

    Cambridge science Paper 1 has 40 questions in 1 hour 15 minutes: about 1 minute 50 seconds each. Economics Papers 1 and 3 are also multiple choice. Edexcel IAL and the UK boards mix multiple-choice questions into some written papers instead.

    • Answer before you look. Cover the options, work out your answer, then find it. This protects you from attractive wrong options.
    • Eliminate. Cross out options you know are wrong. With two options left, even a guess has a 50% chance.
    • Watch for NOT, FALSE and INCORRECT, usually in bold. Underline them.
    • Check powers of ten and units. Wrong options are often your answer with a factor of 10, 1000 or 2 missing.
    • Never leave a blank. There is no penalty for a wrong answer, so guess any question you cannot do.
    • Transfer carefully. If the paper uses a separate answer sheet, check every answer is on the correct line.

    Structured questions

    Structured questions are split into parts: (a), (b)(i), (b)(ii). They usually build on each other, so an answer to (a) is often a hint for (b). Keep answers precise: use the technical term, give one idea per mark, and do not copy out the question. “With reference to the data” or “use the information” means quote figures from the question, with units, in your answer.

    Extended writing

    For long answers in any subject, build each paragraph with Point, Evidence, Explain: make one clear point, support it with evidence or data, and explain why it matters to the question.

    In Business and Economics, the mark scheme rewards four skills in order: knowledge (correct terms and theory), application (using the business or economy in the case), analysis (chains of reasoning: “this means that… so…”) and evaluation (a supported judgement). The top levels require evaluation, so a long, accurate answer with no judgement cannot reach full marks. See Writing top-band business answers.

    Data, graphs and diagrams

    When you describe data, give the overall trend, quote key values with units, describe any change in the rate of change, and point out anomalies. “Sales rose from 1200 to 1800 units, an increase of 50%” earns more than “sales went up”.

    When you draw a graph:

    • label both axes with quantity and unit, for example “time / s”;
    • choose a scale so that the points cover at least half the grid in each direction;
    • plot points with small, precise crosses;
    • draw a single best-fit line or smooth curve, never dot-to-dot;
    • find a gradient using a large triangle, and show the values you read.

    In Business and Economics diagrams, label the axes (price, quantity), every curve (D₁, D₂, S), the equilibrium points, and show shifts with arrows. Then refer to the diagram in your writing.

    Practical and planning papers (Cambridge Papers 3 and 5)

    Cambridge Physics, Chemistry and Biology each have two practical-skills papers.

    Paper 3: Advanced Practical Skills (AS) Paper 5: Planning, Analysis and Evaluation (A Level)
    Format 2 h, 40 marks, hands-on in a laboratory 1 h 15 min, 30 marks, written
    What you do Follow instructions, take measurements, record, graph, calculate Plan an experiment, analyse given data, evaluate conclusions
    Key marks Tables with “quantity / unit” headings; readings to the precision of the instrument; repeats; good graphs; specific improvements Clear variables (independent, dependent, controlled); a workable method; safety; how to analyse (what to plot); uncertainties and error bars

    In Paper 3, improvements must be specific: “use a video camera with a timer to measure the oscillation time” earns marks, “be more careful” never does. Edexcel IAL tests practical skills in written papers instead (Units 3 and 6). The subject guides go further: Physics practical skills, Chemistry practical skills and Biology practical skills.

    Checking and getting unstuck

    Leave about 5 to 10 minutes at the end. Check that every question has an answer, every final answer has a unit and the right significant figures, and that each answer matches its command word. Ask “is this sensible?”: a car is not travelling at 3000 m s−1.

    When you are stuck:

    • Mark the question, move on, and come back with fresh eyes.
    • Write what you know: the relevant formula, a definition, a labelled diagram. These can earn marks on their own.
    • Use every piece of given information; if you have not used something, that is probably the key.
    • Write inside the answer space. If you need more room, use the extra page provided and label the question number clearly.

    Common ways marks are lost

    Mistake Fix
    Describing when asked to explain Underline the command word before writing
    Missing or wrong units Write the unit with every final answer
    Rounding too early Keep full values; round only at the end
    Bare answers with no working Formula, substitution, answer
    Vague words such as “it affects” State the direction and size of the change
    Ignoring the context in a case study Use the business’s name and its data
    One-sided evaluation Weigh both sides, then make a justified judgement
    Running out of time Use your time per mark and checkpoints

    Weak and strong answers compared

    A science “explain” question. Explain why the rate of a reaction increases when the temperature is increased. [3 marks]

    Weak answer (about 1 mark)

    The particles move faster, so they collide more and the reaction goes faster.

    This repeats the question (“goes faster”) and never mentions activation energy or successful collisions.

    Strong answer (3 marks)

    At a higher temperature the particles have more kinetic energy. [B1] So a greater proportion of collisions have energy equal to or greater than the activation energy. [B1] Therefore the frequency of successful collisions increases. [B1]

    A business “evaluate” question. Leyla’s Bakery made a net profit of $18,000 last year. Leyla needs $40,000 to open a second shop. Evaluate whether she should finance it with a bank loan. [12 marks]

    Weak paragraph

    A bank loan is good because you get money quickly and do not lose control. But you have to pay interest. Overall she should take the loan.

    Correct knowledge, but nothing about Leyla’s bakery, no chain of reasoning, and a judgement with no reason.

    Strong paragraph

    A loan lets Leyla keep full ownership and control (knowledge). Her $18,000 profit suggests she could afford repayments from current earnings (application). However, a new shop may take months to build sales, while repayments are fixed, so cash flow could turn negative and put both shops at risk (analysis). A loan is the better choice only if her forecast for the new shop is realistic and she keeps a cash reserve; if sales are uncertain, a partner sharing the risk may be safer (evaluation).

    The strong paragraph is not much longer. It earns more because every sentence does a job: it uses the case, builds a chain of reasoning and ends with a judgement that depends on clear conditions.

    Quick check
    Summary
    • Underline the command word and topic, circle conditions, and check the marks before you write.
    • One mark is roughly one separate point. Use time per mark (for example about 1.5 minutes per mark in Cambridge Maths Paper 1) and move on if a question overruns.
    • Show formula, substitution and answer. Method marks and follow-through marks reward visible working.
    • Round only at the end: 3 significant figures in maths unless told otherwise; match the data in science. Always give units.
    • In multiple choice, answer before looking at the options, eliminate, and never leave a blank.
    • For extended writing, build point, evidence, explain paragraphs; in Business and Economics show knowledge, application, analysis and a justified judgement.
    • In practical papers, use “quantity / unit” headings, good graph technique and specific improvements.

    Next: Command words decoded lists exactly what each instruction word asks for.

    PreviousStudy methods that actually workNextCommand words decoded
    Preparation Roadmap/Skills that raise grades

    Command words decoded

    ReferenceExamsCambridge command words · Edexcel, AQA and OCR notesStudy time30 minutes to read; keep it open while you practise

    Every exam question starts with an instruction word such as describe, explain or evaluate, and each one asks for a different kind of answer. This reference decodes the Cambridge command words and the special phrases used in maths papers.

    Why this matters

    The command word tells you what kind of answer earns the marks. A perfect description scores almost nothing if the question said “explain”, and a long, accurate essay cannot reach the top band if it was asked to “evaluate” and never makes a judgement. Learning these words is one of the fastest ways to raise your marks without learning any new content.

    Use this page alongside Exam technique. Cambridge prints the official list of command words in each syllabus; check the one for your subject, because not every word is used in every subject.

    Three families of command words

    It helps to sort command words by how much thinking they demand.

    Family Words What the answer looks like
    Recall define, state, give, identify, outline, summarise, describe Facts, features or trends, with no reasons needed
    Apply and reason calculate, determine, deduce, predict, show (that), sketch, demonstrate, compare, contrast, examine, analyse, explain, suggest Working, links and reasons; using knowledge in a new situation
    Judge assess, evaluate, discuss, justify, comment, consider, develop Arguments weighed against each other and a supported conclusion

    The Cambridge command words

    Command word What it means What examiners expect Example stem
    Analyse Break a topic into parts and show how they connect Chains of reasoning (cause, effect, consequence) using the data or case Analyse how a fall in interest rates could affect a car manufacturer.
    Assess Weigh something up and reach a reasoned judgement Points on both sides, then a judgement about importance or extent Assess the importance of cash flow forecasting for a new business.
    Calculate Work out a numerical answer Working shown; final answer with unit and sensible significant figures Calculate the resistance of the wire.
    Comment Give an informed opinion on the information A judgement supported by reasons or data, not just a description Comment on the reliability of these results.
    Compare Point out similarities and differences Both things in every point, with words such as “whereas” and “both” Compare the structures of DNA and RNA.
    Consider Think about the given information and respond to it A response that uses the information, often weighing factors Consider whether the firm should accept the new order.
    Contrast Point out differences only Paired differences: “X is…, whereas Y is…” Contrast the properties of simple molecular and giant covalent substances.
    Deduce Reach a conclusion from the information given The conclusion plus the step of logic from the data Use the data to deduce the order of reaction with respect to A.
    Define Give the exact meaning of a term Precise syllabus wording; an example alone is not a definition Define specific heat capacity.
    Demonstrate Show how something works, or give an example A clear illustration, worked case or example Demonstrate how a price cut could reduce total revenue.
    Describe Say what something is like or what happens Main features, steps or trends, with figures from data; no reasons needed Describe the trend shown in Fig. 2.1.
    Determine Find an answer using the information available A value obtained from data, a graph or a calculation, with method Use the graph to determine the gradient.
    Develop Take an idea further or build on given information Extra depth: the next step, consequence or detail Using the data, develop a plan to reduce staff turnover.
    Discuss Write about an issue in depth, in a structured way Arguments for and against, analysed, usually with a conclusion Discuss whether economic growth always raises living standards.
    Evaluate Judge the quality, importance or value of something Strengths and weaknesses weighed, “it depends on…”, a justified final judgement Evaluate the usefulness of ratio analysis to an investor.
    Examine Look closely and in detail at an issue Detailed investigation of causes, assumptions and effects Examine the causes of the fall in exports.
    Explain Give reasons (why) and mechanisms (how) Cause-and-effect links with “because”, “so”, “therefore” Explain why the current in the circuit decreases.
    Give Produce an answer from the source or from memory A short answer; no explanation Give two examples of fixed costs.
    Identify Name, pick out or recognise A word or phrase, often from a diagram or data Identify the structure labelled X.
    Justify Support a choice or conclusion with evidence Reasons and evidence for the answer you gave Justify your choice of indicator for this titration.
    Outline Set out the main points briefly Key points with little detail Outline the stages of the product life cycle.
    Predict Say what is likely to happen, from information or a pattern A specific outcome, sometimes with a brief reason Predict the effect on the position of equilibrium of increasing the pressure.
    Show (that) Give structured evidence that leads to a given result Every step visible, ending exactly at the given result Show that the speed of the ball is about 6 m s−1.
    Sketch Draw a simple diagram or graph of the key features Correct shape, labelled axes and key points; not to scale Sketch a graph to show how enzyme activity varies with pH.
    State Express clearly and briefly A fact, value or name; no explanation State the unit of electric charge.
    Suggest Apply knowledge to a new situation with several possible answers A sensible, reasoned proposal; you are not expected to have learned it Suggest why the two groups gave different results.
    Summarise Select and present the main points only A short overview without detail or examples Summarise the main findings shown in the table.
    Common mistake“Explain” is not “describe”

    Describing a graph when asked to explain it is the most common command-word error. A description says what happens (“the rate rises, then falls”). An explanation says why (“because the active site changes shape”). If your answer has no “because”, “so” or “therefore”, it is probably a description.

    Tip“Suggest” means you can think, not just recall

    “Suggest” questions put you in an unfamiliar situation on purpose. The mark scheme usually accepts several answers. Use your knowledge to propose something sensible, give a short reason, and move on. Do not leave it blank because “we did not learn this”.

    Phrases in mathematics papers

    Maths papers use a small set of phrases with exact meanings. Misreading them costs marks even when the mathematics is right.

    Phrase What it means
    Find, calculate, determine Work it out and show your method
    Write down, state The answer can be seen with little or no working; usually 1 mark
    Show that The answer is given: every step must be shown, ending exactly at the printed result
    Prove A complete general argument, every step justified, ending with a conclusion; examples are not proof
    Hence You must use the previous result; a different method may score nothing
    Hence or otherwise Using the previous result is the intended quick route, but any correct method earns full marks
    Exact No calculator decimals: leave surds, fractions, π, e or ln⁡ in the answer
    Sketch Correct shape with key features labelled (intercepts, turning points, asymptotes); not to scale
    Plot, draw accurately Calculate points and mark them precisely on a grid
    Verify Substitute the given values to confirm a result, then make a conclusion
    In the form… Match the form exactly, for example a(x+p)2+q
    In terms of k Your answer should contain k
    Worked exampleHence versus hence or otherwise3 marks

    (a) Show that x=2 is a root of x3−7x+6=0. (b) Hence solve x3−7x+6=0.

    (a) 23−7(2)+6=8−14+6=0, so x=2 is a root. B1

    (b) “Hence” means use part (a): (x−2) is a factor. Dividing gives x3−7x+6=(x−2)(x2+2x−3). M1 A1(x−2)(x−1)(x+3)=0⇒x=2,  1 or −3

    Writing the three roots straight from a calculator’s equation solver does not use part (a), so it would not earn the method mark. If the question had said “hence or otherwise”, a different valid method with full working would also be accepted.

    Board differences

    Most command words mean the same across boards, but each board publishes its own list and wording, so read the one for your specification.

    • Edexcel and AQA often write “To what extent…?” or “Assess…” in Business and Economics. Treat both as evaluation: the top marks need a supported judgement.
    • AQA and OCR also use “Name” (a word or phrase only) and “Complete” (fill in a table or diagram).
    • Edexcel Mathematics papers sometimes print “Solutions relying on calculator technology are not acceptable”: you must show algebraic working.
    • OCR Mathematics uses “In this question you must show detailed reasoning”: an answer without full working scores nothing.
    • AQA Mathematics uses “Fully justify your answer”: each step needs a reason.

    Describe, explain and evaluate on one topic

    The clearest way to see the difference is to answer three questions on the same data.

    Worked exampleOne set of data, three command words

    A student measured the volume of oxygen released per minute by the enzyme catalase at different temperatures.

    Temperature / °C 10 20 30 40 50 60
    Rate / cm³ min⁻¹ 3.1 5.8 9.9 12.4 6.2 0.8

    Describe the results. The rate increases from 3.1 cm3 min−1 at 10 °C to a maximum of 12.4 cm3 min−1 at 40 °C. Above 40 °C it falls steeply, to 0.8 cm3 min−1 at 60 °C. (Trend, key values, units; no reasons.)

    Explain the results. Up to 40 °C, enzyme and substrate molecules gain kinetic energy, so they collide more often and more enzyme–substrate complexes form per minute. Above the optimum, vibrations break hydrogen and ionic bonds in the enzyme, so its tertiary structure and the shape of the active site change. The substrate is no longer complementary to it, so fewer complexes form: the enzyme is denatured. (Reasons and mechanisms, linked with “so”.)

    Evaluate the conclusion that the optimum temperature is 40 °C. The data support an optimum near 40 °C, because that reading is the highest. However, readings were taken only every 10 °C, so the true optimum could lie anywhere between 30 °C and 50 °C, and with one reading at each temperature, an anomaly could not be spotted. Repeating each reading three times, at intervals of 2 °C between 30 °C and 50 °C, would make the conclusion more reliable. Overall, the conclusion is reasonable but only approximate. (Strengths, limitations, improvement, judgement.)

    Quick check
    Summary
    • The command word decides what kind of answer earns marks: recall, apply and reason, or judge.
    • Describe says what happens; explain says why and how; evaluate weighs evidence and reaches a judgement.
    • State, give, identify and write down need short answers with no explanation.
    • Assess, evaluate, discuss and “to what extent” need both sides and a justified conclusion.
    • Suggest means apply your knowledge to something unfamiliar: several answers can score.
    • In maths, “hence” forces the previous result, “hence or otherwise” allows any method, “exact” means no decimals, and “show that” needs every step.
    • Sketch means shape and key features; plot means accurate points on a grid.
    • Each board publishes its own list: check the one in your syllabus or specification.

    Next: Motivation, sleep and stress looks after the person who has to do all this work.

    PreviousExam techniqueNextMotivation, sleep and stress
    Preparation Roadmap/Skills that raise grades

    Motivation, sleep and stress

    GuideExamsAll boards · all subjectsStudy time30 minutes to read

    Two years of A Level study is a marathon, not a sprint. This chapter shows how to build habits that keep you going, beat procrastination, sleep and move for better memory, handle exam nerves and burnout, and where to get help.

    Why this matters

    Your brain is the tool you study with. Looking after it is part of studying, not a break from it. Students who sleep well, move every day and take real rest usually learn more in fewer hours than students who study until they are exhausted. None of this is about being “strong”. It is about working with your body instead of against it.

    This chapter supports every phase of the road, from Phase 2: Learn the content to Phase 5: Final revision. Pair it with Study methods that actually work.

    Motivation follows action

    Many students wait to feel motivated before they start. But motivation is unreliable: it rises and falls with mood, sleep and the weather. The good news is that it usually follows action. Start a small task, make some progress, and the motivation to continue arrives.

    So do not rely on motivation. Rely on habits and process goals.

    Process goal

    A goal about an action you control, such as “do ten mixed maths questions every weekday”, rather than about a result you cannot fully control, such as “get an A*” (an outcome goal).

    Outcome goal (the destination) Process goals (what you do each day)
    A* in Mathematics Ten mixed questions every weekday; one timed paper every Saturday
    A in Chemistry 15 minutes of flashcards daily; one past-paper question per topic learned
    A place to study medicine Read one article on health science each week; prepare for the admissions test 3 hours a week

    Outcome goals give direction. Process goals give you something to do today, and you can tick them off, which feels good and builds momentum.

    Building habits

    A habit is something you do with little thought because it is attached to a regular cue. To build a study habit:

    • Link it to a cue. Use a specific plan: “After dinner, at my desk, I will do 20 minutes of flashcards.” Plans in the form “when X, I will do Y” are much more likely to happen than “I will study more”.
    • Make it small. Start with a version so easy you cannot say no, even five minutes. You can increase it once the habit is fixed.
    • Make it visible. Mark each day you do it on a calendar. A growing chain of marks is motivating.
    • Never miss twice. Missing one day is normal. Missing two starts a new habit of not doing it.

    Beating procrastination

    Procrastination is not laziness. It is usually a way of avoiding a bad feeling: a task feels too big, too vague, too boring or too frightening. Fix the feeling and the task becomes easier to start.

    Why you are avoiding it What to do
    It feels too big (“revise organic chemistry”) Break it down; do only the first 25-minute piece
    It is vague Write the exact first action: “open the June paper, question 1”
    You fear doing badly Treat the first try as practice; mistakes show you what to fix
    Something more fun is nearby Phone in another room; block distracting sites
    You are tired Drink water and walk for 10 minutes, then start; if you are exhausted, rest properly and plan the morning
    TipThe five-minute rule

    Tell yourself you will work for just five minutes and then you are allowed to stop. Starting is the hardest part. Most of the time, once five minutes have passed, you will keep going.

    Sleep: when memories are saved

    Sleep is not wasted study time. While you sleep, your brain consolidates what you learned during the day: it strengthens new memories and links them to old ones. Students who cut sleep to study more often remember less of what they studied.

    Teenagers need roughly 8 to 10 hours of sleep a night. Teenage body clocks also run later than adults’, so falling asleep early can be hard. The most useful habit is a regular wake-up time, kept within about an hour on weekends too.

    • Stop using screens 30 to 60 minutes before bed, and charge your phone outside the bedroom.
    • Avoid caffeine (coffee, strong tea, energy drinks) from the early afternoon: it stays in your body for many hours.
    • Keep your room cool, dark and quiet.
    • If you nap, keep it to about 20 minutes and not late in the day.
    Common mistakeAll-nighters do not work

    Staying up all night before an exam damages exactly what the exam tests: recall, reasoning and concentration. The night before an exam, a full night’s sleep usually does more for your marks than a few extra hours of revision.

    Exercise and food

    The World Health Organization recommends that young people aged 5 to 17 do an average of 60 minutes a day of moderate to vigorous physical activity. Exercise improves mood, sleep and concentration. You do not need a gym: a brisk walk, a game of football or a bike ride all count. A 10 to 20 minute walk is also one of the best breaks between study blocks.

    There are no magic “brain foods”. What helps is ordinary good eating: regular meals including breakfast, plenty of water, fruit and vegetables, and not living on sugar and energy drinks, which give a short boost followed by a dip. On exam days, eat a familiar breakfast and take water into the exam if your centre allows it.

    Exam nerves: making them work for you

    Some nervousness before an exam is normal and useful: it makes you alert. It becomes a problem when it blocks your thinking. These techniques are simple and backed by evidence.

    • Slow breathing. Breathe in for a count of 4 and out for a count of 6 to 8. A longer out-breath calms your heart rate. Repeat five times.
    • Rename the feeling. A racing heart feels the same whether you are anxious or excited. Telling yourself “this is energy that will help me focus” can improve performance.
    • Write the worry down. In a well-known 2011 study, anxious students who spent ten minutes writing about their worries just before an exam scored better than those who did not. Write the worry, then put it away.
    • Practise under real conditions. Timed past papers at a desk make the real exam feel familiar, and familiar situations are less frightening.
    1. If your mind goes blank in the exam, put your pen down.
    2. Take three slow breaths, with a long out-breath.
    3. Move to a question you know you can do, and do it.
    4. Come back to the hard question and write anything relevant: a formula, a definition, a diagram.

    Perfectionism

    High standards are good. Perfectionism is different: it means feeling that anything less than perfect is failure. Warning signs include spending hours making notes beautiful, not starting until conditions are perfect, avoiding past papers in case you score badly, and feeling worthless after one poor mark.

    Remember that top grades never require perfection. Grade thresholds are set below full marks, so you can drop marks on every paper and still reach an A*. Early past-paper scores are supposed to be low: they show you what to fix. Aim for progress, keep notes “good enough”, and talk to yourself the way you would talk to a friend who made the same mistake.

    Burnout: know the signs

    Burnout is exhaustion that comes from too much pressure for too long. Watch for:

    • tiredness that a night’s sleep does not fix;
    • dreading study, or feeling “what is the point?”;
    • working longer hours but achieving less;
    • irritability, headaches, stomach aches or getting ill often.

    If you notice these, the answer is not to push harder. Reduce your workload for a week, protect your sleep, take a full day off, and talk to someone. Then adjust your plan to be sustainable.

    Think of it like this

    Athletes do not get fitter during training; they get fitter while they recover from it. Training without rest leads to injury. Study is the same: the learning settles during rest and sleep.

    Rest is part of the plan

    Schedule rest the way you schedule study. Take at least one full day, or two half-days, off each week. Make breaks real: a walk, a meal with family, sport or a hobby, not an hour of scrolling. Keep seeing friends. During school holidays, lighten the load for a few days before building up again.

    When to get help

    Everyone has bad days. But talk to someone if worry, low mood or poor sleep lasts more than about two weeks, if you have panic attacks, if you stop eating properly, or if stress stops you doing normal things.

    • A trusted adult: a parent, relative or teacher you feel comfortable with.
    • Your school counsellor or guidance teacher, if you are at a school. Private candidates can ask their exam centre or tutor who to contact.
    • A doctor: your family doctor can help with sleep problems, anxiety and low mood, and can refer you to specialist support.

    If your exam performance is affected by illness or a serious event, your exam centre can apply to the board for special consideration. If you have a diagnosed condition, access arrangements such as extra time must be applied for in advance through your centre. See Phase 6: Exam day, results and resits.

    Common mistakeIf you are in danger

    If you ever think about harming yourself, or feel unsafe, tell someone straight away. Call emergency services (112 in Turkey and across Europe; 999 or 112 in the UK) or go to the nearest hospital emergency department. In the UK and Ireland you can also call Samaritans free, at any time, on 116 123.

    Summary
    • Motivation follows action: start small and rely on habits and process goals, not feelings.
    • Attach study to a cue (“when X, I will do Y”), make it small, track it, and never miss twice.
    • Beat procrastination by making the first step tiny and specific, and use the five-minute rule.
    • Teenagers need roughly 8 to 10 hours of sleep. Sleep is when memories are consolidated, so never sacrifice it before an exam.
    • Aim for about 60 minutes of activity a day, eat regular meals and drink water.
    • Calm exam nerves with slow breathing, renaming the feeling, writing worries down and practising under real conditions.
    • Perfection is not required for an A*. Watch for burnout signs and schedule real rest every week.
    • Talk to a trusted adult, counsellor or doctor if problems last; in an emergency, call 112 (Turkey and Europe) or 999 (UK).

    Next: The best resources by subject shows where to find good material to study from.

    PreviousCommand words decodedNextThe best resources by subject
    Preparation Roadmap/Skills that raise grades

    The best resources by subject

    ReferenceExamsAll boards · Mathematics, sciences, Business and EconomicsStudy time20 minutes to read; come back when you need material

    Where to find trustworthy study material: the official documents every candidate needs, then well-known free and paid websites, videos and books for each subject, and how to judge whether a resource is any good.

    Why this matters

    There is more A Level material online than anyone could use, and much of it is for the wrong board, an old syllabus, or simply wrong. The best students do not collect resources; they use a few good ones deeply. Start with the official documents, which define exactly what you will be asked, then add one or two trusted resources per subject.

    This chapter supports Phase 2: Learn the content and Phase 4: Past papers and mark schemes. Every resource named here was well established at the time of writing, but websites change, so check that a link still works and still covers your syllabus.

    Official resources come first

    These documents come from the exam board itself. Nothing else is as reliable, and most of them are free.

    Resource What it is How to use it
    Syllabus (Cambridge) or specification (UK boards) The official list of everything that can be examined, plus the paper structure Print the content list and use it as a checklist; tick topics as you master them
    Specimen papers Sample papers written for a new syllabus Your first look at the format; good early practice
    Past papers Real papers from earlier exam series The core of your practice from Phase 4 onwards
    Mark schemes Exactly what earned marks in each past paper Mark your own work strictly; learn the wording examiners accept
    Examiner reports Comments by the senior examiners after each series Lists the common mistakes and what the best answers did
    Grade thresholds The raw marks needed for each grade in each series Turn your past-paper scores into grades and set targets
    Formula lists and data booklets The formulas and data you are given in the exam, such as MF19 for Cambridge 9709 Practise with the same booklet so you know what is given and what you must learn

    Where to find them. Cambridge publishes syllabuses, specimen papers and some past papers on each syllabus page of its website; the full archive is on its School Support Hub for registered schools. Pearson (for Edexcel IAL), AQA and OCR publish past papers, mark schemes and examiner reports on their websites, although the newest series is sometimes released to teachers only for a period. If you study privately, ask your exam centre what it can provide.

    Common mistakeCheck the syllabus code and the years

    Each syllabus has a code (for example 9709 for Cambridge Mathematics or 9702 for Physics) and is valid for particular exam years. Old past papers may test topics that have since been removed, and may not include topics that are new. Use papers from the current syllabus first, and check older papers against the content list.

    Mathematics

    Resource Best for Cost
    Physics & Maths Tutor Past papers sorted by board, topic questions and revision notes Free
    ExamSolutions Video lessons by topic and worked solutions to past-paper questions Mostly free
    TLMaths Short videos covering the whole A Level Maths and Further Maths course, in order Free
    Dr Frost Maths Online exam-style questions with instant marking, plus teaching slides Free
    MadAsMaths Large practice booklets, including hard questions for A* and beyond Free
    Underground Mathematics Rich problems from the University of Cambridge that build deep understanding Free
    Khan Academy Filling gaps in GCSE-level algebra, trigonometry and early calculus Free

    Most of these follow UK specifications, but the Pure Mathematics, Statistics and Mechanics content overlaps heavily with Cambridge 9709. Check the topic against your syllabus before you spend time on it.

    Every Maths chapter in this guide ends with a short “Learn more online” list picked for that topic. Maths: learn more online gathers all of them on one page, grouped by topic and by kind.

    Physics and Chemistry

    Resource Best for Cost
    Isaac Science (formerly Isaac Physics) Graded problem-solving practice in physics, plus chemistry and maths skills Free
    Chemguide Clear, detailed explanations of A Level chemistry, especially mechanisms and the reasons behind trends Free
    Physics & Maths Tutor Notes, topic questions and past papers for both sciences Free
    Save My Exams Revision notes and topic questions organised by board, including Cambridge Free and paid
    Seneca Short, interactive revision courses that quiz you as you go Free and paid
    Khan Academy Foundations: mechanics, electricity, atoms and basic chemistry Free

    For the practical papers, use your subject’s practical chapter here and the examiner reports, which describe exactly how marks were lost in tables, graphs and evaluations.

    Physics: learn more online and Chemistry: learn more online gather the links from the chapters in this guide, one page per subject, grouped by topic and by kind.

    Biology

    Resource Best for Cost
    BioNinja Clear diagrams and concise summaries (written for IB Biology, but most topics overlap) Free
    Save My Exams Notes and questions organised by board and topic Free and paid
    Physics & Maths Tutor Notes, flashcards and topic questions for Biology too Free
    Seneca Interactive revision for many A Level specifications Free and paid
    Cognito Short videos and quizzes, mainly at GCSE level; good for fixing foundations Mostly free

    Every Biology topic chapter in this guide ends with a short “Learn more online” list picked for that topic. Biology: learn more online gathers all of them on one page, grouped by topic and by kind.

    Business and Economics

    Resource Best for Cost
    tutor2u Topic notes, videos and up-to-date examples for Business and Economics Free and paid
    Save My Exams Revision notes organised by board and topic Free and paid
    Khan Academy Microeconomics and macroeconomics foundations Free
    Quality news sources Real examples for application and evaluation: business pages of major news organisations Free and paid

    In these subjects, examples from real businesses and economies lift answers into the higher levels. Keep a notebook of examples, including local ones: a Turkish exporter affected by exchange rates is just as good an example as a British one.

    For every subject

    Resource Best for Cost
    Anki Spaced-repetition flashcards (see Study methods that actually work) Free (except the iPhone app)
    This guide Chapters, quizzes, the glossary and formula sheets such as the Physics data sheet Free

    Textbooks

    One good textbook per subject is enough. On each syllabus page, Cambridge lists endorsed textbooks, which have been reviewed against that syllabus. Publishers of endorsed books include Cambridge University Press & Assessment, Hodder Education, Collins and Oxford University Press. For Edexcel IAL, Pearson publishes its own student books for each unit, and the UK boards list approved books for their specifications.

    Before you buy, check that the book names your syllabus code and exam years. A cheap second-hand copy of an older edition may miss content that was added later, so compare its contents with the current syllabus.

    How to judge a resource

    Before you trust a website, video or book, ask:

    • Is it for my board, my syllabus code and the current years?
    • Who made it? An exam board, an experienced teacher or a university is more reliable than an anonymous upload.
    • Does it explain why, not just what? Good resources show full working and reasoning.
    • Does it match the mark scheme? Compare one of its answers with an official mark scheme.
    • Does it make me active? Questions and quizzes beat passive watching.
    TipA small, strong toolkit

    For each subject, aim for: the syllabus, one textbook, past papers with mark schemes, one question website, and your own flashcards. That is enough for an A*. More resources usually means less depth.

    Avoid leaked and pirated material

    Sometimes social media accounts claim to sell “leaked” papers before an exam. These are almost always scams. If a real paper were involved, using or sharing it would be malpractice, and boards can disqualify candidates from a subject or from all their exams. Never buy, download or share them. Equally, do not post or discuss questions online after you sit a paper: candidates in other time zones may not have taken it yet.

    Pirated copies of textbooks and paid courses break copyright law, often carry malware, and are frequently old editions. Use the many free and legal resources above, your school or public library, and the official board websites instead.

    Quick check
    Summary
    • Start with official resources: syllabus, specimen papers, past papers, mark schemes, examiner reports and grade thresholds.
    • Always check the syllabus code and exam years; old papers may not match the current content.
    • Mathematics: Physics & Maths Tutor, ExamSolutions, TLMaths, Dr Frost Maths, MadAsMaths, Underground Mathematics and Khan Academy.
    • Sciences: Isaac Science, Chemguide, BioNinja, Save My Exams, Seneca, Cognito and Physics & Maths Tutor.
    • Business and Economics: tutor2u, Save My Exams and real-world news examples.
    • Use one endorsed textbook per subject and check it matches your syllabus.
    • Judge every resource by board, author, depth, agreement with mark schemes and whether it makes you active.
    • Never use leaked or pirated material: it is malpractice or illegal, and usually a scam.

    Next: Studying as a private candidate explains how to sit A Levels without a school.

    PreviousMotivation, sleep and stressNextStudying as a private candidate
    Preparation Roadmap/Beyond the exams

    Studying as a private candidate

    GuideExamsCambridge International · Edexcel IAL · UK boardsStudy time40 minutes to read, plus a few hours of planning

    How to take A Levels without a school: choosing a board that works for private candidates (especially in the sciences), finding an exam centre, entry deadlines, predicted grades and references, a self-study plan, and exam-day logistics.

    Why this matters

    Many students take A Levels privately: home-educated students, students whose school does not offer A Levels or a particular subject, students resitting, and adults returning to study. It is entirely possible, and universities treat the results exactly like any others. But without a school, all the organisation is your job. The most common disaster is not a bad exam. It is a missed entry deadline, or a subject that no centre nearby can run.

    What being a private candidate means

    Private candidate

    A candidate who takes exams at an approved exam centre without being a student of that centre, and who is responsible for their own preparation and exam entries.

    Exam centre

    A school, college or organisation approved by an exam board to enter candidates for its exams and to run those exams under the board’s rules.

    You prepare on your own, or with tutors or an online school. You then register with a centre that accepts private candidates. The centre enters you for the exams, supervises them, and passes on your results. You pay the centre’s fees.

    Step 1: choose your board

    For Mathematics, Business and Economics, every major board assesses you by written exams only, so choose by syllabus, resources and exam dates. Subjects with practical work or coursework need more care, because a centre must supervise that part. The sciences are the main example, because of how practical skills are assessed.

    Cambridge International A Level Pearson Edexcel International A Level (IAL) UK A Levels (AQA, OCR, Edexcel)
    Exam series May/June and October/November January, May/June and October/November Usually May/June only
    Science practical Paper 3 is a hands-on laboratory exam; Paper 5 is written Units 3 and 6 are written papers testing practical skills Written papers, plus a separate practical endorsement assessed by teachers over the course
    What a private candidate needs A centre with a laboratory that will run Paper 3 for private candidates Any centre that runs IAL exams A centre willing to supervise the required practical activities
    Structure AS can be taken first and carried forward to the full A Level Modular: units can be sat in different series and resat one at a time Linear: all exams at the end

    Because the Cambridge science Paper 3 is a practical exam, you must find a centre that has a suitable laboratory and agrees to run it for private candidates, and many centres do not. Edexcel IAL tests practical skills in written papers (Units 3 and 6), so no laboratory session at the centre is needed. This is one reason many private candidates choose Edexcel IAL for the sciences. It is not a shortcut: the written practical papers test planning, measurement, data analysis and the core practicals in detail, so you still need to understand experiments well, ideally by doing some.

    For UK A Levels in the sciences, the practical endorsement is reported separately from the grade. Private candidates often find it hard to arrange, and some university science and medicine courses expect a pass.

    Common mistakeDo not mix boards within one subject

    An AS result from one board cannot be combined with A Level papers from another board. Choose your board for each subject before you start studying, because syllabuses, textbooks and paper styles differ. You can take different subjects with different boards.

    Before you commit, check that the universities and courses you are interested in accept your combination, especially for medicine, where some universities have rules about resits and modular exams.

    Step 2: find a centre

    • In Turkey, the British Council runs school exams for private candidates in Istanbul and Ankara. Some schools registered with Cambridge or Pearson also accept private candidates.
    • Anywhere, use the centre search tools on the Cambridge International and Pearson websites, contact your local British Council office, and ask international schools near you directly.

    Ask every centre the same questions, and keep the answers in writing:

    • Do you accept private candidates for my board, subjects and series?
    • (Cambridge sciences) Can you run the practical Paper 3 for private candidates?
    • What is your entry deadline for the series I want, and is there a late-entry option?
    • What are your fees, per subject or component, and are there administration charges?
    • What identification do you need, and where exactly will the exams take place?
    • Can you arrange access arrangements, such as extra time, if I am eligible?
    • How and when will I receive my results and certificates?

    Step 3: entry deadlines and fees

    Centres set their own entry deadlines, and they are usually much earlier than the board’s own deadline, often months before the exams. Late entries, where accepted at all, cost considerably more. Put the deadline in your calendar the day you learn it, with a reminder a month before.

    Fees are set by each centre. They usually combine the board’s entry fee for each subject or component with the centre’s own charge, so they vary a lot between centres and countries. Ask for a full written breakdown before you enter.

    When you enter, check the exact entry options: for example AS only or full A Level, and the right components. With Cambridge, an AS result can be carried forward to the full A Level within 13 months, so you can sit AS in one year and complete the A Level the next.

    Predicted grades and references without a school

    When you apply through UCAS, universities expect a reference and predicted grades. Without a school, you still need someone who knows your academic work.

    • Referee: a tutor, a former teacher, or the head of an online school who has taught you. Family members and friends must not write it.
    • Predicted grades: these must come from someone who has seen and marked your work. Sit timed past papers regularly and have a tutor mark them, and keep a dated record of scores. That record is the evidence behind a credible prediction.
    • Apply after your results: many private candidates apply once their grades are known, often taking a gap year. Your offers are then based on real results, which removes the predicted-grade problem completely.

    See University applications and admissions tests for the full process.

    A self-study plan

    Cambridge designs each A Level around roughly 360 guided learning hours per subject (about 180 for AS). On your own, plan for at least that much, plus revision. Over two years, that means a steady weekly routine, not bursts.

    Stage Roughly when What to do
    Set up First month Choose subjects and boards, download syllabuses, buy textbooks, contact centres
    AS content Months 2 to 9 Learn the AS topics in order, with questions after every chapter
    AS consolidation Months 10 to 12 Timed AS past papers; sit AS exams now if you are taking them separately
    A Level content Months 13 to 19 Learn the A Level topics and keep revisiting AS topics with mixed questions
    Exam preparation Months 20 to 24 Full timed papers, error logs and final revision

    To turn this into a week-by-week plan around your exam dates, use Build your personal study plan and the methods in Study methods that actually work.

    Tutors, online schools and study partners

    Tutors are most valuable for the things you cannot do alone: marking long answers against the mark scheme, explaining a topic you are stuck on, setting mock exams, and writing a reference. Choose one with experience of your specific board, ask for a trial lesson, and avoid paying large sums in advance.

    Online schools can provide a full course, regular feedback and a reference. Check what they actually include: most do not enter you for exams, so you still need a centre.

    Study partners keep you accountable. An online partner taking the same subject can quiz you, compare answers and share the motivation.

    Exam-day logistics

    • Identification: private candidates normally have to show valid photo identification, such as a passport or national identity card, at every exam. The name must match your entry exactly.
    • Statement of entry: when the centre sends your list of entries and timetable, check every subject, component, date and the spelling of your name. Report any error immediately.
    • Venue and travel: the exam may be held at a different address from the centre’s office. Visit or plan the route in advance and aim to arrive at least 30 minutes early.
    • Equipment: black pens, pencil, ruler, eraser and an approved calculator in a clear pencil case. Check the board’s calculator rules; calculators with computer algebra are not allowed.
    • Results and certificates: you receive results through your centre. Certificates arrive later; collect them and keep them safe, because universities and employers may ask for them.

    Your checklist

    1. Choose your subjects and check that they fit the university courses you are considering.
    2. Choose a board for each subject, thinking carefully about the practical papers in the sciences.
    3. Download each syllabus and specimen paper, and get one endorsed textbook per subject.
    4. Find a centre that accepts private candidates for your subjects (and runs Paper 3 if you choose Cambridge sciences).
    5. Write down the centre’s entry deadline and fees, and set reminders.
    6. Build a weekly study plan, and find a tutor or teacher who can mark your work and act as referee.
    7. Enter before the deadline and check your statement of entry carefully.
    8. Sit timed mock exams and keep the marked papers as evidence for predicted grades.
    9. Prepare identification, travel and equipment for each exam day.
    10. Collect your results and certificates, and plan any resits early.
    Summary
    • A private candidate studies independently and sits exams at an approved centre that accepts private candidates.
    • Cambridge science Paper 3 is a hands-on laboratory exam, so you need a centre that can run it; Edexcel IAL Units 3 and 6 test practical skills in written papers, one reason many private candidates choose IAL for sciences.
    • Do not mix boards within one subject; choose before you start.
    • Centres set their own entry deadlines, often months ahead, and their own fees. Get everything in writing.
    • For UCAS you need a referee who knows your work and evidence-based predicted grades; applying after results avoids predictions.
    • Plan for about 360 guided learning hours per subject over the course, plus revision.
    • Bring valid photo identification to every exam and check your statement of entry carefully.

    Next: University applications and admissions tests takes you through UCAS and the tests for competitive courses.

    PreviousThe best resources by subjectNextUniversity applications and admissions tests
    Preparation Roadmap/Beyond the exams

    University applications and admissions tests

    GuideExamsUCAS · 2027 entry · all boardsStudy time45 minutes to read; start planning a year before you apply

    How applying to UK universities works through UCAS: choices, the three-question personal statement, references, predicted grades and offers, the 2027-entry deadlines, the admissions tests (ESAT, TMUA, TARA, UCAT, LNAT and STEP), interviews, supercurricular activities, and applying outside the UK.

    Why this matters

    For competitive courses, excellent grades are necessary but not enough. Universities also weigh your predicted grades, personal statement, reference, and often an admissions test and an interview. The deadlines come early and do not move: you apply before you have sat a single final exam. Knowing the system a year ahead gives you time to prepare every part well.

    Before you start, make sure you understand grades and UCAS points: see Grades, marks and UCAS points. If you are applying without a school, read Studying as a private candidate first.

    How UCAS works

    UCAS is the central admissions service for undergraduate degrees in the UK. You make one online application and send it to up to five choices (a course at a university). There are two important limits:

    • For medicine, dentistry and veterinary medicine or science, you can use at most four of your five choices on these courses.
    • You can usually apply to Oxford or Cambridge, but not both, in the same year.
    Part of the application What it is
    Choices Up to five courses; every university sees the same application
    Qualifications Results you already have (for example IGCSE or AS) and those you are taking
    Personal statement Your answers to three questions about why and how you have prepared for the subject
    Reference Written by a teacher or tutor who knows your academic work
    Predicted grades Your referee’s professional estimate of your final grades

    If English is not your first language, most universities also ask for an English test such as IELTS Academic, unless they accept another qualification you hold. Check each university’s requirements early.

    Deadlines for 2027 entry

    Date Deadline
    15 October 2026, 6 pm UK time Oxford, Cambridge, and most courses in medicine, dentistry and veterinary medicine or science
    13 January 2027, 6 pm UK time Equal-consideration deadline for most other courses

    After 13 January you can still apply until the end of June, but universities do not have to consider late applications equally, and popular courses may be full. After that, places are filled through Clearing. Schools often set an internal deadline weeks earlier so that teachers have time to write references, so ask your referee early.

    The personal statement

    From 2026 entry onwards, the personal statement is no longer one free essay. You answer three questions:

    1. Why do you want to study this course or subject?
    2. How have your qualifications and studies helped you to prepare for this course or subject?
    3. What else have you done to prepare outside of education, and why are these experiences useful?

    The total limit is 4,000 characters including spaces, with at least 350 characters for each answer. You can share the characters between the questions however suits you. All your choices see the same statement, so write about the subject, not about one university.

    Question Strong answers include
    1. Why this subject? A specific idea, problem or experience that interests you, and what you did to follow it up
    2. Qualifications and studies Topics from your A Levels that you enjoyed or found challenging, and skills they gave you, linked to the degree
    3. Preparation outside education Reading, competitions, online courses, projects, work or volunteering, and what each taught you
    TipShow, then reflect

    “I am passionate about physics” tells the reader nothing. “After a lesson on the photoelectric effect I read about how solar cells are designed, and tried to estimate the efficiency of the panel on our roof” shows the interest. Then add one sentence on what you learned from it. Specific detail and reflection are what make a statement convincing.

    Common mistakeWrite it yourself

    UCAS checks statements for copied text. A statement written by an AI tool or by someone else is not your own work and can put your application at risk. Use feedback from teachers or tutors, but every sentence should be yours.

    References and predicted grades

    Your reference is written by a teacher or tutor who knows your academic work. It describes your abilities and potential and includes your predicted grades: your referee’s professional judgement of the grades you are likely to achieve. Universities make offers based on these predictions, so strong, consistent work over the year before you apply matters. Private candidates should read the advice on referees in Studying as a private candidate.

    Offers and replying

    Each university replies with one of these decisions:

    • Conditional offer: a place if you achieve certain results, such as “AAA including A in Mathematics”.
    • Unconditional offer: a place with no conditions, usually because you already have your results.
    • Unsuccessful: no offer.

    Some universities state offers in UCAS tariff points instead of grades. Use the calculator to see how your grades convert:

    When all your decisions have arrived, choose a firm choice (your first choice) and an insurance choice (a back-up with lower conditions), and decline the rest. If you meet your firm offer’s conditions, you are committed to it, so only choose a firm offer you really want. Your reply deadline depends on when your last decision arrives, and is shown in your UCAS account.

    If you have no offers, UCAS Extra lets you add further choices one at a time from late February, and Clearing fills remaining places in the summer, especially around results day.

    Admissions tests for 2027 entry

    Many competitive courses require a test. Several tests are run by UAT-UK and taken on computer at Pearson VUE test centres, which exist in many countries.

    Test Used for Format When (2027 entry)
    ESAT Engineering and science courses at Cambridge, Oxford (replacing the PAT), Imperial and UCL Mathematics 1 plus two modules chosen from Mathematics 2, Physics, Chemistry and Biology; each module 40 minutes, 27 multiple-choice questions 12–16 October 2026 or 4–8 January 2027
    TMUA Mathematics, computer science and economics courses at Cambridge, Oxford (replacing the MAT), Imperial, LSE, Warwick, Durham and UCL Two 75-minute papers, each with 20 multiple-choice questions; no calculator 12–16 October 2026 or 4–8 January 2027
    TARA Several Oxford courses (replacing the TSA) and some UCL courses Three 40-minute modules: critical thinking, problem solving and a writing task 12–16 October 2026 or 4–8 January 2027
    UCAT Most UK medical and dental schools, including Oxford and Cambridge (the BMAT no longer runs) Four timed sections (verbal reasoning, decision making, quantitative reasoning, situational judgement), about 2 hours Summer before you apply: 13 July to 24 September 2026 (now closed)
    LNAT Law at Oxford, Cambridge, UCL, KCL, LSE, Durham, Bristol, Glasgow, SOAS and others 42 multiple-choice questions on reading passages (95 minutes), then one essay (40 minutes) From 1 September 2026; by 15 October 2026 for Oxford and Cambridge; by 31 December 2026 for KCL, LSE and UCL; by 25 January 2027 for most others (13 January for Bristol and Durham)
    STEP Mathematics at Cambridge (part of almost all offers); some other universities, such as Warwick, may include it in offers STEP 2 and STEP 3, long problem-solving papers, run by OCR June 2027, after you receive an offer
    Common mistakeBook early, and know which sitting you need

    Most Oxford and Cambridge applicants must take ESAT, TMUA or TARA in the October sitting; January is mainly for other universities. Booking closes weeks before each sitting (the October 2026 window closed in late September), and booking for January opens in late October 2026. Each test can be taken only once per application year. There is a fee for each test, and bursaries exist for some students; check the official websites for current details.

    Prepare with each test’s official practice papers, timed and marked like A Level past papers.

    Interviews

    Oxford interviews the applicants it shortlists, and Cambridge interviews most of its applicants, usually in December; many interviews take place online. They are academic conversations: you will be given unfamiliar problems or texts and asked to think aloud. Interviewers want to see how you reason, not how much you have memorised. Medical schools use panel interviews or multiple mini interviews (MMIs), short stations testing communication and judgement, usually between November and March.

    To prepare, practise talking through hard problems out loud (Underground Mathematics and Isaac Science are excellent for this), reread your personal statement and be ready to discuss anything in it, and ask a teacher or tutor for a mock interview. For an online interview, test your camera, microphone and connection beforehand and have paper and a pen ready.

    Supercurricular activities and the EPQ

    Supercurricular activities take your subject beyond the syllabus, unlike extracurricular hobbies. Universities, especially the most selective, value them because they show genuine interest. Examples:

    • reading books, articles and popular science about your subject;
    • free online courses and recorded university lectures;
    • competitions such as the UKMT mathematics challenges, the British Physics Olympiad, or national science olympiads (in Turkey, run by TÜBİTAK);
    • essay competitions, summer schools and small research projects;
    • for medicine, work experience or volunteering in caring roles.

    What matters is reflection: what you learned and how it changed your thinking, not a long list.

    The Extended Project Qualification (EPQ), offered by UK boards, is an independent project, usually a dissertation of about 5,000 words or an artefact with a report. It is worth up to 28 UCAS points (for an A*), and some universities make lower offers to applicants with a good EPQ. It needs a supervising centre, so ask your school or exam centre whether it offers the EPQ or a similar project qualification.

    Applying outside the UK

    A Levels are recognised by universities around the world, but each country, and often each university, has its own rules. Always check the university’s international admissions page.

    • United States: you apply to each university directly, often through the Common App. Many universities require or recommend SAT or ACT scores, and policies have changed in recent years. Strong A Level results may earn credit for some first-year courses.
    • Europe: many universities accept A Levels, often with requirements on the number of A Levels and specific subjects. English-taught degrees are common in countries such as the Netherlands and Germany.
    • Turkey: your route depends on your citizenship and where you completed secondary school. Many Turkish universities accept A Level results in their international student admissions, often with minimum grades, while most Turkish citizens who finished school in Turkey enter through the national exam (YKS, run by ÖSYM). You may also need a diploma equivalence (denklik) from the Ministry of National Education. Check with the university and official sources before you rely on A Levels alone.

    Your 2026–27 timeline

    When What happens (2027 entry)
    Summer 2026 (passed) UCAT testing; LNAT booking for Oxford and Cambridge closed 15 September; UAT-UK October booking closed in late September
    October 2026 ESAT, TMUA and TARA: 12–16 October. UCAS and LNAT deadline for Oxford, Cambridge and medicine: 15 October. January UAT-UK booking opens late October
    November 2026 Interview invitations; UCAT results reach universities; prepare for interviews
    December 2026 Oxford and Cambridge interviews; LNAT deadline for KCL, LSE and UCL: 31 December; booking for the January UAT-UK sitting closes
    January 2027 ESAT, TMUA and TARA: 4–8 January. UCAS equal-consideration deadline: 13 January. Oxford and Cambridge decisions usually arrive by the end of the month. LNAT deadline for most universities: 25 January
    February–March 2027 Offers arrive; medical school interviews continue; UCAS Extra opens in late February
    April–May 2027 Reply to offers (firm and insurance) by your deadline; arrange finance, scholarships and visas; final revision; May/June exams begin
    June 2027 Final exams; STEP for Cambridge Mathematics offers; end of June: last date to apply before Clearing
    July–August 2027 Clearing; results in mid-August; places confirmed
    September 2027 University begins. If you are applying next year instead, UCAT testing for 2028 entry is in the summer
    Quick check
    Summary
    • UCAS allows up to five choices (at most four in medicine, dentistry or veterinary courses), and normally not both Oxford and Cambridge.
    • 2027 entry deadlines: 15 October 2026 for Oxford, Cambridge, medicine, dentistry and veterinary courses; 13 January 2027 for most others.
    • The personal statement has three questions, 4,000 characters in total and at least 350 per answer. Be specific, reflect, and write it yourself.
    • Your reference and predicted grades come from a teacher or tutor who knows your work.
    • Choose a firm and an insurance offer; use Extra and Clearing if needed.
    • Admissions tests: ESAT, TMUA and TARA (12–16 October 2026 or 4–8 January 2027; most Oxford and Cambridge applicants sit in October), UCAT (summer before applying), LNAT (deadlines by university) and STEP (June).
    • Interviews test thinking aloud; supercurricular activities and the EPQ show genuine interest.
    • Outside the UK, rules differ by country and university, so check each one; in Turkey, your route depends on citizenship and schooling.

    Back to the start of the road: The preparation road at a glance.

    PreviousStudying as a private candidateNextMathematics: the course map
    Computer Science/About this course

    About Computer Science

    Start hereStudy time10 minutes

    What computer science is, what computer scientists do, why it matters, what the Cambridge International AS & A Level Computer Science (9618) course covers, and who it suits. The full course in this guide is planned.

    Why this matters

    Computer science is the study of how information is represented and processed, and how to make machines solve problems for us. Every app, website, game, bank, hospital system and AI model rests on it. This page explains what the subject really is (it is much more than “using computers”) and whether it fits you.

    What is computer science?

    Computer science has two halves that support each other.

    • Theory: how computers store everything as binary, how processors and memory work, how data travels across networks, how databases organise information, and which problems can be solved efficiently at all.
    • Programming and problem solving: turning a problem into an algorithm (a precise list of steps), then into working code, then testing it until it is correct.

    The central skill is computational thinking: breaking a big problem into smaller ones (decomposition), ignoring details that do not matter (abstraction), and spotting patterns you have solved before.

    What do computer scientists do?

    • Software engineers design and build programs, from phone apps to the systems that run airports.
    • Data scientists and AI engineers build models that find patterns in data, recognise speech and images, or make predictions.
    • Security specialists protect systems from attack and design encryption.
    • Network and systems engineers keep the internet, cloud services and company networks running.
    • Researchers work on faster algorithms, new programming languages, quantum computing and the limits of computation.

    Why it matters

    Almost every field now depends on software: medicine, finance, science, transport, entertainment and government. Computer science also raises important questions about privacy, fairness in automated decisions, and the safe use of AI. Understanding how these systems work lets you build them, and also judge them.

    What the A Level covers

    The guide follows Cambridge International AS & A Level Computer Science (9618), the syllabus for exams in 2027–2029. The course in this guide is planned in these parts:

    • AS theory: information representation, communication and networks, hardware and logic gates, processor fundamentals, system software, security, ethics and databases.
    • AS programming: algorithm design, data types and structures, pseudocode, programming and testing.
    • A Level theory: data representation, communication protocols, hardware and virtual machines, encryption and artificial intelligence.
    • A Level programming: computational thinking, recursion, further data structures, object-oriented programming and other paradigms.
    • Exam readiness: paper guides and mock papers.

    Is it for you?

    Computer Science suits you if you enjoy logical puzzles, are patient when something does not work the first time, and like building things. You do not need to have programmed before, but you must be willing to practise regularly: programming is learnt by doing. Good Maths helps a lot, especially for algorithms and the theory papers.

    It combines well with Maths (and Further Maths), Physics, Economics or Business. Many computer science degrees ask for Maths rather than Computer Science itself, so check the requirements.

    The Computer Science course in this guide

    The full course is planned and will be built later in the build order. When it is ready, its card on the home page will open the first chapter instead of this page. Until then, the Mathematics course is complete and builds the logical and algebraic skills that computer science relies on.

    Engineering/About this course

    About Engineering

    Start hereStudy time10 minutes

    What engineering is, what engineers do, why it matters, how to prepare for it at A Level (there is no Cambridge Engineering A Level), and what this guide’s Engineering course will cover. The full course is planned.

    Why this matters

    Engineers use maths and science to design and build things that work: bridges, engines, phones, power stations, medical devices, water systems. If you want to make things rather than only study them, engineering may be for you. This page explains the field and the best A Level route into it.

    What is engineering?

    Engineering is applied science with a purpose. A scientist asks how the world works; an engineer uses that knowledge to solve a practical problem, within limits of cost, safety, materials and time. The main branches are:

    • Mechanical engineering: machines, engines, vehicles, robots.
    • Civil and structural engineering: buildings, bridges, roads, dams, water supply.
    • Electrical and electronic engineering: power grids, circuits, chips, communication systems.
    • Chemical engineering: turning raw materials into fuels, plastics, medicines and food at industrial scale.
    • Aerospace, biomedical, software and environmental engineering, among others.

    Every branch follows the same design process: define the problem, research, design and model, build a prototype, test, and improve.

    What do engineers do?

    Engineers calculate loads and stresses, simulate designs on computers, choose materials, manage projects and teams, test prototypes, and make sure products are safe and meet regulations. They work in industry, consultancy, construction, energy, transport, healthcare and research. Engineering degrees lead to some of the best-paid and most in-demand graduate jobs.

    Why it matters

    Clean energy, safe water, transport, housing, communication and healthcare all depend on engineering. Problems such as climate change will be solved partly by engineers designing better batteries, grids, buildings and materials.

    How to study it at A Level

    There is no Cambridge International Engineering A Level. Engineering degrees almost always ask for:

    • Mathematics (essential), and often Further Mathematics;
    • Physics (essential for most branches);
    • a third subject such as Chemistry (needed for chemical engineering), Computer Science, Design and Technology or Electronics.

    Other routes include OCR Design and Technology: Design Engineering, Eduqas Electronics, and vocational qualifications such as BTECs and T Levels.

    What this guide’s Engineering course will cover

    This guide’s Engineering course is not an exam course. It shows how A Level Maths and Physics become engineering, in these parts:

    • Foundations: units, estimation, the design process, and the maths engineers use most.
    • Mechanics and structures: forces, stress and strain, beams and simple structures.
    • Thermofluids: energy, heat engines and fluid flow.
    • Electrical and electronic: circuits, power and simple electronics.
    • Design and making: materials, manufacturing and testing.
    • Next steps: degrees, apprenticeships and how to prepare.

    Is it for you?

    Engineering suits you if you enjoy maths and physics and like seeing them used on real objects, solving open problems, and working in teams. The Mathematics and Physics courses in this guide are complete, and they are the best preparation you can do now.

    The Engineering course in this guide

    The full course is planned and will be built later in the build order. When it is ready, its card on the home page will open the first chapter instead of this page.

    Economics/About this course

    About Economics

    Start hereStudy time10 minutes

    What economics is, what economists do, why it matters, what the Cambridge International AS & A Level Economics (9708) course covers, and who it suits. The full course in this guide is planned.

    Why this matters

    Economics studies how people, firms and governments make choices when resources are limited. It explains why prices rise, why some countries grow richer than others, why unemployment happens and what governments can do about it. This page explains what you would study and whether it fits you.

    What is economics?

    Every society faces the economic problem: wants are unlimited, but resources (land, labour, capital and enterprise) are scarce, so choices must be made, and every choice has an opportunity cost. Economics studies those choices at two levels:

    • Microeconomics looks at individual markets: how supply and demand set prices, how firms decide what to produce, why some markets fail (pollution, monopolies) and how governments can step in.
    • Macroeconomics looks at the whole economy: growth, inflation, unemployment, the balance of payments, exchange rates, and the government and central bank policies that steer them.

    Economists build models, simplified pictures of how things work (usually as diagrams), and test them against data.

    What do economists do?

    • Government and central banks use economists to set interest rates, design taxes and benefits, and forecast the economy.
    • Banks, investment firms and consultancies hire economists to study markets and advise companies.
    • International organisations work on trade, development and poverty reduction.
    • Researchers study questions such as what reduces inequality or how to price carbon emissions.
    • An economics background is common in business, law, journalism, politics and public policy.

    Why it matters

    Economic decisions affect jobs, prices, housing, the environment and how much governments can spend on schools and hospitals. Understanding economics helps you follow the news, judge politicians’ promises, and make better decisions with your own money.

    What the A Level covers

    The guide follows Cambridge International AS & A Level Economics (9708), the syllabus for exams in 2026–2028. The course in this guide is planned in these parts:

    • AS microeconomics: the basic economic problem, the price system, demand and supply, elasticity and government intervention.
    • AS macroeconomics: national income, aggregate demand and supply, inflation, unemployment, international trade and government policy.
    • A Level microeconomics: consumer behaviour and behavioural economics, costs and revenue, market structures, the growth of firms, efficiency, labour markets, and inequality and redistribution.
    • A Level macroeconomics: the circular flow and the multiplier, money and banking, unemployment and inflation in depth, policy aims and conflicts, economic development and globalisation.
    • Exam readiness: diagram practice, essay and data-response skills, paper guides and mock papers.

    Is it for you?

    Economics suits you if you follow the news, like explaining why things happen, and enjoy a mix of diagrams, numbers and argument. You will write essays that build a chain of reasoning and then evaluate: weighing up when a policy works and when it does not. Comfortable numeracy helps (percentages, index numbers, reading graphs).

    It combines well with Maths (essential for many top economics degrees), Business, Accounting, History, Geography or Politics.

    The Economics course in this guide

    The full course is planned and will be built later in the build order. When it is ready, its card on the home page will open the first chapter instead of this page.

    Business/About this course

    About Business

    Start hereStudy time10 minutes

    What business studies is, what people in business do, why it matters, what the Cambridge International AS & A Level Business (9609) course covers, and who it suits. The full course in this guide is planned.

    Why this matters

    Business studies looks at how organisations are set up, how they make decisions, and how they succeed or fail. It covers the people, the marketing, the operations and the money behind every company, from a corner shop to a global brand. This page explains what you would study and whether it fits you.

    What is business studies?

    A business takes resources and turns them into goods or services that customers will pay for. Business studies follows that process through the main functions of an organisation:

    • People: organisational structure, leadership, motivation and recruitment.
    • Marketing: market research, pricing, promotion and how products reach customers.
    • Operations: producing efficiently, quality, inventory and location.
    • Finance and accounting: costs, profit, cash flow, budgets and investment decisions.
    • Strategy: where the business is going, and how it responds to competitors, the economy, technology and the law.

    The subject is applied: you are usually given a case study of a real or realistic business and asked to analyse its situation and recommend what it should do.

    What do people in business do?

    • Entrepreneurs start new businesses and take risks to grow them.
    • Managers lead teams, plan projects and solve problems in every kind of organisation.
    • Marketing, sales and brand specialists find customers and keep them.
    • Operations and supply-chain managers make sure products are made and delivered on time.
    • Analysts and consultants study data and advise companies on strategy.
    • Charities, hospitals, schools and governments need the same skills as companies.

    Why it matters

    Businesses create most jobs, products and services, and their decisions affect workers, customers, communities and the environment. Understanding business helps you in almost any career, and if you ever start something of your own, it gives you the basic toolkit: a plan, a budget and an understanding of your customers.

    What the A Level covers

    The guide follows Cambridge International AS & A Level Business (9609), the syllabus for exams in 2026–2028. The course in this guide is planned in these parts:

    • AS Level: business and its environment, people in organisations, marketing, operations management and finance and accounting.
    • A Level: the same five areas in more depth, plus business strategy and decision-making tools.
    • Skills: business calculations, case-study technique and extended writing.
    • Exam readiness: paper guides and mock papers.

    Is it for you?

    Business suits you if you are interested in how organisations really work, enjoy discussing real companies, and can build a reasoned recommendation. You will write a lot: analysis that links causes and effects to the business in the case, then a judgement. There are also calculations (profit, ratios, break-even, investment appraisal) that need steady numeracy.

    It combines well with Economics and Accounting, Maths, a language, or Psychology. Business degrees rarely require A Level Business, but many like Maths.

    The Business course in this guide

    The full course is planned and will be built later in the build order. When it is ready, its card on the home page will open the first chapter instead of this page. Until then, the Tools section already has two business calculators (break-even and net present value).

    Accounting/About this course

    About Accounting

    Start hereStudy time10 minutes

    What accounting is, what accountants do, why it matters, what the Cambridge International AS & A Level Accounting (9706) course covers, and who it suits. The full course in this guide is planned.

    Why this matters

    Accounting is the language of business: it records every transaction, turns the records into reports, and uses them to judge how well a business is doing and what it should do next. Every company, charity and government needs it. This page explains what you would study and whether it fits you.

    What is accounting?

    Accounting has two main branches.

    • Financial accounting records what has happened. Every transaction is entered by double-entry bookkeeping (each one affects two accounts), the records are checked, and at the end of the year they become financial statements: the statement of profit or loss (did the business make a profit?) and the statement of financial position (what does it own and owe?).
    • Cost and management accounting looks forward. It works out what products really cost, sets budgets, compares results with plans, and helps managers decide: should we make this product or buy it in, accept this order, invest in this machine?

    Accounting also uses ratios to analyse performance (profitability, liquidity, efficiency) and has rules and principles so that accounts are reliable and comparable.

    What do accountants do?

    • Auditors check that a company’s accounts are true and fair.
    • Management accountants help businesses control costs and plan.
    • Tax advisers help people and companies follow tax law.
    • Financial analysts study accounts to value companies and advise investors.
    • Forensic accountants investigate fraud.
    • Many business leaders started in accounting, because it shows how every part of an organisation fits together.

    Why it matters

    Investors, banks, governments and employees all rely on accounts being accurate. Good accounting helps businesses survive (most failures are really cash-flow failures), and bad or dishonest accounting has caused some of the largest business collapses in history. It is also a useful life skill: budgeting and understanding money.

    What the A Level covers

    The guide follows Cambridge International AS & A Level Accounting (9706), the syllabus for exams in 2026–2028. The course in this guide is planned in these parts:

    • AS financial accounting: the accounting equation, books of prime entry, double entry, the trial balance and errors, bank reconciliation and control accounts, depreciation, accruals and prepayments, inventory, financial statements of sole traders, partnerships, limited companies, clubs, manufacturing accounts, incomplete records and ratio analysis.
    • AS cost and management accounting: costing materials and labour, absorption costing, and marginal costing with break-even.
    • A Level financial accounting: partnership changes, business purchase and mergers, company accounts in depth, statements of cash flows, international accounting standards, auditing and ethics, and interpreting accounts.
    • A Level cost and management accounting: activity-based costing, budgeting, standard costing and variances, and investment appraisal.
    • Exam readiness: paper guides and mock papers.

    Is it for you?

    Accounting suits you if you like order, accuracy and numbers, and get satisfaction from making things balance. Most of the work is calculation and layout, done carefully and in the right format, but there are also written parts where you explain and recommend. You do not need advanced maths: arithmetic, percentages and care are what count.

    It combines well with Business, Economics and Maths. It is a strong choice if you are considering accounting, finance or business degrees, or a professional accounting apprenticeship.

    The Accounting course in this guide

    The full course is planned and will be built later in the build order. When it is ready, its card on the home page will open the first chapter instead of this page.

    Psychology/About this course

    About Psychology

    Start hereStudy time10 minutes

    What psychology is, what psychologists do, why it matters, what the Cambridge International AS & A Level Psychology (9990) course covers, and who it suits. The full course in this guide is planned.

    Why this matters

    Psychology is the scientific study of the mind and behaviour: why we remember some things and forget others, why people obey or conform, what goes wrong in mental illness and how it can be treated. It is a science, with experiments and statistics, not just opinions about people. This page explains what you would study and whether it fits you.

    What is psychology?

    Psychologists try to explain and predict behaviour using evidence. They form a hypothesis, design a study (an experiment, observation, interview or questionnaire), collect data, and test whether the data support the idea. Because people are complicated, psychology looks at behaviour from several approaches:

    • the biological approach (the brain, hormones, genes);
    • the cognitive approach (memory, attention, thinking);
    • the learning approach (how experience and rewards shape behaviour);
    • the social approach (how other people and groups affect us).

    A big part of the subject is judging research: was the sample representative, was the study ethical, could the result be repeated, does it apply to real life?

    What do psychologists do?

    • Clinical psychologists assess and treat mental health problems such as depression, anxiety and phobias.
    • Educational psychologists help children who struggle at school.
    • Health psychologists study stress, pain and how to help people change habits like smoking.
    • Organisational psychologists work with companies on motivation, leadership and wellbeing at work.
    • Forensic psychologists work with courts and prisons; researchers run studies in universities.
    • Psychology is also used in marketing, user-experience design, sport, policing and the design of public services.

    Why it matters

    Mental health problems affect a large share of people at some point in their lives. Psychology research shapes how they are treated, how schools teach, how witnesses are questioned in court, and how products and services are designed. It also helps you understand yourself and others, and spot weak claims about human behaviour.

    What the A Level covers

    The guide follows Cambridge International AS & A Level Psychology (9990), the syllabus for exams in 2028–2030. The course in this guide is planned in these parts:

    • Research methods: experiments, self-reports, observations, case studies, correlations, sampling, ethics and data analysis.
    • Approaches and core studies: the biological, cognitive, learning and social approaches, each through key studies you learn and evaluate, and the issues and debates that run through them.
    • A Level options: clinical psychology, consumer psychology, health psychology and organisational psychology.
    • Exam readiness: paper guides and mock papers.

    Is it for you?

    Psychology suits you if you are curious about people and enjoy arguing from evidence. You will write a lot: describing studies precisely and evaluating them in balanced paragraphs. There is also maths: descriptive statistics, graphs and reading data. Remember that it is a science subject at A Level, and much of it is about how research is done.

    It combines well with Biology, Maths, Sociology, English or Economics. Psychology degrees rarely require A Level Psychology, but many like a science or Maths.

    The Psychology course in this guide

    The full course is planned and will be built later in the build order. When it is ready, its card on the home page will open the first chapter instead of this page.

    Other Subjects/About this course

    About Other subjects

    Start hereStudy time5 minutes

    What the “Other subjects” section of this guide will be: short, honest guides to other popular A Levels (what each subject is, what it involves, who it suits and what it leads to), to help you choose. It is planned.

    Why this matters

    There are dozens of A Level subjects, and the right combination matters more than any single choice. This section will give you a quick, reliable picture of the popular subjects this guide does not teach in full, so you can choose with your eyes open.

    What this section is

    The full courses in this guide (Mathematics, Physics, Chemistry, and soon Biology and others) teach a whole syllabus. Many students take at least one subject this guide does not teach in full. For those, Other subjects will give a short guide to each, covering:

    • what the subject is and what you actually do in it;
    • how it is assessed (exams, essays, coursework, practical work), in general terms;
    • the skills it builds and the kind of student it suits;
    • which subjects it combines well with and which degrees or careers it supports;
    • where to find the official specification and good free resources.

    Subjects planned

    The section is planned in these groups:

    • Sciences and technology: Information Technology, Environmental Management and Marine Science, and Physical Education.
    • English and languages: English Literature, English Language, modern languages, and Classics and Latin.
    • Humanities and social sciences: History, Geography, Sociology, Law, Politics and Government, Religious Studies and Philosophy, and Global Perspectives, Thinking Skills and the EPQ.
    • Creative subjects: Art and Design, Music, Drama and Theatre, and Media and Film Studies.
    • More subjects at a glance: short notes on further subjects.

    How to use it

    Read the guides for every subject you are considering, then use Choosing your subjects to check that your combination keeps the doors open for the degrees and careers you are thinking about.

    The Other subjects section in this guide

    This section is planned and will be built last. When it is ready, its card on the home page will open it instead of this page.

    Toolbox/Interactive tools

    Graph plotter

    Interactive

    Type any function of x and see its graph. Add sliders for a, b, c and d to explore transformations.

    PreviousAbout Other subjectsNextPeriodic table
    Toolbox/Interactive tools

    Periodic table

    Interactive

    All 118 elements with relative atomic masses, groups, periods and electron configurations. Select an element to see its details.

    PreviousGraph plotterNextCalculators
    Toolbox/Interactive tools

    Calculators

    Interactive

    Small tools for the calculations you will do again and again: relative formula mass, UCAS points, break-even, investment appraisal and unit conversions.

    Relative formula mass

    UCAS tariff points

    Break-even

    Investment appraisal

    Units and prefixes

    PreviousPeriodic tableNextGlossary
    Toolbox/Interactive tools

    Glossary

    Interactive

    Every key term defined in the guide, in one searchable list. Each entry links back to the chapter that explains it.

    PreviousCalculatorsNextMaths formula sheet
    Toolbox/Formula sheets

    Maths formula sheet

    Reference

    Every key formula from the course in one place, each linked back to the chapter that teaches it.

    Cambridge gives you the formula list MF19 in every 9709 exam, but it only helps if you already know which formula fits which question and how to use it. This page collects every key box from the Maths chapters in one place.

    Numbers, fractions and percentages

    Key factsPercentage multipliers
    • Increase by p%: multiply by 1+p100. So +15% is ×1.15.
    • Decrease by p%: multiply by 1−p100. So −20% is ×0.8.
    • Repeat a change n times: multiply by the multiplier to the power n.
    • Undo a change: divide by the multiplier.

    Revise: Numbers, fractions and percentages

    Algebra from scratch

    Key factsInequality rule

    Add or subtract anything on both sides: the sign stays. Multiply or divide both sides by a positive number: the sign stays. Multiply or divide both sides by a negative number: the sign reverses. For example 3>1, but −3<−1.

    Revise: Algebra from scratch

    Indices, surds and standard form

    Key factsThe laws of indices

    For any base a≠0 and indices m and n:

    • am×an=am+n
    • am÷an=am−n
    • (am)n=amn
    • (ab)n=anbn and (ab)n=anbn
    • a0=1
    • a−n=1an
    • a1/n=an and am/n=(an)m=amn
    Key factsRules for surds

    For a,b≥0: ab=a b and ab=ab. Also a a=a.

    There is no rule for sums: a+b≠a+b. Check: 9+16=5, but 9+16=7.

    Revise: Indices, surds and standard form

    Graphs and straight lines

    Key factsGradient from two points

    The line through (x1,y1) and (x2,y2) has gradient m=y2−y1x2−x1 A positive gradient goes up from left to right, a negative gradient goes down, a horizontal line has m=0, and a vertical line has no gradient (its equation is x=k). Parallel lines have equal gradients.

    Key factsThe equation y=mx+c

    Every straight line that is not vertical has an equation y=mx+c, where m is the gradient and c is the y-intercept. Every point on the line has x and y that satisfy the equation.

    Revise: Graphs and straight lines

    Geometry and trigonometry basics

    Key factsAngle facts
    • Angles on a straight line add up to 180∘. Angles around a point add up to 360∘.
    • Vertically opposite angles (where two lines cross) are equal.
    • The angles of a triangle add up to 180∘. An exterior angle of a triangle equals the sum of the two interior angles opposite it.
    • With parallel lines: alternate angles (a Z shape) are equal, corresponding angles (an F shape) are equal, and co-interior angles (a C shape) add up to 180∘.
    • An isosceles triangle has two equal sides and two equal base angles. An equilateral triangle has three equal sides and three angles of 60∘.
    • The interior angles of a polygon with n sides add up to (n−2)×180∘. The exterior angles of any polygon add up to 360∘, so each exterior angle of a regular polygon is 360∘n.
    Key factsPythagoras’ theorem

    In a right-angled triangle the square on the hypotenuse (the side opposite the right angle, always the longest) equals the sum of the squares on the other two sides: c2=a2+b2 The converse is also true: if three sides satisfy c2=a2+b2, the triangle has a right angle. The triples 3,4,5, 5,12,13, 8,15,17 and 7,24,25 are worth recognising.

    Key factsSOH CAH TOA

    sin⁡θ=oppositehypotenusecos⁡θ=adjacenthypotenusetan⁡θ=oppositeadjacent To find a side, use the ratio directly. To find an angle, use the inverse function: θ=sin⁡−1(opphyp). Make sure your calculator is in degree mode.

    FormulaSine rule, cosine rule and area

    asin⁡A=bsin⁡B=csin⁡C a2=b2+c2−2bccos⁡Acos⁡A=b2+c2−a22bc Area=12absin⁡C Use the sine rule when you know a side and its opposite angle (a “matching pair”) and one more piece. Use the cosine rule when you know two sides and the angle between them, or all three sides. The area formula needs two sides and the included angle. None of these is on MF19, so learn them.

    Key factsCircle theorems you need
    • A tangent is perpendicular to the radius at the point of contact.
    • The angle in a semicircle is 90∘: if AB is a diameter and C is on the circle, then angle ACB=90∘.
    • The perpendicular from the centre to a chord bisects the chord.
    • Two tangents drawn from the same outside point are equal in length.
    • The angle at the centre is twice the angle at the circumference on the same arc.
    Key factsArc length and sector area (degrees)

    arc length=θ360×2πrsector area=θ360×πr2 Radians give simpler formulas, s=rθ and A=12r2θ: see Circular measure.

    Revise: Geometry and trigonometry basics

    Quadratics

    Key factsThe zero product rule

    If A×B=0, then A=0 or B=0 (or both).

    Key factsCompleting the square (when a=1)

    x2+bx=(x+b2)2−(b2)2 Halve the coefficient of x to get the number in the bracket, then subtract that number squared.

    Key factsThe quadratic formula

    The solutions of ax2+bx+c=0 are x=−b±b2−4ac2a

    Revise: Quadratics

    Functions

    Key factsThe inside/outside rule

    Changes outside the bracket act on y and do what you expect: +a lifts, ×a stretches vertically. Changes inside the bracket act on x and do the opposite: x+a moves left, ax squashes horizontally by 1a.

    Revise: Functions

    Coordinate geometry: lines and circles

    FormulaDistance, midpoint, gradient

    For A(x1,y1) and B(x2,y2): AB=(x2−x1)2+(y2−y1)2 midpoint=(x1+x22, y1+y22)gradient=y2−y1x2−x1 These are not on MF19. Learn them.

    Key factsThree forms of a line
    • y=mx+c: gradient m, y-intercept c.
    • y−y1=m(x−x1): gradient m, passes through (x1,y1). The y1 and x1 change sign inside the brackets.
    • ax+by+c=0 with a, b, c integers: the general form. The gradient is −ab.
    Key factsParallel and perpendicular
    • Parallel: m1=m2.
    • Perpendicular: m1m2=−1, so m2=−1m1. A gradient of 23 becomes −32, and a gradient of −4 becomes 14.
    Key factsEquation of a circle

    Centre (a,b), radius r: (x−a)2+(y−b)2=r2 Expanded: x2+y2+2gx+2fy+c=0, with centre (−g,−f) and radius g2+f2−c. The x2 and y2 terms must have the same coefficient and there is no xy term. The circle equation is not on MF19.

    Key factsLine and circle

    Substitute the line into the circle to get a quadratic.

    • b2−4ac>0: the line cuts the circle at two points.
    • b2−4ac=0: the line touches the circle: it is a tangent.
    • b2−4ac<0: the line misses the circle.

    Revise: Coordinate geometry: lines and circles

    Circular measure

    Key factsRadians and degrees

    2π radians=360∘π radians=180∘

    • Degrees to radians: multiply by π180.
    • Radians to degrees: multiply by 180π. Radian conversion is not on MF19. Remember π=180∘ and you can rebuild everything.
    FormulaArc length and sector area

    For a sector of radius r and angle θ in radians: s=rθA=12r2θ Both are given on MF19, but only for radians. The perimeter of a sector is 2r+rθ (two radii plus the arc).

    Key factsSegment

    Area of segment = area of sector − area of triangle OAB: Asegment=12r2θ−12r2sin⁡θ=12r2(θ−sin⁡θ) Chord length AB=2rsin⁡θ2. The segment formula is not on MF19, but you can rebuild it in seconds from 12r2θ and 12absin⁡C. Use radian mode for sin⁡θ.

    Revise: Circular measure

    Trigonometry

    Key factsRelated angles

    sin⁡(180∘−θ)=sin⁡θcos⁡(360∘−θ)=cos⁡θtan⁡(180∘+θ)=tan⁡θ sin⁡(−θ)=−sin⁡θcos⁡(−θ)=cos⁡θtan⁡(−θ)=−tan⁡θ The values repeat every 360∘ for sine and cosine, and every 180∘ for tangent. In radians replace 180∘ by π and 360∘ by 2π.

    Key factsProperties of the graphs
    • y=sin⁡x and y=cos⁡x: period 360∘ (2π), range −1≤y≤1. Sine passes through (0,0); cosine starts at (0,1).
    • y=tan⁡x: period 180∘ (π), range all real numbers. It has asymptotes (vertical lines it never reaches) at x=90∘, 270∘, and so on. It passes through the origin.
    • The graphs continue in both directions for ever. The curve for negative angles is found by the symmetry: sine is odd (rotates 180∘ about the origin), cosine is even (mirrors in the y-axis).
    Key factsTransformations

    For y=asin⁡(bx+c)+d (and similarly for cosine):

    • a: stretch parallel to the y-axis, scale factor a. The amplitude (half the height from top to bottom) becomes ∣a∣.
    • b: stretch parallel to the x-axis, scale factor 1b. The period becomes 360∘b (or 2πb).
    • c: translation by −cb in the x-direction. y=sin⁡(x+30∘) moves 30∘ to the left.
    • d: translation by d up the y-axis. The tangent graph has period 180∘b instead.
    Key factsPrincipal values

    sin⁡−1x: −90∘≤y≤90∘(−π2 to π2) cos⁡−1x: 0∘≤y≤180∘(0 to π) tan⁡−1x: −90∘<y<90∘(−π2 to π2) The ranges are on MF19. The calculator answer is the principal value; other solutions come from the symmetry of the graph.

    Key factsAll solutions in a given interval
    • sin⁡θ=k: θ=α or 180∘−α, then add or subtract multiples of 360∘.
    • cos⁡θ=k: θ=α or 360∘−α (equivalently ±α), then add or subtract multiples of 360∘.
    • tan⁡θ=k: θ=α, then add or subtract multiples of 180∘. In radians use π and 2π. Sketching the graph of the function and marking k shows how many solutions to expect.
    Key factsThe two identities

    tan⁡θ≡sin⁡θcos⁡θsin⁡2θ+cos⁡2θ≡1 Rearranged: sin⁡2θ≡1−cos⁡2θ and cos⁡2θ≡1−sin⁡2θ. Both are on MF19. The second is Pythagoras on the unit circle: cos⁡θ and sin⁡θ are the legs of a right-angled triangle with hypotenuse 1.

    Revise: Trigonometry

    Series: binomial expansion and progressions

    Key factsBinomial theorem

    For a positive integer n: (a+b)n=an+(n1)an−1b+(n2)an−2b2+⋯+(nr)an−rbr+⋯+bn The general term (the term in br) is (nr)an−rbr. The powers of a fall from n to 0, the powers of b rise from 0 to n, and the powers in each term add up to n. The expansion has n+1 terms. It is on MF19.

    FormulaArithmetic progression

    un=a+(n−1)dSn=n2(2a+(n−1)d)=n2(a+l) a is the first term, d the common difference, n the number of terms and l the last term. Both are on MF19. Three numbers a,b,c are in AP if and only if b−a=c−b, that is 2b=a+c.

    FormulaGeometric progression

    un=arn−1Sn=a(1−rn)1−r=a(rn−1)r−1(r≠1) Use the first form when ∣r∣<1 and the second when r>1, which avoids negatives. Three numbers a,b,c are in GP if ba=cb, that is b2=ac. All are on MF19.

    Key factsSum to infinity

    A geometric series converges if and only if −1<r<1, and then S∞=a1−r If ∣r∣≥1, the series does not converge and S∞ does not exist. The formula is on MF19, but you must state or check ∣r∣<1.

    Revise: Series: binomial expansion and progressions

    Differentiation

    Key factsDifferentiating powers

    If y=xn then dydx=nxn−1(n any rational number)

    • A constant multiple stays: if y=kxn then dydx=knxn−1.
    • Differentiate a sum or difference term by term.
    • A constant on its own differentiates to 0 (the graph y=c is flat).
    • In particular ddx(kx)=k: the gradient of a straight line.
    Key factsThe chain rule

    If y is a function of u, and u is a function of x, then dydx=dydu×dudx For powers of a bracket this gives the shortcut ddx[(f(x))n]=n(f(x))n−1×f′(x) “Differentiate the outside, keep the inside, then multiply by the derivative of the inside.”

    Key factsTangent and normal at (x1,y1)
    • Gradient of the tangent: m=dydx evaluated at x=x1.
    • Gradient of the normal: −1m (perpendicular gradients multiply to −1).
    • Equation of either line: y−y1=(gradient)(x−x1).
    Key factsFinding and classifying stationary points
    1. Find dydx and solve dydx=0 for x.
    2. Substitute each x into the equation of the curve to find y.
    3. Find d2ydx2 at each point:
      • d2ydx2<0: a maximum (the gradient is decreasing through zero).
      • d2ydx2>0: a minimum (the gradient is increasing through zero).
      • d2ydx2=0: the test gives no answer; check the sign of dydx on each side.
    Key factsConnected rates of change

    dAdt=dAdr×drdt Find dAdr by differentiating the formula that links A and r. Then multiply by the rate you are given, or divide by it if the unknown rate is drdt.

    Revise: Differentiation

    Integration

    Key factsIntegrating powers of x

    ∫xn dx=xn+1n+1+c(n≠−1) ∫k f(x) dx=k∫f(x) dx∫(f(x)+g(x)) dx=∫f(x) dx+∫g(x) dx This works for every rational n except −1 (the integral of 1x is a logarithm, see Logarithms and exponentials). A constant k integrates to kx, since k=kx0. MF19 lists ∫xn dx, but not the rule for (ax+b)n below. Never forget + c.

    Key factsIntegral of (ax+b)n

    ∫(ax+b)n dx=(ax+b)n+1a(n+1)+c(n≠−1) In words: raise the power by one, then divide by the new power and by the coefficient of x inside the bracket. This applies to a linear bracket only.

    Key factsDefinite integral

    ∫abf(x) dx=[F(x)]ab=F(b)−F(a) where F is any integral of f. Reversing the limits changes the sign: ∫baf(x) dx=−∫abf(x) dx.

    Key factsImproper integrals
    • Infinite limit: ∫1∞f(x) dx=lim⁡t→∞∫1tf(x) dx. For a power xn, if n<−1 the term tn+1→0, and the integral exists; if n>−1 the term grows and the integral does not exist (it diverges).
    • Undefined at a limit: ∫01f(x) dx=lim⁡t→0+∫t1f(x) dx. For xn with −1<n<0 this converges, as tn+1→0.
    Key factsVolume of revolution

    About the x-axis (from x=a to x=b): V=π∫aby2 dx

    About the y-axis (from y=c to y=d): V=π∫cdx2 dy Square the function before integrating (y2 is not 2y); the factor π stays outside. These are not on MF19.

    Revise: Integration

    Modulus and polynomials

    Key factsSquaring with a modulus

    ∣a∣=∣b∣  ⟺  a2=b2 For ∣a∣=b, where b may contain x, square both sides to get a2=b2, then check every solution in the original equation: squaring can add false answers (those where b<0).

    Key factsModulus inequalities (b>0)

    ∣x−a∣<b  ⟺  a−b<x<a+b ∣x−a∣>b  ⟺  x<a−b  or  x>a+b For ∣ax+b∣<c, write −c<ax+b<c and solve. For ∣f(x)∣<∣g(x)∣, square both sides to get f(x)2<g(x)2, rearrange to 0 on one side and use a sketch of the quadratic.

    Key factsRemainder theorem and factor theorem
    • Remainder theorem: when f(x) is divided by (x−a), the remainder is f(a). When divided by (ax+b), the remainder is f(−ba).
    • Factor theorem: (x−a) is a factor of f(x) if and only if f(a)=0. Similarly (ax+b) is a factor if and only if f(−ba)=0. Reason: f(x)=(x−a)q(x)+r. Put x=a and the bracket vanishes, leaving f(a)=r.

    Revise: Modulus and polynomials

    Logarithms and exponentials

    Key factsLaws of logarithms (any valid base)

    log⁡ax+log⁡ay=log⁡a(xy)log⁡ax−log⁡ay=log⁡a(xy) log⁡a(xk)=klog⁡axlog⁡a(1x)=−log⁡ax log⁡aa=1log⁡a1=0alog⁡ax=xlog⁡a(ax)=x The laws work only when the bases are the same. They are not on MF19. There is no law for log⁡a(x+y).

    Key factsProperties of ex and ln⁡x
    • y=ex: domain all real x, range y>0, passes through (0,1), increasing, asymptote y=0.
    • y=ln⁡x: domain x>0, range all real y, passes through (1,0), increasing, asymptote x=0.
    • y=ekx with k>0 is exponential growth; with k<0 it is exponential decay. Both pass through (0,1) and have asymptote y=0.
    • The laws of logs apply to ln⁡: ln⁡a+ln⁡b=ln⁡ab, ln⁡ak=kln⁡a, ln⁡e=1.
    Key factsReducing to linear form
    • y=kxn: take ln⁡ of both sides, ln⁡y=nln⁡x+ln⁡k. Plot ln⁡y against ln⁡x: the gradient is n and the ln⁡y-intercept is ln⁡k.
    • y=kax: ln⁡y=xln⁡a+ln⁡k. Plot ln⁡y against x: the gradient is ln⁡a and the intercept is ln⁡k.
    • To find k from the intercept c: k=ec. To find a from the gradient m: a=em.

    Revise: Logarithms and exponentials

    Further trigonometry

    Key factsFeatures of the reciprocal graphs
    • y=sec⁡x: period 360∘, symmetrical about the y-axis, passes through (0,1); ∣sec⁡x∣≥1, so the range is y≤−1 or y≥1. Minimum points at the maxima of cosine.
    • y=cosec⁡x: period 360∘; ∣cosec⁡x∣≥1. It is the graph of sec⁡x shifted 90∘ to the right.
    • y=cot⁡x: period 180∘, range all real values, decreasing between asymptotes.
    • Where the original function is ±1, its reciprocal is also ±1: the curves touch there.
    Key factsIdentities

    sec⁡2θ≡1+tan⁡2θcosec⁡2θ≡1+cot⁡2θ Together with sin⁡2θ+cos⁡2θ≡1, these are on MF19. Use sec⁡2θ to turn an equation containing both sec⁡θ and tan⁡θ into a quadratic in sec⁡θ alone (and cosec⁡θ, cot⁡θ likewise).

    FormulaCompound angles

    sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B Take care with the signs: in the cosine formula the sign in the middle is opposite to the sign in the bracket (cos⁡(A+B) has a minus). All three are on MF19.

    FormulaDouble angles

    sin⁡2A=2sin⁡Acos⁡A cos⁡2A=cos⁡2A−sin⁡2A=2cos⁡2A−1=1−2sin⁡2A tan⁡2A=2tan⁡A1−tan⁡2A There are three forms of cos⁡2A. Choose the one that leaves the same function as the rest of your equation. All are on MF19.

    Key factsHarmonic form

    For a>0 and b>0: asin⁡θ+bcos⁡θ=Rsin⁡(θ+α)R=a2+b2, tan⁡α=ba acos⁡θ+bsin⁡θ=Rcos⁡(θ−α)R=a2+b2, tan⁡α=ba With a minus sign in the expression, expand Rsin⁡(θ−α) or Rcos⁡(θ+α) and match in the same way. Choose R>0 and an acute α. To get α right, write Rcos⁡α=… and Rsin⁡α=… first, then check the quadrant.

    Revise: Further trigonometry

    Further differentiation

    Key factsDerivatives to know (radians)

    ddx(ex)=exddx(ln⁡x)=1x ddx(sin⁡x)=cos⁡xddx(cos⁡x)=−sin⁡xddx(tan⁡x)=sec⁡2x These are on MF19. With a linear inside function, the chain rule multiplies by the coefficient a of x: ddxeax+b=aeax+bddxln⁡(ax+b)=aax+b ddxsin⁡(ax+b)=acos⁡(ax+b)ddxcos⁡(ax+b)=−asin⁡(ax+b)ddxtan⁡(ax+b)=asec⁡2(ax+b)

    Key factsProduct rule

    If y=uv, where u and v are functions of x, then dydx=udvdx+vdudx In words: first times the derivative of the second, plus second times the derivative of the first. It is on MF19, written dydx =udvdx+vdudx.

    Key factsQuotient rule

    If y=uv, then dydx=vdudx−udvdxv2 The numerator is “bottom times derivative of top, minus top times derivative of bottom”. The order matters because of the minus sign. It is on MF19.

    Key factsParametric differentiation

    If x=f(t) and y=g(t), then dydx=dy/dtdx/dt This is the chain rule dydx =dydt×dtdx with dtdx=1dx/dt. The answer is usually in terms of t. It is on MF19.

    Key factsImplicit differentiation
    • Differentiate each term with respect to x. For a function of y, ddxf(y) =f′(y)dydx.
    • ddx(xy) =xdydx+y.
    • Collect the terms containing dydx on one side, factorise, and divide. The answer usually contains both x and y.
    Key factsDerivative of tan⁡−1x (Paper 3 only)

    ddx(tan⁡−1x)=11+x2 This is on MF19. With a linear inside function, ddxtan⁡−1(ax) =a1+a2x2. The derivatives of sin⁡−1x and cos⁡−1x are not required.

    Revise: Further differentiation

    Further integration

    Key factsStandard integrals (radians)

    ∫ex dx=ex+c∫1x dx=ln⁡∣x∣+c ∫sin⁡x dx=−cos⁡x+c∫cos⁡x dx=sin⁡x+c∫sec⁡2x dx=tan⁡x+c With a linear inside function ax+b, divide by the coefficient a: ∫eax+b dx=1aeax+b+c∫1ax+b dx=1aln⁡∣ax+b∣+c ∫sin⁡(ax+b) dx=−1acos⁡(ax+b)+c∫cos⁡(ax+b) dx=1asin⁡(ax+b)+c ∫sec⁡2(ax+b) dx=1atan⁡(ax+b)+c The basic forms are on MF19; the versions with ax+b follow from the chain rule. The modulus in ln⁡∣x∣ allows for negative x. Never forget + c.

    Key factsIdentities for integration

    sin⁡2x≡12(1−cos⁡2x)cos⁡2x≡12(1+cos⁡2x)tan⁡2x≡sec⁡2x−1 Also useful: sin⁡xcos⁡x≡12sin⁡2x. These identities are the key step in many questions, so learn to spot them.

    FormulaTrapezium rule

    To estimate ∫aby dx with n strips of equal width h=b−an, use the n+1 ordinates y0,y1,…,yn: ∫aby dx≈12h[y0+2(y1+y2+⋯+yn−1)+yn] The first and last ordinates are counted once, and every other ordinate twice. It is not on MF19, so learn it.

    Key factsIntegral of 1x2+a2 (Paper 3 only)

    ∫1x2+a2 dx=1atan⁡−1(xa)+c This is on MF19. For 1px2+q, take out the factor p first: 1p⋅1x2+q/p.

    Revise: Further integration

    Numerical solution of equations

    Key factsSign change rule

    If f(x) is continuous on the interval [a,b] and f(a) and f(b) have opposite signs, then f(x)=0 has at least one root between a and b. To use it in an exam: (1) write the equation as f(x)=0, (2) calculate f(a) and f(b) and quote their values, (3) state “there is a change of sign, and f is continuous, so there is a root in the interval.”

    Key factsIterative formula

    xn+1=F(xn) Start with a first estimate x0 (or x1), compute x1=F(x0), then x2=F(x1), and so on. If the sequence x0,x1,x2,… converges to a number α, then α=F(α), so α is a root of the original equation.

    Revise: Numerical solution of equations

    Partial fractions and the binomial series

    Key factsPartial fraction templates

    For a proper fraction (numerator degree lower than the denominator):

    • Distinct linear factors: f(x)(ax+b)(cx+d) =Aax+b+Bcx+d, and likewise for three factors.
    • A repeated linear factor: f(x)(ax+b)(cx+d)2 =Aax+b+Bcx+d+C(cx+d)2. A squared factor needs two terms, one for each power.
    • A quadratic factor that does not factorise: f(x)(ax+b)(cx2+d) =Aax+b+Bx+Ccx2+d. The numerator over the quadratic is linear.
    Key factsBinomial series

    For any rational number n and ∣x∣<1: (1+x)n=1+nx+n(n−1)2!x2+n(n−1)(n−2)3!x3+⋯ It is on MF19. If n is a positive integer the series stops and is valid for all x. Otherwise it has infinitely many terms and only converges for ∣x∣<1. You never need the general term at Cambridge level: calculate the first few terms.

    Revise: Partial fractions and the binomial series

    Integration techniques: parts, substitution and partial fractions

    Key factsReverse chain rule for a logarithm

    ∫f′(x)f(x) dx=ln⁡∣f(x)∣+c It applies when the numerator is the derivative of the denominator, or a constant multiple of it. This result is on MF19.

    Key factsIntegration by parts

    ∫u dvdx dx=uv−∫v dudx dx For a definite integral, apply the limits to the uv term as well: ∫abu dvdx dx =[uv]ab−∫abv dudx dx. This formula is on MF19.

    Key factsThe substitution routine
    1. Differentiate the substitution to find dudx (or dxdu) and replace dx.
    2. Rewrite the whole integrand in terms of u.
    3. For a definite integral, change the limits to u-values and do not go back to x.
    4. Integrate in u. For an indefinite integral, substitute back to finish in x.

    Revise: Integration techniques: parts, substitution and partial fractions

    Differential equations

    Key factsSeparation of variables

    For dydx=f(x) g(y), rearrange and integrate: ∫1g(y) dy=∫f(x) dx Add one constant c, on one side only. Then use the initial condition to find c, and solve for y if the question asks for y in terms of x.

    Revise: Differential equations

    Vectors

    Key factsMagnitude and unit vector

    For 𝐚=x𝐢+y𝐣+z𝐤, the magnitude is ∣𝐚∣=x2+y2+z2 A unit vector in the direction of 𝐚 is 𝐚∣𝐚∣, which has magnitude 1.

    Key factsDisplacement and midpoint

    AB→=𝐛−𝐚 The midpoint M of AB has position vector 12(𝐚+𝐛). If OABC is a parallelogram, then OB→=OA→+OC→.

    Key factsVector equation of a line

    𝐫=𝐚+t𝐛 where 𝐚 is the position vector of a point on the line, 𝐛 is a direction vector and t is a scalar parameter. Each value of t gives one point on the line.

    Key factsRelationship between two lines
    1. Parallel: the direction vectors are multiples of each other.
    2. Not parallel, and they meet: intersecting, at one point.
    3. Not parallel, and they never meet: skew. This can only happen in three dimensions; in two dimensions, lines that are not parallel always meet.
    Key factsFinding the foot of the perpendicular
    1. Write a general point X on the line, with position vector 𝐚+t𝐛.
    2. Find PX→ in terms of t.
    3. Solve PX→⋅𝐛=0 for t.
    4. Put t back to get F; the perpendicular distance is ∣PF→∣.

    Revise: Vectors

    Complex numbers

    Key factsConjugate pairs

    If a polynomial has real coefficients and a+ib is a root, then its conjugate a−ib is also a root. The two roots give the real quadratic factor (z−(a+ib))(z−(a−ib))=z2−2az+(a2+b2)

    Key factsMultiplying and dividing in polar form

    ∣z1z2∣=∣z1∣∣z2∣arg⁡(z1z2)=arg⁡z1+arg⁡z2 ∣z1z2∣=∣z1∣∣z2∣arg⁡z1z2=arg⁡z1−arg⁡z2 In exponential form: r1eiθ1×r2eiθ2 =r1r2ei(θ1+θ2).

    Key factsThe three loci
    • ∣z−a∣=k: the distance from z to the point a is k. It is a circle, centre a, radius k.
    • ∣z−a∣=∣z−b∣: z is equally far from a and b. It is the perpendicular bisector of the line joining a and b.
    • arg⁡(z−a)=α: the line from a to z makes angle α with the real axis direction. It is a half-line starting at a (not including a itself). For inequalities, ∣z−a∣<k is the inside of the circle, > the outside; a solid line means ≤ or ≥, a dashed line means < or >.

    Revise: Complex numbers

    Representing and summarising data

    Key factsFive-number summary and the box plot

    The five-number summary is the minimum, Q1, the median, Q3 and the maximum. A box plot (box-and-whisker diagram) draws a box from Q1 to Q3 with a line at the median, and whiskers out to the minimum and maximum. The width of the box is the IQR. Draw it on a scale.

    Key factsMean, variance and standard deviation

    xˉ=∑xnvariance=∑x2n−xˉ2standard deviation=variance For a frequency table, use ∑fx and ∑fx2 with n=∑f. The mean and standard deviation formulae are given on MF19, but the form ∑x2n−xˉ2 is the quicker one to use, so practise it.

    Key factsFrequency density

    frequency density=frequencyclass widthsofrequency=frequency density×class width The vertical axis is labelled “frequency density”. Bars touch for continuous data.

    Revise: Representing and summarising data

    Permutations and combinations

    Key factsMultiplication principle

    If one task can be done in m ways and, for each of these, a second task can be done in n ways, then the two tasks together can be done in m×n ways. This extends to any number of tasks. If the tasks are alternatives (either one or the other, not both), add the counts.

    Key factsArrangements with repeats

    The number of arrangements of n items, of which p are identical of one kind, q are identical of another kind, and so on, is n!p! q!⋯

    Key factsTechniques for restrictions
    • Together: glue the items into one block, arrange the block with the others, then multiply by the number of arrangements inside the block.
    • Not together (apart): count all arrangements and subtract those with the items together.
    • Fixed positions (at the ends, first digit nonzero, last digit even): fill the restricted slots first.

    Revise: Permutations and combinations

    Probability

    Key factsProbability of “A or B”

    P(A∪B)=P(A)+P(B)−P(A∩B) A∪B means “A or B or both”; A∩B means “A and B”. Subtract the overlap so it is not counted twice.

    Key factsConditional probability

    P(A∣B)=P(A∩B)P(B) This is read “the probability of A given B”. The condition B shrinks the sample space to B alone. Rearranged, it gives the multiplication rule P(A∩B)=P(B)×P(A∣B), which is what a tree diagram uses.

    Revise: Probability

    Discrete random variables: binomial and geometric

    Key factsExpectation and variance

    E(X)=∑x P(X=x)E(X2)=∑x2 P(X=x) Var(X)=E(X2)−[E(X)]2 E(X) is the long-run average (the mean of the distribution). The standard deviation is Var(X).

    Key factsBinomial formulae

    For X∼B(n,p), with q=1−p: P(X=r)=(nr)prqn−rE(X)=npVar(X)=npq The probability formula and the mean and variance are on MF19.

    Key factsGeometric formulae

    For X∼Geo(p), with q=1−p: P(X=r)=qr−1pP(X>r)=qrE(X)=1p The first formula and the mean are on MF19. To have r−1 failures and then a success, multiply qr−1 by p. To need more than r trials, the first r trials must all fail: qr.

    Key factsBinomial or geometric?
    • Binomial: a fixed number of trials, and you count the successes. “How many of the 20 bulbs are faulty?”
    • Geometric: no fixed number of trials, and you count the trials until the first success. “How many bulbs must I test before I find a faulty one?”

    Revise: Discrete random variables: binomial and geometric

    The normal distribution

    Key factsStandardising

    Z=X−μσ∼N(0,1) Φ(z)=P(Z≤z) is the area to the left of z. The normal table gives Φ(z) for z≥0. By symmetry, Φ(−z)=1−Φ(z) and P(Z>z)=1−Φ(z).

    Key factsInverse normal

    If P(X<x)=p, find z with Φ(z)=p, then x=μ+zσ. If p<0.5, z is negative: use z=−Φ−1(1−p). Common values: Φ−1(0.9)=1.2816, Φ−1(0.95)=1.6449, Φ−1(0.975)=1.9600.

    Revise: The normal distribution

    The Poisson distribution

    Key factsThe Poisson distribution

    If X∼Po(λ), then P(X=r)=e−λ λrr!,r=0,1,2,3,… E(X)=λandVar(X)=λ

    Key factsScaling the mean

    If events occur at a constant average rate of k per unit of time (or length, or area), the number of events in an interval of size t has distribution Po(kt).

    Key factsPoisson approximation to the binomial

    If X∼B(n,p) with n>50 and np<5, then X is approximately Po(np).

    Key factsNormal approximation to the Poisson

    If X∼Po(λ) with λ>15, then X is approximately N(λ,λ). The mean is λ, the variance is λ and the standard deviation is λ.

    Revise: The Poisson distribution

    Linear combinations of random variables

    Key factsLinear change of a variable

    For any random variable X and constants a and b: E(aX+b)=aE(X)+bVar(aX+b)=a2 Var(X)

    • The constant b affects the mean only.
    • The multiplier a is squared in the variance. If a is negative, a2 is still positive.
    • Standard deviation: sd(aX+b)=∣a∣ sd(X). You may use these results without proof. Learn them: MF19 does not give the rule for Var(aX+bY).
    Key factsLinear combination of two variables

    For any random variables X and Y: E(aX+bY)=aE(X)+bE(Y) If X and Y are independent, then Var(aX+bY)=a2 Var(X)+b2 Var(Y)

    • The mean rule always holds, independent or not.
    • The variance rule needs independence. Knowing the value of one variable must tell you nothing about the other.
    • Variances add even when you subtract the variables: Var(X−Y) =Var(X)+Var(Y), because (−1)2=1.
    • Both rules extend to three or more variables.
    Key factsNormal variables

    If X∼N(μX,σX2) and Y∼N(μY,σY2) are independent, then aX+bY∼N ⁣(aμX+bμY,  a2σX2+b2σY2) Also, aX+b is normal when X is normal. The same holds for any number of independent normal variables.

    Key factsSum of independent Poisson variables

    If X∼Po(λ) and Y∼Po(μ) are independent, then X+Y∼Po(λ+μ) This extends to any number of independent Poisson variables.

    Revise: Linear combinations of random variables

    Continuous random variables

    Key factsProperties of a pdf

    Let X have pdf f(x), defined on one interval and taken as 0 outside it.

    • f(x)≥0 everywhere.
    • Total area 1: ∫−∞∞f(x) dx=1, which in practice means integrating over the interval where f is not zero.
    • P(a<X<b)=∫abf(x) dx.
    • P(X=a)=0 for any single value a, because a single point has no width. So P(X<a) and P(X≤a) are equal, and strict and non-strict inequalities make no difference.
    • f(x) is not a probability. It can be larger than 1 (a narrow, tall curve), but the area never is.
    Key factsMean and variance of a continuous variable

    E(X)=∫x f(x) dxE(X2)=∫x2f(x) dx Var(X)=E(X2)−[E(X)]2

    • Integrate over the interval where f is not zero.
    • More generally E(g(X))=∫g(x) f(x) dx.
    • Both formulas are on MF19. The standard deviation is Var(X).
    • If the pdf is symmetrical about a line x=c, then E(X)=c and c is also the median.
    Key factsFinding a median or percentile

    Integrate from the lower end of the interval up to an unknown value m and set the area equal to the required probability: ∫amf(x) dx=0.5(median)∫aqf(x) dx=p100(pth percentile) Then solve for m or q. The median is not the midpoint of the interval unless the pdf is symmetrical.

    Revise: Continuous random variables

    Sampling and estimation

    Key factsDistribution of the sample mean

    For a random sample of size n from a population with mean μ and variance σ2: E(Xˉ)=μVar(Xˉ)=σ2nstandard deviation of Xˉ=σn

    • If X is normal, then Xˉ∼N ⁣(μ,σ2n) exactly, for any n.
    • Central limit theorem: if n is large, Xˉ is approximately N ⁣(μ,σ2n) whatever the shape of the population. A sample of 30 or more is usually treated as large.
    • The variance of Xˉ shrinks as n grows: means of big samples are more reliable than single values.
    Key factsUnbiased estimates from a sample

    For a sample x1,…,xn: xˉ=∑xns2=1n−1∑(x−xˉ)2=1n−1(∑x2−(∑x)2n)

    • xˉ is an unbiased estimate of μ. s2 is an unbiased estimate of σ2.
    • The sample variance with divisor n is not the unbiased estimate.
    • Both estimators are on MF19. On a calculator, s is the “σn−1” or “sx” key.
    • Use s2 in place of σ2 when σ2 is unknown (for large samples).
    Key factsConfidence interval for a population mean

    For a sample of size n with mean xˉ, the c% confidence interval for μ is xˉ−zσn  <  μ  <  xˉ+zσn where z satisfies P(−z<Z<z)=c/100.

    Confidence 90% 95% 98% 99%
    z 1.645 1.960 2.326 2.576
    • Use when the population is normal with known σ (any n), or when n is large (use s for σ if σ is unknown).
    • The formula is not on MF19: learn it, and learn how to find z from the table of the normal distribution.
    • The margin of error is zσ/n; the interval has width 2zσ/n.
    Key factsApproximate confidence interval for a proportion

    From a large sample with sample proportion p^, an approximate c% confidence interval for p is p^−zp^(1−p^)n  <  p  <  p^+zp^(1−p^)n with z from the same table as before. This interval is not on MF19: learn it.

    Revise: Sampling and estimation

    Hypothesis testing

    Key factsWriting the conclusion
    • If you reject H0: “There is sufficient evidence at the 5% significance level to suggest that [H1 in context].”
    • If you do not reject H0: “There is insufficient evidence at the 5% significance level to suggest that [H1 in context].”
    • Never write “H0 is true”, “H0 is proved” or “accept H0” as a definite fact. A test cannot prove H0; it can only fail to find evidence against it.
    Key factsProbabilities of the errors
    • Type I: P(reject H0∣H0 true). This is the probability that the test statistic falls in the critical region when H0 holds, which is the actual significance level.
    • Type II: P(do not reject H0∣H1 true). This is the probability that the test statistic falls in the acceptance region. The question gives a specific value of the parameter that makes H1 true, and you calculate with that value.

    Revise: Hypothesis testing

    Correlation and regression (UK boards)

    Key factsSums of squares and the PMCC

    For n pairs (x,y): Sxx=∑x2−(∑x)2nSyy=∑y2−(∑y)2nSxy=∑xy−∑x∑yn r=SxySxx Syy

    • −1≤r≤1.
    • r=1: all points lie exactly on a line with positive gradient. r=−1: exactly on a line with negative gradient.
    • r near 0: no linear correlation. (There may still be a curved relationship.)
    • r is unchanged if you change the units of x or y (for example cm to m), or add a constant to every value.
    • r says nothing about the gradient of the line: r=0.9 does not mean the gradient is 0.9.
    Key factsRegression line of y on x

    y=a+bxb=SxySxxa=yˉ−bxˉ

    • The line always passes through the mean point (xˉ,yˉ).
    • b is the gradient: for each increase of 1 in x, the predicted y changes by b.
    • a is the intercept: the predicted y when x=0. It may have no sensible meaning if x=0 is outside the data.
    • Use it to predict y from x, where x is the explanatory variable. The line of x on y is a different line and should not be used for that.

    Revise: Correlation and regression (UK boards)

    Forces and equilibrium

    Key factsResolving a force

    A force F at angle θ to a given direction has components

    • Fcos⁡θ along that direction, and
    • Fsin⁡θ perpendicular to it.

    The angle must be measured from the direction you are resolving along. If you are given the angle to the other direction, swap cos⁡ and sin⁡.

    Key factsThe friction law

    F≤μR where R is the normal reaction and μ is the coefficient of friction between the two surfaces. When the object is about to slip (limiting equilibrium) or is sliding, F=μR, and friction acts against the motion (or the direction it would move).

    Key factsWeight on a slope inclined at angle θ
    • Component of weight down the slope: mgsin⁡θ
    • Component of weight into the slope: mgcos⁡θ
    • With no other forces perpendicular to the slope, R=mgcos⁡θ.
    • A block just about to slide down a rough slope, with no other forces, has μ=tan⁡θ.

    Revise: Forces and equilibrium

    Kinematics: describing motion

    Key factsReading a velocity–time graph
    • The gradient is the acceleration.
    • The area under the graph is the displacement. Area above the t-axis counts as positive and area below as negative.
    • Distance travelled is the total of all the areas taken as positive.
    • A horizontal line means constant velocity (zero acceleration). A straight sloping line means constant acceleration.
    Key factsThe constant acceleration equations

    v=u+at s=12(u+v)t s=ut+12at2 v2=u2+2as

    • Use them only when a is constant, in a straight line.
    • All four are on MF19, but you must know when to choose each one.
    • Each equation leaves out one of the five quantities. Pick the one that has the three you know and the one you want.
    Key factsCalculus in kinematics

    v=dsdta=dvdt=d2sdt2 v=∫a dts=∫v dt

    • Each integration needs a constant, found from a given value such as “s=0 when t=0” or “v=5 when t=0”.
    • The object is at rest when v=0. It is at its greatest speed or turns round where a=0 or v=0 respectively, so solve v=0 or a=0 as the question needs.
    • Displacement between two times: ∫t1t2v dt. Distance travelled needs the integral split wherever v changes sign.

    Revise: Kinematics: describing motion

    Newton’s laws and connected particles

    Key factsNewton’s laws
    1. First law. An object stays at rest, or moves in a straight line at constant speed, unless a resultant force acts on it.
    2. Second law. The resultant force on an object equals its mass times its acceleration, in the direction of the resultant: F=ma
    3. Third law. If object A exerts a force on object B, then B exerts an equal and opposite force on A.
    Key factsConnected particles
    • The tension in a light string is the same throughout, including either side of a smooth pulley.
    • The acceleration is the same size for both particles, because the string is inextensible: one goes up as the other comes down.
    • Write F=ma separately for each particle, and solve the equations together. Alternatively, treat the particles as one system when the tension is not needed.

    Revise: Newton’s laws and connected particles

    Momentum and collisions

    Key factsConservation of linear momentum

    In a direct collision (along a straight line) of two particles A and B, with velocities measured in the same positive direction: mAuA+mBuB=mAvA+mBvB total momentum before = total momentum after, where u is the velocity before and v the velocity after.

    • Momentum is a vector, so signs matter. A particle moving in the negative direction has a negative u or v.
    • If the particles coalesce (stick together), they move with a common velocity V afterwards and the mass of the combined body is mA+mB: mAuA+mBuB=(mA+mB)V.
    • Kinetic energy is generally not conserved in a collision, so do not use it as an equation here.

    Revise: Momentum and collisions

    Work, energy and power

    Key factsWork done

    W=Fdcos⁡θ

    • F is the size of the force (N), d the distance moved (m) and θ the angle between the force and the direction of motion.
    • If θ=0∘, W=Fd. If θ=90∘, W=0: a force at right angles to the motion (the normal reaction, or the weight on a horizontal floor) does no work.
    • Work done against a force such as friction or a resistance is Fd (positive energy lost). The work done by the friction force on the object is negative.
    • This formula is not on MF19: learn it.
    Key factsWork–energy principle

    The total work done on a particle equals the change in its kinetic energy. Including the change in potential energy, for motion with driving forces and resistances: work done by driving forces−work done against resistances=ΔKE+ΔGPE that is, energy put in, less energy lost to resistances, equals the energy gained.

    • With no driving force and no resistance (smooth surface, no friction or air resistance), energy is conserved: KE+GPE stays constant.
    • It is the same principle whether the path is straight or curved, because only the change in height matters for GPE.
    Key factsPower, force and speed

    P=Fv

    • P is the power in watts, F is the driving force in newtons and v is the speed in m s−1. 1 kW =1000 W.
    • For a vehicle, F is the driving force of the engine. It is not the resultant force: the resultant force is F minus resistances (and minus the weight component on a slope).
    • Use F=ma with the resultant force to find acceleration: Pv−Rresistance−mgsin⁡θ=ma.
    • At the maximum speed, the acceleration is 0, so the driving force equals the total resistance (plus the weight component on a hill).

    Revise: Work, energy and power

    Projectiles and moments (UK boards)

    Key factsProjectile results (launched from ground level)
    • Time of flight (back to the launch height): T=2usin⁡αg
    • Greatest height: H=u2sin⁡2α2g, reached at t=usin⁡αg
    • Range: R=u2sin⁡2αg, which is greatest when α=45∘
    • Trajectory: y=xtan⁡α−gx22u2cos⁡2α, a parabola
    • Speed at time t: (ucos⁡α)2+(usin⁡α−gt)2; direction tan⁡θ=usin⁡α−gtucos⁡α Do not memorise these without understanding them: in exams, it is safer to build each result from the table.
    Key factsEquilibrium of a rigid body

    A body is in equilibrium when:

    1. the resultant force is zero (forces balance in two perpendicular directions), and
    2. the total moment about any point is zero: clockwise moments = anticlockwise moments (the principle of moments).
    • The weight of a uniform rod acts at its centre.
    • A force acting at the pivot has no moment about that pivot: choose the pivot where an unknown force acts to remove it from the equation.
    • If a force makes an angle with the rod, use the perpendicular component (or perpendicular distance).

    Revise: Projectiles and moments (UK boards)

    Proof: how mathematicians know

    Key factsBuilding blocks for number proofs
    • An even number is 2n, where n is an integer. An odd number is 2n+1.
    • Consecutive integers are n, n+1, n+2, and so on. Two consecutive even numbers are 2n and 2n+2.
    • A multiple of k has the form k×(integer). To show a number is a multiple of k, take a factor k out of the algebra.
    • Use a different letter for each separate number (2a+1 and 2b+1), otherwise you accidentally force them to be equal.
    • End by saying what you have shown, in words.

    Revise: Proof: how mathematicians know

    PreviousGlossaryNextPhysics formula and data sheet
    Toolbox/Formula sheets

    Physics formula and data sheet

    Reference

    Every key formula from the course in one place, each linked back to the chapter that teaches it.

    Cambridge prints a list of data and formulae at the start of every 9702 paper. Knowing what each symbol means and when each equation applies is what earns the marks. This page collects every key box from the Physics chapters.

    Physics from zero

    Key factsStandard form

    a×10nwith 1≤a<10 and n a whole number A large number has a positive n (move the decimal point to the left to get a). A number smaller than 1 has a negative n (move the point to the right).

    Key factsSignificant figures in answers
    • When you multiply or divide, give the answer to the same number of s.f. as the least precise value you used.
    • Carry the full calculator value through every step and round only at the end.
    • Never give fewer than 2 s.f. in a final answer, and always give the unit.
    Key factsMethod for rearranging
    1. Write the formula and underline the quantity you want.
    2. Move the other terms away from it one at a time, undoing the operations in the reverse order to the one that built the formula (undo adding or subtracting first, then multiplying or dividing, then powers or roots).
    3. Do exactly the same to both sides each time.
    4. Substitute numbers last, in base units, and check the unit of your answer.
    Key factsGradient

    m=ΔyΔx=y2−y1x2−x1 Draw the gradient triangle as large as the line allows: use two points on the line that are far apart (more than half of its length), not two points from a table. The gradient has a unit: the unit of y divided by the unit of x.

    FormulaPythagoras and the three ratios

    c2=a2+b2sin⁡θ=opphypcos⁡θ=adjhyptan⁡θ=oppadj c is the hypotenuse, a and b the other two sides. The memory aid is SOH CAH TOA. Put your calculator in degree mode. To find an angle, use the inverse: θ=sin⁡−1(opp/hyp).

    FormulaResolving a force

    Fx=Fcos⁡θFy=Fsin⁡θ θ is the angle between the force and the horizontal direction of Fx. If the angle is given to the other direction, sin⁡ and cos⁡ swap. Always draw the triangle and use SOH CAH TOA, rather than remembering which is which.

    FormulaAreas and volumes

    Circle: A=πr2=πd24C=2πr Sphere: A=4πr2V=43πr3 Cylinder: V=πr2hcurved area=2πrh Rectangular block: V=l×w×h r: radius · d: diameter =2r · h: height or length · volume of any prism or cylinder is cross-sectional area × length.

    FormulaThe wave equation

    v=fλ v: wave speed (m s−1) · f: frequency (Hz) · λ: wavelength (m). In a transverse wave the oscillations are at right angles to the direction the wave travels (light, water ripples). In a longitudinal wave they are parallel to it (sound).

    Revise: Physics from zero

    Physical quantities, units and uncertainty

    Key factsWhat homogeneity can and cannot tell you

    It can show that an equation is wrong: if the two sides have different base units, the equation cannot be correct. It can also find the unit of an unknown constant, and it can suggest the form of an equation.

    It cannot prove an equation is right. A homogeneous equation can still be false, because the test cannot see a number or a missing factor. The equation s=ut+at2 is homogeneous, but the true equation has 12at2. The test also cannot see the difference between two quantities with the same unit, such as work and torque.

    Arguments of functions such as sin⁡, ln⁡ and ex must have no unit, so in N=N0e−λt the product λt must be a pure number.

    FormulaAbsolute, fractional and percentage uncertainty

    fractional uncertainty=Δxxpercentage uncertainty=Δxx×100% x: the measured value · Δx: the absolute uncertainty, in the same unit as x. The fractional and percentage uncertainties have no unit.

    Key factsHow to write a final answer

    Give the uncertainty to one significant figure (two at most), and give the value to the same number of decimal places as its uncertainty.

    (2.70±0.05) g cm−3 is correct. (2.7043±0.05) g cm−3 has too many figures in the value, and (2.7±0.0512) g cm−3 has too many in the uncertainty. Both lose marks.

    FormulaCombining uncertainties

    Sum or difference z=x+y or z=x−y: add the absolute uncertainties. Δz=Δx+Δy.

    Product or quotient z=xy or z=x/y: add the percentage (or fractional) uncertainties. Δzz=Δxx+Δyy.

    Power z=xn: multiply the percentage uncertainty by ∣n∣. Δzz=∣n∣Δxx.

    A constant such as π or 2 has no uncertainty. A quantity in a product that is repeated (x and x again) is the same as a power, so the percentage is counted twice.

    Key factsComponents

    A vector of magnitude F at angle θ to a chosen direction has components F∥=Fcos⁡θ(along that direction)F⊥=Fsin⁡θ(at 90° to it) The component along the direction from which θ is measured uses cosine: the cosine is the side next to the angle. If θ is measured from the other direction, the cosine and sine swap. The vector and its components form a right-angled triangle with the vector as the hypotenuse, so F2=F∥2+F⊥2.

    Revise: Physical quantities, units and uncertainty

    Kinematics

    Key factsSigns and “deceleration”

    Acceleration is positive if it points in the positive direction, whatever the object is doing. An object slows down when its acceleration points opposite to its velocity. So a car moving in the positive direction while braking has a negative acceleration, and a car moving in the negative direction while braking has a positive one. “Deceleration” just means “acceleration opposite to the velocity”. In calculations, write a=−5.0 m s−2 and not “a deceleration of 5.0”.

    Key factsGraphs of motion
    Graph Gradient Area under graph
    displacement–time (s–t) velocity not used
    velocity–time (v–t) acceleration displacement
    acceleration–time (a–t) not used change in velocity

    Gradient = rise over run = (change in the y quantity) / (change in the x quantity). On a curve, use the gradient of the tangent at the point, which gives the instantaneous value.

    FormulaEquations of uniformly accelerated motion

    v=u+ats=12(u+v)ts=ut+12at2v2=u2+2as s: displacement (m) · u: initial velocity (m s−1) · v: final velocity (m s−1) · a: acceleration (m s−2) · t: time (s). These apply only to motion in a straight line with constant acceleration. The data sheet gives s=ut+12at2 and v2=u2+2as; you must remember the other two, or derive them.

    Key factsThe two independent motions

    Horizontal: no force acts horizontally, so the horizontal velocity is constant. sx=uxt.

    Vertical: gravity gives a constant downward acceleration g, so the vertical motion is free fall. Use the equations for constant acceleration with a=±9.81 m s−2.

    The two motions are independent and share only the time. Find the time from the direction where you know the most, then use it in the other.

    Revise: Kinematics

    Dynamics and momentum

    FormulaWeight

    W=mg W: weight (N) · m: mass (kg) · g: acceleration of free fall, 9.81 m s−2 on the Earth (the data-sheet value).

    FormulaNewton’s second law (constant mass)

    F=ma F: resultant force (N) · m: mass (kg) · a: acceleration (m s−2). The acceleration is always in the same direction as the resultant force.

    FormulaMomentum

    p=mv p: momentum (kg m s−1, which is the same as N s) · m: mass (kg) · v: velocity (m s−1). Momentum is a vector: it has the direction of the velocity, so its sign matters.

    FormulaForce and momentum

    F=ΔpΔt=Δ(mv)Δt F: resultant force (N) · Δp: change of momentum (kg m s−1) · Δt: time over which it changes (s).

    FormulaForce from a flow of mass

    F=ΔmΔt v Δm/Δt: mass arriving per second (kg s−1) · v: the change of velocity of that mass (m s−1), here from v to zero.

    FormulaConservation of momentum (two bodies, one dimension)

    m1u1+m2u2=m1v1+m2v2 m1,m2: masses (kg) · u1,u2: velocities before (m s−1) · v1,v2: velocities after. Directions matter: use + and − signs.

    Key factsElastic collision in one dimension

    Total kinetic energy before = total kinetic energy after, and

    relative speed of approach=relative speed of separation u1−u2=v2−v1 Use momentum together with this relative-speed equation: two linear equations are much easier than momentum together with a quadratic kinetic-energy equation.

    Revise: Dynamics and momentum

    Forces, density and pressure

    FormulaMoment of a force

    moment=F×d⊥ F: force (N) · d⊥: perpendicular distance from the pivot to the line of action of the force (m). The unit of moment is the newton metre (N m). A moment is clockwise or anticlockwise.

    FormulaTorque of a couple

    torque=F×d F: size of one of the two forces (N) · d: perpendicular distance between their lines of action (m). The unit is N m.

    Key factsThree coplanar forces in equilibrium

    Draw the three force vectors head to tail, each to scale and in the correct direction. If the forces are in equilibrium, the triangle closes. You can then find unknown sides and angles by scale drawing, or by the sine rule and cosine rule, or by simple trigonometry if the triangle is right-angled.

    Three non-parallel forces in equilibrium also have lines of action that meet at one point. If they did not, the forces would have a resultant moment.

    FormulaDensity

    ρ=mV ρ: density (kg m−3) · m: mass (kg) · V: volume (m3).

    FormulaPressure

    p=FA p: pressure (pascal, 1 Pa=1 N m−2) · F: force normal to the surface (N) · A: area (m2).

    FormulaHydrostatic pressure

    Δp=ρgΔh Δp: increase of pressure with depth (Pa) · ρ: density of the fluid (kg m−3) · g: 9.81 m s−2 · Δh: the vertical depth difference (m).

    FormulaUpthrust (Archimedes’ principle)

    F=ρgV F: upthrust (N) · ρ: density of the fluid (kg m−3) · V: volume of fluid displaced, which is the submerged volume of the object (m3).

    Revise: Forces, density and pressure

    Work, energy and power

    FormulaWork done

    W=Fscos⁡θ W: work done (J) · F: force (N) · s: displacement (m) · θ: angle between the force and the displacement. Fcos⁡θ is the component of the force along the displacement.

    FormulaChange in gravitational potential energy

    ΔEP=mgΔh ΔEP: change in GPE (J) · m: mass (kg) · g: acceleration of free fall, 9.81 m s−2 · Δh: change in vertical height (m). Valid only where g is uniform, which is true near the Earth’s surface. For large changes in height, see Gravitational fields.

    FormulaKinetic energy

    EK=12mv2 EK: kinetic energy (J) · m: mass (kg) · v: speed (m s−1). Energy is a scalar, so it has no direction.

    FormulaKinetic energy from momentum

    EK=12m(pm)2=p22m p: momentum (kg m s−1) · m: mass (kg). For the same momentum, the lighter object has the larger kinetic energy.

    FormulaEfficiency

    efficiency=useful energy outputtotal energy input=useful power outputtotal power input Give it as a ratio (0 to 1) or multiply by 100 for a percentage. It has no unit.

    FormulaPower

    P=Wt=ΔEt P: power (W) · W: work done or energy transferred (J) · t: time (s). One watt is one joule per second: 1 W=1 J s−1.

    FormulaPower and velocity

    P=Fv P: power (W) · F: force in the direction of motion (N) · v: speed (m s−1). At a changing speed, Fv gives the power at that instant.

    Revise: Work, energy and power

    Deformation of solids

    FormulaHooke’s law and the spring constant

    F=kx F: load (N) · x: extension or compression (m) · k: spring constant (N m−1). The graph of F against x is a straight line through the origin, with gradient k.

    FormulaStress, strain and the Young modulus

    σ=FAε=xLE=σε=FLAx F: tensile force (N) · A: cross-sectional area (m2) · x: extension (m) · L: original length (m) · E: Young modulus (Pa). For a wire of diameter d, A=πd2/4.

    FormulaElastic potential energy

    EP=12Fx=12kx2 EP: elastic potential energy (J) · F: final force (N) · x: extension (m) · k: spring constant (N m−1). Valid only within the limit of proportionality. For any other graph, find the area under it.

    Revise: Deformation of solids

    Waves

    FormulaPeriod and frequency

    f=1T f: frequency (Hz) · T: period (s). A source of frequency 50 Hz has T=1/50=0.020 s, and a period of 5.0 ms means f=200 Hz.

    FormulaPhase difference

    ϕ=xλ×360∘orϕ=2πxλ ϕ: phase difference between two points on a wave (° or rad) · x: their separation along the wave (m) · λ: wavelength (m). In the same way, two oscillations of period T that reach their crests a time Δt apart have ϕ=(Δt/T)×360∘.

    FormulaThe wave equation

    v=fλ v: wave speed (m s⁻¹) · f: frequency (Hz) · λ: wavelength (m). This holds for every progressive wave.

    FormulaIntensity

    I=PA I: intensity (W m⁻²) · P: power carried by the wave (W) · A: area at right angles to the direction of travel (m²).

    Key factsIntensity and amplitude

    I∝A2 A here is the amplitude of the wave. If the amplitude is multiplied by 3, the intensity is multiplied by 9. Equivalently, the amplitude is proportional to I.

    FormulaIntensity from a point source

    I=P4πr2 P: power radiated by the source (W) · r: distance from the source (m). So I∝1/r2: the inverse square law. This assumes none of the energy is absorbed on the way.

    FormulaThe Doppler effect for sound

    fo=fs vv±vs fo: observed frequency (Hz) · fs: frequency of the source (Hz) · v: speed of sound (m s⁻¹) · vs: speed of the source (m s⁻¹). Use v−vs (minus) when the source moves towards the observer, and v+vs (plus) when it moves away.

    FormulaMalus’s law

    I=I0cos⁡2θ I0: intensity of the plane-polarised light incident on the filter (W m⁻²) · I: intensity transmitted (W m⁻²) · θ: angle between the plane of polarisation of the incident light and the transmission axis of the filter.

    Revise: Waves

    Superposition

    Key factsConditions for interference at a point (sources in phase)

    constructive: Δx=nλdestructive: Δx=(n+12)λ Δx: path difference (m) · λ: wavelength (m) · n=0,1,2,… A path difference of 12λ,112λ,212λ… gives destructive superposition.

    FormulaStationary waves on a string

    L=nλ2⇒λ=2Lnandfn=vλ=nv2L L: length of the string (m) · n: number of loops = 1, 2, 3, … (the harmonic number) · v: speed of the wave on the string (m s⁻¹). The resonant frequencies are whole-number multiples of the lowest frequency f1.

    FormulaResonance in pipes

    closed at one end: L=λ4, 3λ4, 5λ4,…open at both ends: L=λ2, λ, 3λ2,… A closed pipe resonates only at odd multiples of its lowest frequency: f1,3f1,5f1,… An open pipe resonates at f1,2f1,3f1,…

    FormulaDouble-slit interference

    λ=axDorx=λDa λ: wavelength of the light (m) · a: separation of the centres of the two slits (m) · x: fringe separation on the screen (m) · D: distance from the slits to the screen (m). Valid for D≫a (small angles).

    FormulaThe diffraction grating

    dsin⁡θ=nλd=1N d: distance between adjacent slits (m) · θ: angle between the straight-through direction and the nth order · n: order of the maximum (0, 1, 2, …) · λ: wavelength (m) · N: number of lines per unit length (m⁻¹). The highest order that can be seen is the largest whole number with n<d/λ, because sin⁡θ cannot be more than 1.

    Revise: Superposition

    Electricity

    FormulaCharge and current

    Q=It Q: charge (C) · I: current (A) · t: time (s). Convert minutes and hours to seconds, and mA and µA to A, before you substitute.

    Key factsQuantisation of charge

    Q=ne(n=0,±1,±2,… ) Any charge you can isolate is a whole number of times e. (Quarks carry ±13e and ±23e, but they are never found on their own: see Particle physics.)

    FormulaCurrent and drift speed

    I=Anvq I: current (A) · A: cross-sectional area (m2) · n: number density of charge carriers (m−3) · v: mean drift speed (m s−1) · q: charge on each carrier (C). For electrons, q=e=1.60×10−19 C.

    FormulaPotential difference and energy

    V=WQsoW=QV V: p.d. (V) · W: energy transferred (J) · Q: charge (C). The source that provides the energy has an e.m.f., which is defined in D.C. circuits.

    FormulaElectrical power

    P=VI=I2R=V2R P: power (W) · V: p.d. across the component (V) · I: current in it (A) · R: its resistance (Ω). Use I2R when the current is known (series circuits) and V2/R when the p.d. is known (parallel circuits and mains).

    FormulaResistance

    R=VIsoV=IR R: resistance (Ω) · V: p.d. across the component (V) · I: current in it (A).

    FormulaResistivity

    R=ρLAsoρ=RAL R: resistance (Ω) · ρ: resistivity (Ω m) · L: length (m) · A: cross-sectional area (m2). For a round wire, A=πd2/4.

    Revise: Electricity

    D.C. circuits

    FormulaE.m.f. and internal resistance

    E=I(R+r)V=E−Ir E: e.m.f. (V) · I: current (A) · R: external (load) resistance (Ω) · r: internal resistance (Ω) · V=IR: terminal p.d. (V). The term Ir is often called the lost volts.

    FormulaCombined resistance

    R=R1+R2+…(series)1R=1R1+1R2+…(parallel) For two resistors in parallel, R=R1R2R1+R2 (“product over sum”). For n identical resistors R in parallel, the total is R/n.

    FormulaThe potential divider

    Vout=R2R1+R2×VinV1V2=R1R2 Vin: p.d. across the whole divider · Vout: p.d. across R2 · V1,V2: p.d.s across R1,R2. This assumes nothing connected to the output draws a significant current.

    FormulaComparing e.m.f.s with a potentiometer

    E1E2=l1l2 E1,E2: e.m.f.s of the two cells · l1,l2: their balance lengths, measured from A on the same wire with the same driver current.

    Revise: D.C. circuits

    Particle physics

    Key factsTwo conservation rules for every nuclear process

    Nucleon number is conserved: the top numbers on each side add up to the same total. Charge is conserved: the bottom numbers (proton number, or charge in units of e) add up to the same total.

    Key factsBeta decay at the quark level

    β− decay: d→u+−1  0e+νˉ, so a neutron (udd) becomes a proton (uud). β+ decay: u→d++1  0e+ν, so a proton (uud) becomes a neutron (udd).

    Revise: Particle physics

    Motion in a circle

    FormulaRadians, degrees and arc length

    2π rad=360∘θ/rad=θ/∘×π180s=rθ s: arc length (m) · r: radius (m) · θ: angle at the centre (in radians). The equation s=rθ is the definition of the radian rewritten: an angle of θ radians cuts off an arc of θ radius-lengths.

    FormulaAngular speed, period and linear speed

    ω=2πT=2πfv=rω ω: angular speed (rad s−1) · T: period (s) · f: frequency (Hz) · v: linear speed along the circle (m s−1) · r: radius of the circle (m).

    FormulaCentripetal acceleration

    a=v2r=rω2(=vω) a: centripetal acceleration (m s−2), directed towards the centre · v: speed · r: radius · ω: angular speed. Which form to use depends on what you are given: v with r, or ω with r.

    FormulaCentripetal force

    F=mv2r=mrω2 F: resultant force towards the centre (N) · m: mass (kg) · v: speed (m s−1) · r: radius (m) · ω: angular speed (rad s−1). The force is always directed towards the centre.

    FormulaConical pendulum

    vertical: Tcos⁡θ=mghorizontal: Tsin⁡θ=mrω2⇒tan⁡θ=rω2g Divide the two equations to remove T and m. The radius is r=Lsin⁡θ for a string of length L.

    FormulaVertical circle (ball of mass m on a string, radius r)

    bottom: Tb−mg=mvb2rtop: Tt+mg=mvt2r energy: 12mvb2=12mvt2+mg(2r)⇒vb2=vt2+4gr The tension is largest at the bottom and smallest at the top. The string stays taut only if Tt≥0, which needs vt2≥gr. So the minimum speed at the top is vt=gr, and the minimum speed at the bottom for the ball to complete the circle is vb=5gr.

    Revise: Motion in a circle

    Gravitational fields

    FormulaNewton’s law of gravitation

    F=Gm1m2r2 F: force on each mass (N) · G: gravitational constant, 6.67×10−11 N m2 kg−2 · m1,m2: the two masses (kg) · r: distance between the centres (m).

    FormulaField strength of a point mass or uniform sphere

    g=GMr2 g: gravitational field strength (N kg−1) · M: mass producing the field (kg) · r: distance from the centre of M (m), directed towards the centre. Outside the planet, g∝1/r2.

    FormulaCircular orbit

    GMmr2=mv2r=mrω2⇒v=GMr M: mass of the body at the centre · m: mass of the orbiting object · r: orbital radius, from the centre of M. The satellite’s mass cancels: all satellites at the same radius move at the same speed, whatever their mass. A larger orbit means a smaller speed.

    FormulaGravitational potential of a point mass

    ϕ=−GMr ϕ: gravitational potential (J kg−1) · M: mass producing the field (kg) · r: distance from its centre (m). Potential is negative everywhere, rises towards zero as r increases, and is zero at infinity.

    FormulaChange in gravitational potential energy

    ΔEP=m Δϕ Δϕ: the potential at the final point minus the potential at the starting point (J kg−1). If the mass moves to a higher potential (further out), Δϕ>0 and the mass gains energy.

    FormulaGravitational potential energy of two point masses

    EP=mϕ=−GMmr EP: gravitational potential energy (J) · M,m: the two masses (kg) · r: distance between their centres (m). It is negative, with zero at infinite separation, and it increases (becomes less negative) as the masses move apart.

    FormulaEnergies of a satellite in a circular orbit

    EK=GMm2rEP=−GMmrEtotal=EK+EP=−GMm2r Derived from GMmr2=mv2r, which gives mv2=GMmr and so EK=12mv2=GMm2r. Notice EK=−12EP, and the total energy is negative: the satellite is bound to the Earth.

    Revise: Gravitational fields

    Temperature

    Key factsDirection of energy transfer

    Thermal energy is transferred from a region of higher temperature to a region of lower temperature. Energy does not flow the other way by itself, however much more total energy the cooler body holds.

    Key factsThermometric properties

    Any physical property that varies with temperature can be used to measure it. The syllabus names four: the density of a liquid (equivalently its volume, which is the length of the liquid in a narrow tube), the volume of a gas at constant pressure, the resistance of a metal, and the e.m.f. of a thermocouple. A thermistor, a semiconductor whose resistance changes sharply with temperature, works in the same way.

    FormulaConverting between kelvin and degrees Celsius

    T/K=θ/∘C+273.15 T: thermodynamic temperature (K) · θ: Celsius temperature (∘C). Zero kelvin is −273.15 ∘C. A temperature change is the same number in both units: ΔT=Δθ. Never add 273 to a temperature change.

    FormulaEnergy to change the temperature

    E=mc Δθ E: energy transferred to the substance (J) · m: mass (kg) · c: specific heat capacity (J kg−1 K−1) · Δθ: rise in temperature (K or °C). For a falling temperature, Δθ is negative and energy leaves the substance.

    FormulaEnergy for a change of state

    E=mL E: energy transferred (J) · m: mass that changes state (kg) · L: specific latent heat (J kg−1), either Lf (fusion) or Lv (vaporisation). The same energy is given out when the change is reversed (freezing, condensing).

    Revise: Temperature

    Ideal gases

    FormulaEquation of state of an ideal gas

    pV=nRTpV=NkT p: pressure (Pa) · V: volume (m3) · n: amount of gas (mol) · R: molar gas constant 8.31 J K−1 mol−1 · T: thermodynamic temperature (K) · N: number of molecules · k: Boltzmann constant 1.38×10−23 J K−1. The two forms are linked by k=R/NA.

    Key factsBasic assumptions of the kinetic theory of an ideal gas
    1. A gas contains a very large number of molecules, in continuous random motion.
    2. The volume of the molecules is negligible compared with the volume of the container.
    3. There are no forces between molecules, except during a collision.
    4. Collisions between molecules, and with the walls, are perfectly elastic.
    5. The time of a collision is negligible compared with the time between collisions.
    6. Newton’s laws of motion apply to the molecules.
    FormulaPressure of an ideal gas from the kinetic theory

    pV=13Nm⟨c2⟩p=13ρ⟨c2⟩ N: number of molecules · m: mass of one molecule · ⟨c2⟩: mean-square speed (m2 s−2) · ρ=Nm/V: density of the gas.

    FormulaMean translational kinetic energy of a molecule

    ⟨EK⟩=12m⟨c2⟩=32kTcr.m.s.=3kTm T: thermodynamic temperature (K) · m: mass of one molecule (kg) · k=1.38×10−23 J K−1. The mean kinetic energy per molecule depends only on T, not on the gas. For n moles the total translational kinetic energy is 32nRT.

    Revise: Ideal gases

    Thermodynamics

    FormulaInternal energy of a monatomic ideal gas

    U=32NkT=32nRT=32pV U: internal energy (J) · N: number of molecules · n: amount of gas (mol) · T: thermodynamic temperature (K) · p: pressure (Pa) · V: volume (m³). U depends only on T for a given amount of gas: ΔU=32nR ΔT. The factor 32 is for a monatomic gas such as helium or argon. Gases of two-atom molecules also have rotational energy, so their U is larger, but it is still proportional to T.

    FormulaWork done by a gas at constant pressure

    work done by the gas=p ΔVW=−p ΔV p: pressure (Pa) · ΔV: increase in volume (m³) · W: work done on the gas (J), the quantity used in the first law below. When the gas expands, ΔV>0, the gas does work and W is negative. When the gas is compressed, ΔV<0 and W is positive.

    Key factsThe area under a p–V graph is the work

    The work done by a gas as its volume changes equals the area under the p–V graph between the initial and final volumes. For an expansion the gas does work; for a compression the area is the work done on the gas. With p in kPa and V in litres, the area comes out in joules.

    FormulaThe first law (Cambridge form)

    ΔU=q+W ΔU: increase in internal energy (J) · q: energy transferred to the system by heating (J) · W: work done on the system (J). For a gas at constant pressure W=−p ΔV.

    Key factsWork and cycles

    On a p–V graph, the net work in a cycle is the area enclosed by the loop. If the loop is traced clockwise the gas does net work on its surroundings (a heat engine, as in a car). If it is traced anticlockwise, net work is done on the gas and energy is pumped out by heating (a refrigerator). Over one cycle, ΔU=0, so net q = −(net W).

    Revise: Thermodynamics

    Oscillations

    FormulaPeriod, frequency and angular frequency

    f=1Tω=2πf=2πT T: period (s) · f: frequency (Hz) · ω: angular frequency (rad s−1).

    FormulaPhase difference from a time lag

    Δϕ=2π ΔtT(radians)Δϕ=360∘×ΔtT Δt: time by which one oscillator lags the other · T: their common period. A phase difference of 0 means in phase; π rad (180°) means in antiphase (they always move in opposite directions).

    Key factsThe defining equation of SHM

    a=−ω2x a: acceleration (m s−2) · x: displacement from equilibrium (m) · ω: angular frequency (rad s−1). The minus sign says that a points opposite to x. The constant of proportionality is written ω2 because it turns out to equal the square of the angular frequency of the motion.

    FormulaPeriods of a spring and a pendulum

    T=2πmkT=2πlg m: mass (kg) · k: force constant of the spring (N m−1) · l: pendulum length to the centre of the bob (m) · g: acceleration of free fall. The pendulum result is only valid for small angles. These two formulae are not on the Cambridge data sheet: a question that needs them gives them, or asks you to derive ω2 as above.

    Key factsDisplacement, velocity and acceleration in SHM

    x=x0sin⁡ωtv=v0cos⁡ωta=−ω2x0sin⁡ωt=−ω2x v0=ωx0amax⁡=ω2x0 x0: amplitude (m) · v0: maximum speed (m s−1) · ωt is an angle in radians. The velocity is the gradient of the x–t graph and the acceleration is the gradient of the v–t graph.

    FormulaSpeed at displacement x

    v=± ωx02−x2 The ± gives the two directions of travel through the point. The speed is greatest (ωx0) at x=0 and zero at x=±x0. Do not write v=ωx: the speed is not proportional to the displacement (it is zero at x=±x0 and greatest at x=0).

    Key factsEnergy in SHM

    EK=12mv2=12mω2(x02−x2)EP=12mω2x2E=EK+EP=12mω2x02 E: total energy (J) · it depends on the mass, on the square of the amplitude and on the square of the frequency. For a mass on a spring k=mω2, so EP=12kx2 and E=12kx02, as in Deformation of solids. Also E=12mv02.

    Revise: Oscillations

    Electric fields

    FormulaElectric field strength and the force on a charge

    E=FqF=qE E: electric field strength (N C−1 or V m−1) · F: force on the charge (N) · q: the charge (C), with its sign.

    FormulaUniform field between parallel plates

    E=ΔVΔd E: field strength (V m−1) · ΔV: p.d. between the plates (V) · Δd: separation of the plates (m). The field points from the plate at higher potential (positive) to the plate at lower potential (negative).

    Key factsEnergy gained in an electric field

    qV=12mv2−12mu2 For a particle starting from rest, qV=12mv2, so v=2qV/m. This works for any path between two points with p.d. V, but only if you know the p.d. If you know the distance instead, use a=qE/m and the equations of motion.

    Key factsDeflection in a uniform field

    y=qE2m(Lv)2=qEL22mv2 y: deflection after travelling a distance L along the plates (m) · v: the constant speed along the plates. The deflection is proportional to the field strength, so doubling the p.d. doubles y.

    FormulaCoulomb’s law

    F=Q1Q24πε0r2 F: force on each charge (N) · Q1,Q2: the charges (C) · r: separation of the charges (m) · ε0: permittivity of free space, 8.85×10−12 F m−1. The constant 14πε0=8.99×109 m F−1. The formula is for point charges in free space (air is close enough).

    FormulaElectric field strength of a point charge

    E=Q4πε0r2 Q: the charge producing the field (C) · r: distance from the point charge, or from the centre of a charged sphere (m). The field points away from a positive Q and towards a negative Q. It is an inverse-square law: doubling r divides E by 4.

    FormulaElectric potential of a point charge

    V=Q4πε0r V: electric potential (V, or J C−1) · Q: the charge producing the field, with its sign (C) · r: distance from the point charge or from the centre of a charged sphere (m). The potential is zero at infinity.

    FormulaWork done in moving a charge

    W=q ΔV=q(V2−V1) W: work done on the charge by the external force (J) · q: the charge, with its sign (C). It is also the increase in the charge’s electric potential energy. A negative answer means the field does the work and the charge’s kinetic energy increases if it is free to move.

    FormulaField strength and potential gradient

    E=−ΔVΔxor, at a point,E=−dVdx E: field strength in the direction of x (V m−1) · ΔV/Δx: the gradient of a graph of V against distance. The minus sign means the field points in the direction in which the potential falls.

    FormulaElectric potential energy of two point charges

    EP=qV=Qq4πε0r EP: electric potential energy of the pair (J) · Q, q: the two charges, with signs · r: their separation (m). Zero at infinite separation. Positive for like charges (they repel, so work was done to bring them together). Negative for unlike charges (they attract).

    Revise: Electric fields

    Capacitance

    FormulaCapacitance

    C=QV C: capacitance (F) · Q: charge on one plate (C) · V: p.d. across the capacitor (V). Rearranged: Q=CV and V=Q/C.

    FormulaCombined capacitance

    C=C1+C2+…(parallel)1C=1C1+1C2+…(series) Both are on the data sheet. In parallel the p.d. is shared by all, and the charges add. In series the charge is the same on all, and the p.d.s add. For two capacitors in series, C=C1C2C1+C2. For n identical capacitors C in series, the total is C/n.

    FormulaEnergy stored in a capacitor

    W=12QV=12CV2=Q22C W: energy stored (J) · Q: charge (C) · V: p.d. (V) · C: capacitance (F). Use 12CV2 when you know C and V, and Q2/2C when the charge is fixed.

    Key factsHalf the energy is wasted when you charge through a resistor

    When a supply of p.d. V charges a capacitor through a resistor, the supply transfers energy QV=CV2. The capacitor stores only 12CV2. The other half is dissipated in the resistor as thermal energy, whatever the value of R. For the 470 μF capacitor at 12 V: the supply transfers 470×10−6×144=6.77×10−2 J, the capacitor stores 3.38×10−2 J, and 3.38×10−2 J heats the resistor and wires.

    FormulaDischarge of a capacitor

    x=x0 e−t/RC x can be the charge Q, the p.d. V across the capacitor or the current I; x0 is its value at t=0; R is the resistance in the discharge circuit (Ω); C is the capacitance (F); t is the time since discharge began (s). The equation is on the data sheet.

    Revise: Capacitance

    Magnetic fields

    FormulaForce on a current-carrying conductor

    F=BILsin⁡θ F: force (N) · B: magnetic flux density (T) · I: current (A) · L: length of wire in the field (m) · θ: angle between the wire and the field direction. For a wire at right angles to the field, F=BIL.

    FormulaForce on a moving charge

    F=BQvsin⁡θ F: force (N) · B: magnetic flux density (T) · Q: charge (C) · v: speed (m s−1) · θ: angle between the velocity and the field. For a charge moving at right angles to the field, F=BQv.

    Key factsCharged particle moving at right angles to a uniform magnetic field

    r=mvBQT=2πmBQ The path is a circle at constant speed. A faster particle moves on a larger circle but takes the same time to go round, and a heavier particle moves on a larger circle than a lighter one at the same speed. Compare with a charge entering an electric field, which follows a parabola and gains speed.

    FormulaThe Hall voltage

    VH=BIntq VH: Hall voltage (V) · B: flux density (T) · I: current (A) · n: number density of charge carriers (m−3) · t: thickness of the slab, measured along the field (m) · q: charge on each carrier (C).

    Key factsForces between parallel currents

    Currents in the same direction attract. Currents in opposite directions repel. The force on each wire is equal in size and opposite in direction to the force on the other, even if the currents are different, because the two forces are an interaction pair under Newton’s third law. Each wire sits in the field of the other.

    Revise: Magnetic fields

    Electromagnetic induction

    FormulaMagnetic flux

    Φ=BAΦ=BAcos⁡θ Φ: magnetic flux (Wb) · B: magnetic flux density (T) · A: area of the surface (m2) · θ: angle between the field and the normal (the line at right angles) to the surface. Φ=BA holds when the surface is at right angles to the field (θ=0). The second form is the general one: only the part of the area at right angles to B, which is Acos⁡θ, counts.

    Key factsWhat the experiments show
    1. An e.m.f. is induced in a circuit only while the magnetic flux linkage is changing. A steady flux, however large, induces nothing. In experiment 3 nothing moves: the changing current changes the field, and so the flux through the second coil.
    2. The e.m.f. is induced whether the flux changes because the magnet moves, the coil moves, the field changes strength, or the coil turns.
    3. The direction of the e.m.f. reverses when the change reverses (pushing in against pulling out, switching on against switching off).
    4. The size of the e.m.f. increases when the flux changes faster, when there are more turns, and when the field is stronger or the area larger.
    FormulaFaraday’s and Lenz’s laws combined

    E=−Δ(NΦ)ΔtorE=−d(NΦ)dt E: induced e.m.f. (V) · NΦ: flux linkage (Wb) · t: time (s). The size is ∣E∣=Δ(NΦ)/Δt, which is the gradient of a graph of flux linkage against time. The minus sign is Lenz’s law: the induced e.m.f. opposes the change. For constant N, E=−N ΔΦ/Δt. If the flux linkage changes at a steady rate, Δ(NΦ)/Δt is the e.m.f. itself; if the rate varies, it gives the average e.m.f. over the interval.

    FormulaE.m.f. of a rod moving through a uniform field

    E=BLv E: induced e.m.f. (V) · B: flux density (T) · L: length of rod in the field (m) · v: speed (m s−1), with B, L and v all at right angles to each other. If the rod is in a closed circuit of resistance R, the current is I=E/R=BLv/R.

    Key factsThe area under an e.m.f.–time graph

    Since E Δt=−Δ(NΦ), the area under an e.m.f.–time graph is minus the change in flux linkage, so it does not depend on how fast the change happens. In the graph above, each pulse has an area of 0.15×0.20=0.030 V s, which is the whole flux linkage of the coil. (Note that 1 V s=1 Wb.)

    Revise: Electromagnetic induction

    Alternating currents

    FormulaA sinusoidal alternating current or voltage

    V=V0sin⁡ωtI=I0sin⁡ωtω=2πf=2πT V,I: the value at time t · V0,I0: peak values · ω: angular frequency (rad s−1). The calculator must be in radian mode. For a current through a resistor, I=V/R at every instant, so the current is in phase with the voltage.

    FormulaMean power in a resistive load

    ⟨P⟩=12P0whereP0=I02R=V02R=I0V0 For a sinusoidal alternating current in a resistor, the mean power is half the maximum (peak) power. ⟨P⟩ means “mean value of P”.

    FormulaR.m.s. values and mean power

    Ir.m.s.=I02Vr.m.s.=V02⟨P⟩=Ir.m.s.Vr.m.s.=Ir.m.s.2R=Vr.m.s.2R 1/2=0.707. These apply to a sinusoidal alternating current or voltage only. With r.m.s. values, all the d.c. power formulae work for the mean power.

    FormulaRipple on a smoothed supply

    ΔV≈V0 tRC=ItC(t≪RC) ΔV: ripple (V) · V0: peak p.d. (V) · R: load resistance (Ω) · C: capacitance (F) · I: load current (A) · t: time between peaks, T for half-wave and T/2 for full-wave (s). The ripple gets smaller as RC gets larger.

    Revise: Alternating currents

    Quantum physics

    FormulaEnergy of a photon

    E=hf=hcλ E: energy of one photon (J) · h: the Planck constant, 6.63×10−34 J s · f: frequency (Hz) · c: speed of light, 3.00×108 m s−1 · λ: wavelength in a vacuum (m). Photon energy depends only on frequency: higher frequency (shorter wavelength) means more energy per photon.

    Key factsIntensity of a photon beam

    P=N hfI=PA=NA hf N: number of photons per second · N/A: the photon flux, the number of photons crossing unit area each second. For a fixed frequency, doubling the intensity means doubling the number of photons per second, not the energy of each one.

    FormulaElectronvolt conversion

    1 eV=1.60×10−19 J To change eV into J, multiply by 1.60×10−19. To change J into eV, divide by it. Prefixes work as usual: 1 keV=103 eV and 1 MeV=106 eV.

    FormulaMomentum of a photon

    p=Ec=hfc=hλ p: momentum (kg m s⁻¹) · E: photon energy (J) · c: speed of light · h: Planck constant · λ: wavelength (m). A shorter wavelength means a larger momentum.

    Key factsObservations of photoelectric emission
    1. Photoelectrons are emitted only if the frequency of the light is at least a certain minimum value, the threshold frequency, which depends on the metal. Below it, no electrons are emitted, however intense the light and however long it shines.
    2. Emission begins immediately, with no measurable delay, even for very dim light above the threshold.
    3. The maximum kinetic energy of the photoelectrons increases with the frequency of the light but does not depend on its intensity.
    4. Above the threshold, the number of photoelectrons emitted per second, and so the photoelectric current, is proportional to the intensity.
    FormulaThe photoelectric equation

    hf=Φ+12mvmax⁡2 hf: energy of the photon (J) · Φ: work function of the metal (J) · 12mvmax⁡2: maximum kinetic energy of a photoelectron (J). Energy is conserved: photon energy = energy to escape + kinetic energy.

    FormulaThreshold frequency and wavelength

    f0=Φhλ0=cf0=hcΦ Light with f≥f0 (equivalently λ≤λ0) ejects electrons. A larger work function means a higher threshold frequency and a shorter threshold wavelength.

    FormulaThe de Broglie wavelength

    λ=hp=hmv λ: de Broglie wavelength (m) · h: Planck constant · p: momentum of the particle (kg m s⁻¹) · m: mass (kg) · v: speed (m s⁻¹). The more momentum, the shorter the wavelength.

    FormulaPhoton emitted or absorbed in a transition

    hf=E1−E2=hcλ E1: energy of the higher level · E2: energy of the lower level · f,λ: frequency and wavelength of the photon. The photon energy equals the energy gap, so a large drop gives a short wavelength.

    Revise: Quantum physics

    Nuclear physics

    FormulaMass–energy equivalence

    E=mc2ΔE=c2 Δm E: energy (J) · m: mass (kg) · c: speed of light, 3.00×108 m s−1 · Δm: the change in mass (kg). Because c2 is so large, a small mass change is a large energy change.

    Key factsBalancing a nuclear equation

    The nucleon number A (the top number) adds up to the same total on both sides. The proton number Z (the bottom number, which gives the charge) adds up to the same total on both sides.

    FormulaMass defect and binding energy

    Δm=Zmp+(A−Z)mn−mnucleusEB=c2 Δmbinding energy per nucleon=EBA Z: proton number · A: nucleon number · mp,mn: masses of a proton and a neutron · mnucleus: mass of the nucleus alone. If Δm is in u, then EB in MeV is Δm×934.

    Key factsThe shape of the curve
    • The curve rises steeply from hydrogen-2 (about 1.1 MeV per nucleon), with a spike at helium-4 (about 7.1 MeV): this nucleus is unusually stable.
    • It reaches a broad maximum of about 8.8 MeV near A=56 (iron and nickel). These are the most stable nuclei.
    • It then falls slowly, to about 7.6 MeV for uranium-235.
    • A nucleus with a larger binding energy per nucleon is more stable.
    Key factsWhy fusion and fission release energy

    The products are more tightly bound, so the total binding energy rises. Energy equal to this rise is released, and since the energy comes from mass, the total rest mass of the products is less than that of the reactants: Ereleased=c2 ΔmΔm=(mass of reactants)−(mass of products) Δm is positive when energy is released. The energy appears as kinetic energy of the products and as gamma photons.

    FormulaActivity and the decay constant

    A=λN A: activity (Bq) · λ: decay constant (s−1) · N: number of undecayed nuclei in the sample. A sample with twice as many nuclei has twice the activity. The activity is also the rate at which N falls: A=−ΔN/Δt.

    FormulaExponential decay

    x=x0 e−λt x can be the number of undecayed nuclei N, the activity A, or the corrected count rate C. x0 is its value at t=0, λ is the decay constant and t is the time. All three decay with the same λ.

    FormulaHalf-life and decay constant

    λ=0.693t1/2equivalentlyt1/2=0.693λ A large decay constant means a short half-life. Keep the units consistent: if t1/2 is in hours, λ is in h−1. Convert to seconds when you need an activity in becquerel.

    Revise: Nuclear physics

    Medical physics

    FormulaDepth from an echo

    d=vt2 d: depth of the boundary (m) · v: speed of sound in the tissue (m s−1) · t: time between sending the pulse and receiving the echo (s). The factor 2 is there because the pulse goes there and back.

    FormulaIntensity reflection coefficient

    IRI0=(Z1−Z2)2(Z1+Z2)2 I0: intensity of the incident wave · IR: intensity of the reflected wave · Z1,Z2: specific acoustic impedances of the two media. The rest, I0−IR, is transmitted through the boundary.

    FormulaAttenuation law

    I=I0e−μx I0: intensity entering the medium · I: intensity after a thickness x · μ: attenuation coefficient (m−1, or cm−1) · x in the same length unit as 1/μ. This law holds for ultrasound and for X-rays.

    FormulaMinimum wavelength

    λmin⁡=hceV h: Planck constant · c: speed of light · e: elementary charge · V: accelerating p.d. of the tube. The maximum photon energy is eV, which is V electronvolts.

    FormulaPosition along the line from the arrival times

    x=c Δt2 x: distance of the annihilation from the midpoint of the line between the two detectors (m) · Δt: difference between the arrival times of the two photons (s) · c: speed of light. The photon that arrives first came from the half of the line nearer to its detector.

    Revise: Medical physics

    Astronomy and cosmology

    FormulaInverse square law for radiant flux intensity

    F=L4πd2 F: radiant flux intensity at the observer (W m−2) · L: luminosity of the star (W) · d: distance from the star to the observer (m). This assumes that nothing absorbs the light on the way.

    FormulaWien’s displacement law

    λmax⁡T=2.9×10−3 m K λmax⁡: wavelength of peak intensity (m) · T: thermodynamic temperature of the surface (K). The constant is not on the data sheet: a question gives it, or gives a reference star to compare with.

    FormulaStefan–Boltzmann law for a star

    L=4πσr2T4 L: luminosity (W) · σ: Stefan–Boltzmann constant, 5.67×10−8 W m−2 K−4 · r: radius of the star (m) · T: thermodynamic temperature of its surface (K). The power per unit area is σT4.

    FormulaRedshift

    Δλλ≈Δff≈vc Δλ: observed wavelength minus emitted wavelength (m) · λ: emitted (laboratory) wavelength · Δf: size of the change in frequency · v: recession speed of the source (m s−1) · c: speed of light. Valid only when v≪c.

    FormulaHubble’s law

    v≈H0d v: recession speed (m s−1) · d: distance to the galaxy (m) · H0: the Hubble constant, ≈2.3×10−18 s−1 (about 70 km s−1 Mpc−1 in astronomers’ units). Only SI units are required in calculations.

    Revise: Astronomy and cosmology

    Practical skills: measurement, tables, graphs and uncertainty

    Key factsThe rules for a table of results
    • One table for the readings, with raw and calculated columns in the order you take or work them out.
    • Every heading is quantity / unit: I / mA. The forms “I (mA)” and “I in mA” are also accepted. “I mA” and “mA” alone are not. A column with no unit (a ratio, or a log) needs no unit, but the heading still names the quantity.
    • Raw readings are recorded to the resolution of the instrument, and every value in a column has the same number of decimal places. A millimetre rule gives 0.300 m, not 0.3 m and not 0.30000 m.
    • Calculated values are given to n or n+1 significant figures, where n is the smallest number of significant figures in the raw data used in that calculation. If V has 2 s.f. and I has 4 s.f., then R is given to 2 or 3 s.f.
    • Repeat readings are shown: both timings, then the mean.
    Key factsThe rules for a graph
    • Axes labelled quantity / unit (T2 / s², l / m). The independent variable (the one you changed) goes on the horizontal axis, unless the question says otherwise.
    • Scales of 1, 2 or 5 units per large square (or 10, 20, 50 or 0.1, 0.2, 0.5 and so on). Never 3, 4, 6, 7, 8 or 9. Number the axes at least every two large squares.
    • The points fill at least half the grid in both directions. If they do not, use a false origin: start an axis at a value other than zero.
    • Plot every point to better than half a small square as a small cross (×) or dot with a diameter below 1 mm. Never a large blob.
    • A line of best fit: a single thin straight line (or smooth curve) with points balanced either side along its whole length. Use a transparent ruler, and do not force it through the origin or through the first and last points.
    • One anomalous point may be ignored in drawing the line, if you circle it and label it.
    Key factsEstimating and combining uncertainties
    • One reading, digital: at least ±1 in the last digit shown. One reading, analogue: about half the smallest division at each reading. A length on a millimetre rule needs two readings, so ±1 mm.
    • Repeated readings: half the range, 12(xmax−xmin), or the resolution if that is larger.
    • Absolute Δx (same unit as x); fractional Δx/x; percentage Δxx×100%.
    • Product or quotient: add the percentage uncertainties. Power xn: multiply the percentage uncertainty by n. Sum or difference: add the absolute uncertainties.
    • Give the uncertainty to 1 s.f. (2 at most) and the value to the same number of decimal places.

    Revise: Practical skills: measurement, tables, graphs and uncertainty

    Planning, analysis and evaluation

    Key factsPlanning checklist
    1. Define the problem. Name the independent variable (what you change), the dependent variable (what you measure as a result), and the control variables (what you keep the same).
    2. Methods of data collection. A labelled diagram of a workable arrangement. How you change the independent variable (and over what range, with at least six values). The instrument for each measured quantity, named and suited to its size. How each control variable is kept constant.
    3. Method of analysis. What equation to expect, how to rearrange it into y=mx+c, what to plot, what the gradient and intercept equal, and what result would show the relationship is valid.
    4. Additional detail. A specific safety risk and its precaution; repeated readings and means; a trial run; a way to reduce a named source of error.
    Key factsThree standard forms
    Equation Rearranged Plot (y against x) Gradient Intercept
    y=mx+c already straight y against x m c
    y=axn lg⁡y=nlg⁡x+lg⁡a lg⁡y against lg⁡x n lg⁡a
    y=aekx ln⁡y=kx+ln⁡a ln⁡y against x k ln⁡a
    Key factsLogarithm columns
    • The unit goes inside the logarithm. Write the heading as lg⁡(L/cm) or ln⁡(V/V). The logarithm itself has no unit, because you can only take the logarithm of a number, and L/cm is a pure number.
    • The number of decimal places in a logarithm equals the number of significant figures in the quantity. One more decimal place is also accepted. So L=76.5 cm (3 s.f.) gives lg⁡(L/cm)=1.884 (or 1.8837).
    • An uncertainty goes beside every value, in its own column, in the same unit as the value (so in a log column, no unit).
    FormulaUncertainties from the two lines

    Δ(gradient)=∣mbest−mworst∣Δ(intercept)=∣cbest−cworst∣ m: gradient · c: y-intercept. Draw the worst acceptable line dashed (or label it). If the graph has a false origin, find each intercept by substituting a point from that line and its gradient into y=mx+c.

    Revise: Planning, analysis and evaluation

    PreviousMaths formula sheetNextChemistry data and key equations
    Toolbox/Formula sheets

    Chemistry data and key equations

    Reference

    Every key formula from the course in one place, each linked back to the chapter that teaches it.

    Cambridge gives you a data booklet in 9701 exams, with values such as bond energies and electrode potentials. The equations below are the ones you must know how to use. This page collects every key box from the Chemistry chapters.

    Chemistry from zero

    Key factsCharges add to zero

    In an ionic formula, (number of cations × charge) + (number of anions × charge) = 0. Choose the smallest numbers of each ion that make the total charge zero. Write the cation first. Put a small number after an ion for how many are needed, and put brackets around a polyatomic ion if you need more than one.

    Key factsAcid reactions to learn
    • acid + metal → salt + hydrogen
    • acid + metal oxide or hydroxide (a base) → salt + water
    • acid + carbonate → salt + water + carbon dioxide
    Key factsThe Cambridge rule for answers

    Give a calculated answer to the same number of significant figures as the least precise data in the question, or one more. If a question gives masses to 3 s.f., give the answer to 3 s.f. (4 s.f. is also acceptable). Never round in the middle of a calculation: keep extra digits in your working, and round only the final answer.

    Revise: Chemistry from zero

    Atomic structure

    Key factsCounting particles

    For ZAXq (charge q, written 3+ or 2−):

    • number of protons = Z
    • number of neutrons = A−Z
    • number of electrons = Z−q, where a positive charge means electrons lost and a negative charge means electrons gained.
    Key factsFilling the orbitals
    1. Fill the lowest energy orbital first (the order above).
    2. An orbital holds at most two electrons, with opposite spins (↑↓).
    3. In a sub-shell with more than one orbital (p or d), put one electron in each orbital, all with the same spin, before pairing any of them.
    Key factsFactors that control ionisation energy
    • Nuclear charge: more protons, stronger attraction, higher ionisation energy.
    • Atomic radius: the further the electron is from the nucleus, the weaker the attraction, so the lower the ionisation energy.
    • Shielding: more inner electrons (in inner shells and sub-shells) repel the outer electron, reducing the attraction, so the ionisation energy is lower.
    • Spin-pair repulsion: two electrons paired in one orbital repel one another more strongly than two electrons in separate orbitals, so a paired electron is easier to remove.

    Revise: Atomic structure

    Moles and stoichiometry

    FormulaRelative atomic mass from isotopic abundances

    Ar=∑(isotopic mass×percentage abundance)100 Add up (isotopic mass × abundance) for every isotope, then divide by 100 if the abundances are percentages (or by the sum of the abundances).

    FormulaThe mole equations

    n=mMN=nL n is the amount in mol, m the mass in g, M the molar mass in g mol⁻¹, and N the number of particles. Rearranged: m=nM and M=m/n.

    Key factsThe method for every reacting-mass question
    1. Write the balanced equation.
    2. Convert the given mass to moles: n=m/M.
    3. Use the equation ratio to find the moles of the substance you want.
    4. Convert to mass (or volume, or concentration) as asked.
    FormulaPercentage yield

    percentage yield=actual yieldtheoretical yield×100 Both yields must be in the same unit (grams, or moles).

    FormulaGas volume

    V=n×24.0 dm3(room conditions) V is in dm³. To get cm³, multiply by 1000. Gases measured at the same temperature and pressure react in the ratio of their mole coefficients, so a volume ratio equals a mole ratio.

    FormulaConcentration

    c=nVn=cVc (g dm−3)=c (mol dm−3)×M c is in mol dm⁻³, n in mol and V in dm³. Convert cm³ to dm³ by dividing by 1000.

    Revise: Moles and stoichiometry

    Chemical bonding

    Key factsElectronegativity trends
    • Across a period it increases. The nuclear charge increases, the electrons go into the same shell so shielding is about the same, and the atomic radius decreases. The bonding electrons feel a stronger attraction.
    • Down a group it decreases. The atomic radius increases and there are more inner shells, so the shielding increases. The bonding electrons are further from the nucleus and held less strongly.
    Key factsCounting hybridisation

    For an atom in a molecule, count the regions of electron density: σ bonds plus lone pairs (a double or triple bond counts as one σ region). 4 regions: sp³ (109.5° family), 3 regions: sp² (120°), 2 regions: sp (180°). The π bonds are made by the p orbitals that are left over.

    Revise: Chemical bonding

    Shapes of molecules and intermolecular forces

    Key factsThe VSEPR rules
    1. Electron pairs (regions of electron density) around a central atom repel each other and arrange themselves as far apart as possible.
    2. A double or triple bond is one region.
    3. Lone pairs repel more strongly than bonding pairs: lone pair–lone pair repulsion > lone pair–bonding pair repulsion > bonding pair–bonding pair repulsion.
    4. The shape of a molecule is the arrangement of its atoms only. Lone pairs affect the shape but are not part of its name.
    FormulaWhere does 109.5° come from?

    (Optional, uses A Level maths: skip it if the vectors are new to you; you only need to remember 109.5°.) A regular tetrahedron has its four corners at (1,1,1), (1,−1,−1), (−1,1,−1) and (−1,−1,1) from its centre. The angle between two of these directions satisfies cos⁡θ=(1)(1)+(1)(−1)+(1)(−1)3=−13, so θ=109.47∘, which is written 109.5°.

    Key factsThe tetrahedral family
    Bonding pairs Lone pairs Shape Bond angle Example
    4 0 tetrahedral 109.5° CHX4, NHX4X+
    3 1 pyramidal 107° NHX3, HX3OX+
    2 2 non-linear 104.5° HX2O

    All have 4 pairs of electrons in a tetrahedral arrangement. Each lone pair reduces the angle by about 2.5°: 109.5°, then 107°, then 104.5°.

    Key factsShapes and bond angles
    Pairs Bonding Lone Shape Bond angle Example
    2 2 0 linear 180° COX2, BeClX2
    3 3 0 trigonal planar 120° BFX3, COX3X2−, NOX3X−
    3 2 1 non-linear a little under 120° SOX2
    4 4 0 tetrahedral 109.5° CHX4, NHX4X+, SiClX4, AlClX4X−, SOX4X2−
    4 3 1 pyramidal 107° NHX3, HX3OX+, PClX3
    4 2 2 non-linear 104.5° HX2O
    5 5 0 trigonal bipyramidal 120° and 90° PFX5, PClX5
    6 6 0 octahedral 90° SFX6
    Key factsWhen is a molecule polar?

    A molecule is non-polar if its bonds are non-polar (such as ClX2 or OX2) or if its polar bonds are arranged symmetrically so that the dipoles cancel (COX2, BFX3, CHX4, CClX4, SFX6, PClX5, BeClX2). A molecule is polar if it has polar bonds that do not cancel. A lone pair on the central atom usually makes a molecule polar (NHX3, HX2O, SOX2, PClX3).

    Key factsStrength of the types of bonding

    Ionic, covalent and metallic bonds are generally much stronger than intermolecular forces. Among intermolecular forces: hydrogen bonds > pd–pd forces > id–id forces, for molecules of similar size.

    Revise: Shapes of molecules and intermolecular forces

    States of matter

    Key factsWhen does a gas behave most like an ideal gas?

    At low pressure (the molecules are far apart, so their own volume is negligible) and high temperature (the molecules move so fast that the attractions between them have little effect).

    FormulaThe ideal gas equation

    pV=nRT p = pressure in pascals (Pa), V = volume in cubic metres (m³), n = amount of gas in moles, R = the gas constant, 8.31 J K⁻¹ mol⁻¹, T = temperature in kelvin (K).

    FormulaRelative molecular mass of a gas

    Mr=mRTpV m = mass of the gas in grams, with p in Pa, V in m³ and T in K. The result is in g mol⁻¹, and it is the relative molecular mass when written without units.

    Key factsProperties of the four types
    Structure Examples m.p. and b.p. Conductivity Solubility in water
    giant ionic NaCl, MgO high solid: no; molten or dissolved: yes often soluble
    simple molecular IX2, CX60, ice low no polar: soluble; non-polar: insoluble
    giant molecular diamond, graphite, SiOX2 very high graphite only (solid) insoluble
    giant metallic Cu high yes (solid and molten) insoluble

    Revise: States of matter

    Chemical energetics

    FormulaHeat energy transferred

    q=mcΔT q = heat energy transferred in J, m = mass of water or solution being heated or cooled in g, c = specific heat capacity, 4.18 J g⁻¹ K⁻¹ for water and dilute solutions, ΔT=Tfinal−Tinitial in K.

    FormulaEnthalpy change from an experiment

    ΔH=−mcΔTn Divide by 1000 to give kJ mol⁻¹. n = amount in mol of the limiting reactant (or the substance in the definition, such as the water formed). The minus sign appears because the energy gained by the solution is lost by the reaction.

    Key factsUsing bond energies

    ΔH=∑(bond energies of bonds broken)−∑(bond energies of bonds formed) ∑ (sigma) means “the sum of”: add up the values for every bond (or, later, every substance) listed. Add up one side for every bond that breaks (each molecule’s bonds, multiplied by the number of molecules in the equation), and subtract the total for every bond that forms. Reactants and products must all be gases for the result to be reliable.

    Key factsHess’s law with enthalpies of formation

    ΔHr⊖=∑ΔHf⊖(products)−∑ΔHf⊖(reactants) Multiply each ΔHf⊖ by the number of moles in the equation. Elements in their standard states have ΔHf⊖=0.

    Key factsHess’s law with enthalpies of combustion

    ΔHr⊖=∑ΔHc⊖(reactants)−∑ΔHc⊖(products) Notice that the order is the reverse of the formation rule: reactants minus products.

    Revise: Chemical energetics

    Redox and oxidation numbers

    Key factsRules for oxidation numbers

    Apply these in this order. The sum of oxidation numbers is 0 in a neutral compound and equals the charge in an ion.

    1. An atom in an uncombined element has oxidation number 0 (Na, OX2, PX4, SX8).
    2. A simple monatomic ion has an oxidation number equal to its charge (FeX3+ is +3, OX2− is −2).
    3. Fluorine is always −1 in compounds.
    4. Group 1 metals are +1 and Group 2 metals are +2 in compounds. Aluminium is +3.
    5. Hydrogen is +1, except in metal hydrides such as NaH and CaHX2, where it is −1.
    6. Oxygen is −2, except in peroxides such as HX2OX2 (−1), and when combined with fluorine in OFX2 (+2).
    7. The atom of the more electronegative element in a bond takes the negative oxidation number (chlorine is −1 in CClX4).
    Key factsRedox by oxidation number
    • Oxidation: the oxidation number of an element increases.
    • Reduction: the oxidation number of an element decreases.
    • If no element changes its oxidation number, the reaction is not redox (acid–base and precipitation reactions are not).

    Revise: Redox and oxidation numbers

    Chemical equilibria

    Key factsLe Chatelier’s principle: the effects
    Change Position of equilibrium moves Value of K
    increase the concentration of a reactant right (towards products) unchanged
    increase the concentration of a product left (towards reactants) unchanged
    increase the pressure (gases) towards the side with fewer moles of gas unchanged
    increase the temperature in the endothermic direction changes
    add a catalyst no change (equilibrium reached faster) unchanged
    FormulaEquilibrium constant Kc

    a A+b B⇌c C+d DKc=[C]c [D]d[A]a [B]b [X] is the concentration of X in mol dm⁻³ at equilibrium. Products go on the top and reactants on the bottom, and each concentration is raised to the power of its coefficient in the balanced equation.

    FormulaEquilibrium constant Kp

    a A(g)+b B(g)⇌c C(g)+d D(g)Kp=pC c pD dpA a pB b pX is the partial pressure of X at equilibrium. Its unit is the unit of pressure (Pa or atm) to the power Δn, where Δn is moles of gaseous products minus moles of gaseous reactants. 1 atm = 101 kPa (101 325 Pa); Kp may be given in Pa, kPa or atm, so use the unit you are given.

    Key factsWhat changes the value of K
    • Only a change of temperature changes the value of K.
    • Changing the concentration of a species, the pressure or the volume changes the position of equilibrium, but the system settles at a new set of concentrations that give the same K.
    • A catalyst changes neither the position nor K: it only shortens the time to reach equilibrium.
    Key factsTemperature and K
    • Forward reaction exothermic (ΔH negative): increasing T decreases K, so the position moves to the left.
    • Forward reaction endothermic (ΔH positive): increasing T increases K, so the position moves to the right.

    Revise: Chemical equilibria

    Acids and bases

    Key factsThe pH scale

    Pure water and neutral solutions: pH 7. Acidic solutions: pH below 7 (the lower the number, the more acidic). Alkaline solutions: pH above 7. The scale usually runs from 0 to 14.

    Key factsStrong and weak at a glance

    Strong acids: HCl, HNOX3, HX2SOX4. Weak acid: CHX3COOH. Strong bases: NaOH, KOH. Weak base: NHX3. At AS treat sulfuric acid as a strong, diprotic acid: each molecule can supply two HX+ ions in a reaction.

    Key factsWhat to say about each curve
    Combination Starts at Steep part pH at equivalence Suitable indicator
    strong acid + strong base low (about 1) long, pH 3 to 11 exactly 7 any in the list: methyl orange, bromophenol blue, thymol blue, thymolphthalein
    weak acid + strong base higher (about 3) pH 6.5 to 11 above 7 (about 8.7) thymol blue, thymolphthalein
    strong acid + weak base low (about 1) pH 3 to 7.5 below 7 (about 5.3) methyl orange, bromophenol blue
    weak acid + weak base higher (about 3) none: gradual rise about 7 none: use a pH meter
    Key factsChoosing an indicator

    The indicator’s pH range must lie within the steep (vertical) section of the titration curve. The end point is then very close to the equivalence point.

    Revise: Acids and bases

    Reaction kinetics

    Key factsConcentration and pressure

    Increasing the concentration of a reactant in solution, or the pressure of a gas, increases the number of particles in a given volume. They collide more frequently, so the frequency of effective collisions increases and the rate increases. The activation energy is the same.

    Key factsSurface area

    The same mass of a solid with a larger surface area (a powder instead of lumps) has more of its particles exposed, so there are more collisions per second between them and the other reactant, and the rate increases. Eₐ and the energy of the particles are unchanged.

    Key factsFeatures of the Boltzmann distribution
    • The curve starts at the origin: no molecules have zero energy.
    • It rises to a peak (the most likely energy) and then falls gradually.
    • It never touches the horizontal axis at high energy: there is no maximum energy, since a very few molecules can have a very high energy.
    • The area under the curve is the total number of molecules. It stays the same if the amount of gas and its volume are fixed.
    • The activation energy Eₐ is marked on the energy axis. The area to the right of Eₐ is the number of molecules with energy of at least Eₐ, the ones that can react on collision.
    Key factsWhy temperature has such a large effect

    The main reason is that, at a higher temperature, a greater proportion of molecules has energy greater than or equal to Eₐ, so the frequency of effective collisions increases sharply. The extra collision frequency is a minor contribution. Eₐ does not change.

    Key factsWhat a catalyst does and does not do

    A catalyst provides a different mechanism with a lower Eₐ, so a greater proportion of collisions are effective and the rate increases. It does not change ΔH, the products, the position of equilibrium or the amount of product formed. In a reversible reaction it speeds up the forward and reverse reactions equally, so equilibrium is reached sooner but at the same position.

    Revise: Reaction kinetics

    Periodicity: Period 3

    Key factsRadius across Period 3

    Atomic radius decreases from Na to Cl: more protons, the same shell, about the same shielding. Positive ions are smaller than their atoms and shrink from NaX+ to AlX3+. Negative ions are larger than their atoms and shrink from PX3− to ClX−. There is a big jump up from AlX3+ to PX3−, because the negative ions have one more shell.

    Key factsStructure and bonding explain melting point and conductivity

    Giant metallic (Na, Mg, Al): high melting point, rising across the period; conduct. Giant covalent (Si): highest melting point; semiconductor. Simple molecular (P₄, S₈, Cl₂, Ar): low melting point from weak id-id forces, higher for the molecules with more electrons; do not conduct.

    Key factsFrom ionic to covalent across Period 3

    The electronegativity of the elements rises from Na (0.9) to Cl (3.0), so the difference from oxygen or chlorine falls from left to right. Large differences give ionic bonding (Na₂O, MgO, NaCl, MgCl₂), which gives giant ionic structures that are basic oxides, that conduct when molten, and that dissolve in water without being hydrolysed. Small differences give covalent bonding (SiO₂, P₄O₁₀, SO₂, SO₃, SiCl₄, PCl₅), giving acidic oxides, and chlorides that are hydrolysed by water. Aluminium is in the middle: the oxide and the chloride show properties of both.

    Revise: Periodicity: Period 3

    Group 2

    Key factsGroup 2 in one sentence

    Down Group 2 the atomic radius increases and the ionisation energies decrease, so the metals lose their two outer electrons more easily and become more reactive.

    Key factsThermal stability of Group 2 carbonates and nitrates

    Thermal stability increases down the group. MgCOX3 decomposes at a lower temperature than CaCOX3, then SrCOX3, then BaCOX3, which needs the highest temperature. The nitrates follow the same trend.

    Key factsSolubility trends
    • The hydroxides increase in solubility down the group: Mg(OH)X2 is almost insoluble and Ba(OH)X2 is fairly soluble.
    • The sulfates decrease in solubility down the group: MgSOX4 is soluble and BaSOX4 is insoluble.
    Key factsWhy thermal stability increases down Group 2

    Down the group the cation radius increases while the charge stays at 2+, so the charge density and the polarising power fall. The carbonate (or nitrate) ion is distorted less, its C–O (or N–O) bonds are weakened less, and a higher temperature is needed to decompose it. So MgCOX3 decomposes at a lower temperature than BaCOX3.

    Key factsExplaining the solubility trends

    Hydroxides: the lattice energy falls more than the hydration enthalpy down the group, so ΔHsol becomes less endothermic and the solubility rises. Sulfates: the lattice energy changes little (the large anion dominates the ion distance) but the cation’s hydration enthalpy falls a lot, so ΔHsol becomes more endothermic and the solubility falls.

    Revise: Group 2

    Group 17: the halogens

    Key factsHalogens as oxidising agents

    The oxidising power decreases down the group: ClX2 > BrX2 > IX2. A halogen displaces any halide below it in the group. Chlorine displaces bromide and iodide; bromine displaces iodide; iodine displaces neither.

    Key factsThermal stability of the hydrogen halides

    Thermal stability decreases down the group: HCl > HBr > HI. The H–X bond becomes longer and weaker (H–Cl 431, H–Br 366, H–I 299 kJ mol⁻¹), so less energy is needed to break it. Hydrogen iodide is decomposed easily by heating, which you see as purple iodine vapour.

    Key factsHalide ions as reducing agents

    The reducing power increases down the group: IX− > BrX− > ClX−. A halide ion is the weaker reducing agent when the halogen it makes is the stronger oxidising agent.

    Key factsChlorine and water purification

    ClX2+HX2O⇌HCl+HOCl and HOCl⇌HX++ClOX−. Chlorine disproportionates in water (0 → −1 in HCl, 0 → +1 in HOCl). The active species that kill bacteria are HOCl and ClOX−.

    Revise: Group 17: the halogens

    Nitrogen and sulfur

    Key factsWhy N₂ is unreactive

    (1) The N≡N triple bond is very strong (944 kJ mol⁻¹), so a very large activation energy is needed to break it. (2) NX2 is non-polar, so it has no δ+ or δ− atoms to attract attacking reagents.

    Key factsNO₂ as a catalyst in acid rain

    SOX2+NOX2→SOX3+NO, then NO+12 OX2→NOX2 (the NOX2 is regenerated), then SOX3+HX2O→HX2SOX4. Overall: SOX2+12 OX2→SOX3 with NOX2 as the catalyst. Because everything is a gas, NOX2 is a homogeneous catalyst.

    Revise: Nitrogen and sulfur

    Introduction to organic chemistry

    Key factsReading a skeletal formula
    • Each corner and each line end is a carbon atom.
    • Each line is a bond. A double bond is two parallel lines, a triple bond is three.
    • Hydrogen atoms attached to carbon are not drawn. Each carbon makes four bonds in total, so count the bonds shown and add enough hydrogen atoms to make four. (The one common exception: the H of an aldehyde group is often drawn, as –CHO, to make the group clear.)
    • Atoms other than carbon and hydrogen are drawn as their symbols (O, N, Cl, Br). A hydrogen atom attached to one of these is drawn: write OH and NH₂, not just O and N.
    • Draw the zigzag so that bond angles look about 120°. An alkyne or nitrile unit (C≡C, C≡N) is drawn as a straight line with its neighbours.
    Key factsCurly arrows

    A curly arrow starts at a bond or at a lone pair of electrons and ends at the atom or bond where that pair goes. It never starts at a charge symbol or at an atom. A radical step (one electron moving) is written as an equation in this course, not drawn with arrows.

    Revise: Introduction to organic chemistry

    Isomerism

    Key factsWhen does cis–trans isomerism occur?

    Both conditions must be met:

    1. There is a C=C double bond, so rotation is restricted.
    2. Each carbon of the C=C is bonded to two different groups. If either carbon has two identical groups, swapping them gives the same molecule and there is no isomerism.
    Key factsFinding a chiral centre

    For each carbon atom with four single bonds, write the four groups attached to it. If all four are different, it is a chiral centre. Mark it with an asterisk. Check each carbon in turn; ignore carbons with a double bond or two identical groups (such as CHX3 or CHX2).

    Revise: Isomerism

    Hydrocarbons: alkanes and alkenes

    Key factsWhy alkanes are unreactive
    1. Strong bonds. The C–H bond (410 kJ mol⁻¹) and the C–C bond (350 kJ mol⁻¹) are strong, so a lot of energy is needed to break them.
    2. Almost no polarity. The electronegativities of carbon (2.5) and hydrogen (2.1) are close, so the bonds are almost non-polar. There is no δ+ atom for a nucleophile to attack and no δ− atom for an electrophile to attack, and there are no lone pairs or π bonds to start a reaction.
    Key factsAddition reactions of ethene (and other alkenes)
    Reaction and reagent Conditions Product (from ethene)
    hydrogenation: HX2(g) Pt or Ni catalyst, heat ethane, CHX3CHX3
    hydration: steam, HX2O(g) HX3POX4 catalyst, high temperature and pressure ethanol, CHX3CHX2OH
    hydrogen halide: HBr(g) or HCl(g) room temperature bromoethane, CHX3CHX2Br
    halogen: BrX2 (or ClX2) room temperature 1,2-dibromoethane, CHX2BrCHX2Br
    oxidation: cold, dilute, acidified KMnOX4 room temperature ethane-1,2-diol, CHX2(OH)CHX2(OH)
    Key factsTest for an alkene

    Shake the compound with orange bromine water (aqueous bromine) at room temperature. An alkene decolourises it quickly, from orange to colourless. An alkane does not.

    Key factsHot concentrated acidified KMnO₄ splits C=C
    Carbon of the double bond Product
    =CHX2 (two H atoms) COX2 and HX2O
    =CHR (one H atom, one alkyl group) carboxylic acid, RCOOH
    =CRRX′ (two alkyl groups) ketone, RCORX′

    Revise: Hydrocarbons: alkanes and alkenes

    Halogenoalkanes

    Key factsNucleophilic substitution reactions of halogenoalkanes
    Nucleophile Reagent and conditions Organic product
    OHX− NaOH(aq), heat (under reflux) alcohol
    CNX− KCN in ethanol, heat (under reflux) nitrile (one more carbon atom)
    NHX3 excess NHX3 in ethanol, heated under pressure primary amine
    HX2O aqueous AgNOX3 in ethanol, warm alcohol (the halide ion is precipitated as AgX)
    Key factsWhich mechanism?

    Primary halogenoalkanes tend to react by SN2, tertiary halogenoalkanes by SN1, and secondary halogenoalkanes by a mixture of the two, depending on their structure.

    Key factsReactivity of halogenoalkanes

    Rate of hydrolysis: iodoalkane > bromoalkane > chloroalkane, because the C–X bond energy decreases from C–Cl (340 kJ mol⁻¹) to C–I (240 kJ mol⁻¹). Bond strength matters more than bond polarity.

    Revise: Halogenoalkanes

    Alcohols

    Key factsAlcohol to halogenoalkane
    Reagent and conditions Equation (ethanol)
    HX(g) CHX3CHX2OH+HBr→CHX3CHX2Br+HX2O
    KCl with concentrated HX2SOX4 or HX3POX4 the acid makes HCl, which then reacts as above
    PClX3 and heat 3 CHX3CHX2OH+PClX3→3 CHX3CHX2Cl+HX3POX3
    PClX5 CHX3CHX2OH+PClX5→CHX3CHX2Cl+POClX3+HCl
    SOClX2 CHX3CHX2OH+SOClX2→CHX3CHX2Cl+SOX2+HCl
    Key factsOxidation of alcohols
    Alcohol Product with the oxidising agent What the oxidant does to the alcohol
    primary (1°) aldehyde, then carboxylic acid lose H from O–H and from the C–OH carbon; the aldehyde is oxidised further
    secondary (2°) ketone lose H from O–H and from the C–OH carbon; the ketone is not oxidised further
    tertiary (3°) no reaction no H on the C–OH carbon, so the orange colour stays
    Key factsAcidity of alcohols compared with water

    Alcohols are weaker acids than water. The electron-donating alkyl group makes the alkoxide ion, ROX−, less stable than OHX− (the negative charge on oxygen is intensified), so the alcohol loses HX+ less easily and sodium reacts less vigorously with it.

    Revise: Alcohols

    Aldehydes and ketones

    Key factsNaming aldehydes and ketones
    Family Ending Numbering Examples
    aldehyde -al the carbon of CHO is always carbon 1, so it needs no number ethanal CHX3CHO, butanal CHX3CHX2CHX2CHO
    ketone -one number the chain from the end that gives the C=O carbon the lower number; no number is needed in propanone or butanone propanone CHX3COCHX3, pentan-2-one CHX3COCHX2CHX2CHX3, pentan-3-one CHX3CHX2COCHX2CHX3
    Key factsReduction of aldehydes and ketones
    Starting compound Product Example
    aldehyde primary alcohol CHX3CHX2CHO+2 [H]→CHX3CHX2CHX2OH
    ketone secondary alcohol CHX3COCHX3+2 [H]→CHX3CH(OH)CHX3

    Reagent: NaBHX4 or LiAlHX4 (dry ether). Two hydrogen atoms add across C=O: one to the carbon and one to the oxygen.

    Key factsDistinguishing aldehydes from ketones
    Reagent Aldehyde Ketone
    2,4-DNPH orange precipitate orange precipitate
    Tollens’ reagent, warm silver mirror (or black precipitate) no change
    Fehling’s solution, warm blue solution gives an orange-red precipitate solution stays blue
    acidified KX2CrX2OX7, warm orange gives green stays orange

    Revise: Aldehydes and ketones

    Carboxylic acids, esters and nitriles

    Key factsCarboxylic acids as acids
    Reacts with Products Example
    reactive metal, e.g. Mg salt + hydrogen (a redox reaction) 2 CHX3COOH+Mg→(CHX3COO)X2Mg+HX2
    alkali, e.g. NaOH salt + water CHX3COOH+NaOH→CHX3COONa+HX2O
    carbonate, e.g. NaX2COX3 salt + water + carbon dioxide 2 CHX3COOH+NaX2COX3→2 CHX3COONa+HX2O+COX2
    Key factsHydrolysis of nitriles
    Conditions Products Equation
    dilute acid (e.g. HCl), heat carboxylic acid + ammonium ion RCN+2 HX2O+HX+→RCOOH+NHX4X+
    dilute alkali (e.g. NaOH), heat carboxylate ion + ammonia RCN+HX2O+OHX−→RCOOX−+NHX3

    After alkaline hydrolysis, acidify (dilute strong acid) to obtain the carboxylic acid.

    Revise: Carboxylic acids, esters and nitriles

    Organic synthesis (AS)

    FormulaOverall yield of a route

    overall yield=y1×y2×y3×⋯ where each y is the fractional yield of one step (percentage yield ÷ 100). Convert the answer back to a percentage.

    Revise: Organic synthesis (AS)

    Infrared spectroscopy and mass spectrometry

    FormulaRelative atomic mass from isotopes

    Ar=∑(isotopic mass×abundance)∑abundance Abundances can be percentages, relative heights or fractions: the units cancel. The sum of the abundances in the denominator is 100 when percentages are used.

    FormulaNumber of carbon atoms from the [M+1]⁺ peak

    n=100×abundance of [M+1]+1.1×abundance of M+ n is the number of carbon atoms in the molecule. Round to the nearest whole number. Other isotopes (2H, 15N, 17O) are ignored at this level.

    Revise: Infrared spectroscopy and mass spectrometry

    Lattice energy, entropy and Gibbs free energy

    Key factsBorn–Haber equation

    ΔHf⊖ = (sum of the atomisation enthalpies) + (sum of the ionisation energies) + (sum of the electron affinities) + ΔHlatt⊖. Rearrange it to find whichever term is missing. Every term is for the amount of substance in the formation equation of one mole of the compound.

    Key factsLattice energy: charge and radius

    The lattice energy is more exothermic (a larger negative number) when the ions have higher charges and when the ions are smaller, so that the distance between their centres is shorter. Charge has by far the bigger effect.

    Key factsEnthalpy change of solution

    ΔHsol⊖=−ΔHlatt⊖+∑ΔHhyd⊖ ∑ΔHhyd is the hydration enthalpy of every ion in the formula: for MgClX2 it is one MgX2+ and two ClX−. The lattice energy is negative, so −ΔHlatt is positive.

    FormulaEntropy change of a reaction

    ΔS⊖=∑S⊖(products)−∑S⊖(reactants) Multiply each S⊖ by its coefficient in the balanced equation. The unit is J K⁻¹ mol⁻¹.

    FormulaGibbs free energy equation

    ΔG⊖=ΔH⊖−T ΔS⊖ ΔG⊖ is in kJ mol⁻¹, ΔH⊖ in kJ mol⁻¹, T is the temperature in kelvin, and ΔS⊖ must be in kJ K⁻¹ mol⁻¹: divide the value in J K⁻¹ mol⁻¹ by 1000.

    Key factsFeasibility

    A reaction is feasible (it can happen) when ΔG is negative. When ΔG is zero the reaction is just on the point of becoming feasible, and when ΔG is positive it is not feasible at that temperature.

    Revise: Lattice energy, entropy and Gibbs free energy

    Electrochemistry: electrolysis and electrode potentials

    Key factsNaming the electrodes

    Anode: oxidation happens here (ions lose electrons); in electrolysis it is positive. Cathode: reduction happens here (ions gain electrons); in electrolysis it is negative. Remember “An Ox, Red Cat”. The names follow the process, not the sign: in the next half of the chapter the cathode of a working cell will turn out to be the positive electrode.

    FormulaCharge and current

    Q=I t Q = charge in coulombs (C), I = current in amperes (A), t = time in seconds.

    FormulaCell potential

    Ecell⊖=E⊖(positive)−E⊖(negative) The positive electrode is the one with the more positive E⦵, so E⦵cell is a positive number for a working cell. Use the values exactly as in the table; never multiply them.

    Key factsFeasibility from E⦵

    A reaction is feasible under standard conditions if E⦵cell is positive, where E⦵cell = E⦵(the half-equation being reduced) − E⦵(the half-equation being oxidised). The oxidising agent is on the left of the more positive half-equation.

    FormulaNernst equation (298 K)

    E=E⊖+0.059z log⁡10[oxidised][reduced] z is the number of electrons in the half-equation. Concentrations are in mol dm⁻³. A solid metal counts as 1, so for CuX2+/Cu the equation is E=E⊖+0.0592log⁡10[CuX2+].

    FormulaFree energy and cell potential

    ΔG⊖=− n F Ecell⊖ n = number of moles of electrons transferred in the equation as written, F=9.65×104 C mol−1, E⦵cell in volts. This gives ΔG⊖ in J mol⁻¹; divide by 1000 for kJ mol⁻¹.

    Revise: Electrochemistry: electrolysis and electrode potentials

    pH, buffers, solubility product and partition

    FormulapH of a weak acid

    [HX+]=Ka×c Ka is the acid dissociation constant (mol dm⁻³) and c the concentration of the acid (mol dm⁻³). Then pH=−lg⁡[HX+].

    Key factsWhen the approximations hold

    The weak-acid formula needs (1) [HX+]=[AX−], which fails if the acid is so dilute that the HX+ from water matters, and (2) [HA]≈c, which fails if a large fraction (more than a few per cent) of the acid has dissociated. In 9701 questions the approximations are allowed unless the question says otherwise: state them in your answer.

    FormulapH of a buffer

    [HX+]=Ka×[HA][AX−] [HA] is the concentration of the weak acid and [AX−] that of its conjugate base. Then pH=−lg⁡[HX+].

    Key factsThe half-equivalence point

    Half the acid has been neutralised when the volume of alkali added is half the equivalence volume. Then [HA]=[AX−], the ratio is 1, [HX+]=Ka, and pH=pKa So the pKa of a weak acid is the pH read from the titration curve at half the volume needed to reach the equivalence point.

    Key factsWhich pH method?

    Decide what is in the flask first, then choose the method.

    What is in the solution Method Example in this chapter
    strong monoprotic acid [HX+]=c 0.0500 mol dm⁻³ HCl: pH 1.30
    strong alkali [OHX−]=c (times the number of OHX− per formula unit), then [HX+]=Kw/[OHX−] 0.0200 mol dm⁻³ NaOH: pH 12.30
    weak acid alone [HX+]=Kac 0.100 mol dm⁻³ ethanoic acid: pH 2.88
    weak acid with its salt, or part-neutralised by alkali [HX+]=Ka[HA]/[AX−] with amounts ethanoic acid + ethanoate: pH 4.46
    strong acid and strong alkali mixed amounts, find the excess, divide by the total volume pH 1.60
    weak acid with alkali beyond the equivalence point excess strong alkali decides the pH 30.0 cm³ of alkali: pH 11.96
    Key factsPredicting precipitation
    • Ionic product < Ksp: the solution is unsaturated; no precipitate forms.
    • Ionic product = Ksp: exactly saturated.
    • Ionic product > Ksp: too many ions; a precipitate forms until the ionic product falls back to Ksp.
    Key factsExtraction

    A fixed volume of solvent removes more solute if it is used in several small portions than in one large portion. For a solute that prefers the organic solvent, shake with a fresh portion each time and combine the organic layers.

    Revise: pH, buffers, solubility product and partition

    Rate equations and catalysis

    Key factsThe rate equation comes from experiment

    The orders cannot be read from the balanced equation. The reaction 2 NO(g)+2 HX2(g)→NX2(g)+2 HX2O(g) has a rate equation rate=k[NO]2[HX2]: second order in NO but first order in HX2, even though the equation shows two of each. Only experiments (or a known mechanism) give the order. The coefficients in the equation tell you how much of each substance reacts, not how the rate depends on it.

    Key factsRecognising the order from a concentration–time graph
    • Zero order: a straight line with constant negative gradient; successive half-lives get shorter (each half the last).
    • First order: an exponential curve; every half-life is the same.
    • Second order: a curve that flattens more slowly; successive half-lives get longer (each double the last).
    FormulaFirst-order half-life and rate constant

    t12=ln⁡2k=0.693ksok=0.693t12 k is in s⁻¹ if t12 is in seconds. Convert minutes or hours to seconds first. This applies to first-order reactions only. (ln 2 is just the number 0.693; you do not need the ln button.)

    Key factsWhat changes k?

    k changes with temperature and with a catalyst. It does not change when the concentrations change. A reaction can go faster because the concentration is higher (the rate equation) or because k is larger (temperature, catalyst): do not mix the two up.

    Key factsIntermediate or transition state?

    An intermediate sits in a valley of the pathway diagram: it is a real species with a measurable life. A transition state sits at a peak: it is the arrangement of atoms at the point of no return, not a species you can isolate. A mechanism with n steps has n transition states and n − 1 intermediates.

    Key factsThe rule for the rate-determining step

    The rate equation contains the species that take part in the rate-determining step (and any steps before it). Species that only react after the slow step do not appear in the rate equation, however much of them is needed overall. The power of each species in the rate equation is the number of its particles in that step.

    Revise: Rate equations and catalysis

    Transition elements

    Key factsCommon oxidation states in the first row
    Element Oxidation states and examples
    V +5 (VOX2X+), +4 (VOX2+), +3 (VX3+), +2 (VX2+)
    Cr +6 (CrX2OX7X2−), +3 (CrX3+), +2 (CrX2+)
    Mn +7 (MnOX4X−), +4 (MnOX2), +2 (MnX2+)
    Fe +3 (FeX3+), +2 (FeX2+)
    Co +3, +2 (CoX2+)
    Cu +2 (CuX2+), +1 (CuX+)
    Key factsWhy transition elements are good catalysts
    1. They have more than one stable oxidation state, so they can give electrons to a reactant and take them back later.
    2. They have vacant d orbitals that are energetically accessible, so they can accept lone pairs from reactant molecules and form dative bonds (or, on a surface, weak bonds) to them.
    Key factsElectrode potentials used in this chapter (V)

    MnOX4X−/MnX2+ +1.52 · CrX2OX7X2−/CrX3+ +1.33 · VOX2X+/VOX2+ +1.00 · FeX3+/FeX2+ +0.77 · IX2/IX− +0.54 · CuX2+/CuX+ +0.15 · VOX2+/VX3+ +0.34 · VX3+/VX2+ −0.26 · VX2+/V −1.20 · ZnX2+/Zn −0.76

    Key factsThe three redox titrations
    Titration Overall equation Mole ratio
    MnOX4X− with FeX2+ MnOX4X−+8 HX++5 FeX2+→MnX2++5 FeX3++4 HX2O MnOX4X−:FeX2+=1:5
    MnOX4X− with CX2OX4X2− 2 MnOX4X−+16 HX++5 CX2OX4X2−→2 MnX2++10 COX2+8 HX2O MnOX4X−:CX2OX4X2−=2:5
    CuX2+ with IX−, then SX2OX3X2− 2 CuX2++4 IX−→2 CuI+IX2 and IX2+2 SX2OX3X2−→2 IX−+SX4OX6X2− CuX2+:SX2OX3X2−=1:1

    Revise: Transition elements

    Complexes: colour, isomerism and stability

    Key factsColour of a complex ion
    1. The ligands split the d orbitals into two sets with an energy gap ΔE.
    2. An electron in the lower set absorbs light of frequency f=ΔE/h and is promoted to the higher set.
    3. The colour you see is the complementary colour of the light absorbed.
    Key factsStability constant: how to use it
    • Write the expression as [complex] over [metal ion in solution] × [free ligand] raised to its coefficient. Leave out HX2O as solvent.
    • The larger Kstab, the more the equilibrium lies towards the complex, so the more stable the complex.
    • If two ligands compete for a metal ion, the ligand that forms the complex with the larger Kstab replaces the other.

    Revise: Complexes: colour, isomerism and stability

    Arenes: benzene and its compounds

    Key factsThe structure of benzene

    Benzene is a planar hexagon. Each carbon is sp² hybridised, with σ bonds at 120° to two carbons and one hydrogen. The six p orbitals overlap sideways to give a delocalised π system above and below the ring. All six C–C bonds are equal in length (0.140 nm), between a single and a double bond. We draw benzene as a hexagon with a circle.

    Key factsReactions of arenes
    Reaction Reagent and conditions Product from benzene
    Halogenation (ring) ClX2 with AlClX3, or BrX2 with AlBrX3, room temperature chlorobenzene, bromobenzene
    Nitration concentrated HNOX3 with concentrated HX2SOX4, 25–60 °C nitrobenzene
    Friedel–Crafts alkylation CHX3Cl with AlClX3, heat methylbenzene
    Friedel–Crafts acylation CHX3COCl with AlClX3, heat 1-phenylethanone
    Hydrogenation HX2, Pt or Ni catalyst, heat cyclohexane
    Side-chain oxidation (methylbenzene) hot alkaline KMnOX4, then dilute acid benzoic acid
    Key factsDirecting effects of groups on the ring
    Group already on the ring New group goes to Effect on rate
    −NHX2, −OH, alkyl (−R, such as −CHX3) the 2- and 4-positions (and 6) activates: the ring reacts faster
    −NOX2, −COOH, −COR (such as −COCHX3) the 3-position (and 5) deactivates: the ring reacts more slowly

    Positions are counted from the carbon that already carries the group (carbon 1): 2 is next to it, 3 is two bonds away, 4 is opposite, and 6 is the other carbon next to it.

    Key factsHalogenoarene reactivity

    Chlorobenzene is much less reactive than chloroethane towards nucleophiles, because the chlorine lone pair delocalises into the ring (partial double-bond character, a stronger C–Cl bond), and the electron-rich ring repels nucleophiles.

    Revise: Arenes: benzene and its compounds

    Phenol and the acidity of organic compounds

    Key factsReactions that show phenol is acidic

    Phenol reacts with Na(s) (giving HX2) and with NaOH(aq), forming sodium phenoxide, CX6HX5OX−NaX+. It does not react with NaX2COX3 or NaHCOX3 to give COX2.

    Key factsWhy phenol is a stronger acid than ethanol and water

    The phenoxide ion is stabilised because a lone pair on oxygen overlaps with the ring’s π system, delocalising the negative charge over the ring. The alkoxide ion is destabilised by the electron-donating alkyl group. Order: phenol > water > ethanol.

    Key factsAcidity of chlorine-substituted ethanoic acids

    CHX3COOH<CHX2ClCOOH<CHClX2COOH<CClX3COOH in acid strength. Chlorine is electron-withdrawing, so it spreads the negative charge of the carboxylate ion and stabilises it. More Cl atoms, or Cl nearer the COOH group, give a stronger acid.

    Key factsAzo coupling

    Benzenediazonium salt + phenol (or 2-naphthol) in NaOH(aq), below 10 °C → azo compound, Ar–N=N–Ar′, coloured (yellow or orange for phenol). The diazonium ion attacks the 4-position of phenol (opposite OH). The azo group joins the two rings into one delocalised system that absorbs visible light.

    Revise: Phenol and the acidity of organic compounds

    Acyl chlorides and A Level carboxylic acids

    Key factsHydrolysis of chlorine compounds

    Rate: RCOCl (fast, cold water) > RCl (slow, heat with NaOH(aq)) > CX6HX5Cl (no reaction under these conditions). The acyl chloride carbon is very δ+ because it is bonded to both O and Cl; chlorobenzene’s C–Cl bond is strengthened by delocalisation of a chlorine lone pair into the ring.

    Revise: Acyl chlorides and A Level carboxylic acids

    Amines, amides and amino acids

    Key factsPreparation of amines
    Route Reagent and conditions Product
    halogenoalkane + ammonia excess NHX3 in ethanol, heated under pressure (sealed tube) primary amine
    halogenoalkane + primary amine primary amine in ethanol, heated under pressure (sealed tube) secondary amine
    nitrile + hydrogen LiAlHX4 in dry ether, or HX2 with a nickel catalyst, heated primary amine
    amide + hydride LiAlHX4 in dry ether primary or secondary amine
    Key factsRelative basicity

    ethylamine > ammonia > phenylamine. An alkyl group pushes electron density onto N (the lone pair is more available); the benzene ring draws the lone pair into the ring by delocalisation (the lone pair is less available).

    Key factsHydrolysis of amides
    Conditions Products from CHX3CONHX2 Equation
    dilute acid (e.g. HCl), heat ethanoic acid + ammonium ion CHX3CONHX2+HX++HX2O→CHX3COOH+NHX4X+
    NaOH(aq), heat ethanoate ion + ammonia CHX3CONHX2+OHX−→CHX3COOX−+NHX3

    An N-substituted amide gives an amine instead of ammonia: CHX3CONHCHX3+OHX−→CHX3COOX−+CHX3NHX2 in alkali, and CHX3NHX3X+ in acid.

    Key factsPredicting electrophoresis
    • Direction: positive ions (cations) move towards the negative electrode (cathode); negative ions (anions) move towards the positive electrode (anode).
    • Charge: the amino acid’s charge depends on pH. Below the isoelectric point it is positive, above it negative; at the isoelectric point it has no net charge and does not move.
    • Speed: the greater the charge and the smaller the species, the faster it moves.

    Revise: Amines, amides and amino acids

    Polymerisation

    Key factsAddition and condensation polymers
    Addition Condensation
    Monomer a C=C bond (usually one kind) two reactive groups per molecule (e.g. –OH, –COOH, –NH₂, –COCl)
    Products the polymer only the polymer and a small molecule (HX2O or HCl)
    Backbone all carbon atoms contains ester or amide links
    Examples poly(ethene), PVC polyesters, polyamides, proteins
    Key factsHydrolysis of condensation polymers
    Polymer With dilute acid, heat With aqueous alkali, heat
    polyester dicarboxylic acid and diol salt of the dicarboxylic acid (carboxylate ions) and diol
    polyamide dicarboxylic acid and a diammonium salt (the amine is protonated) salt of the dicarboxylic acid and the free diamine

    Revise: Polymerisation

    Organic synthesis (A Level)

    Key factsReducing agents in 9701
    Reducing agent Conditions Reduces Does not reduce
    NaBHX4 in water or ethanol aldehydes → primary alcohols, ketones → secondary alcohols C=C, carboxylic acids, esters, amides, nitriles
    LiAlHX4 dry ether (reacts violently with water) aldehydes, ketones, carboxylic acids (→ primary alcohols), amides (→ amines), nitriles (→ amines) C=C, the benzene ring
    HX2 with Pt or Ni heat C=C (→ C–C), the benzene ring (→ cyclohexane), nitriles (→ amines) acids, amides (not used for these in the syllabus)
    hot Sn with concentrated HCl, then NaOH(aq) heat under reflux a nitro group on a ring (→ NHX2) carboxylic acids (COOH is left alone)
    Key factsDirecting effects: which group goes where
    Group already on the ring New group goes to Effect on the ring
    −NHX2, −OH, alkyl (−R, e.g. −CHX3) the 2- and 4-positions (and 6) more reactive (activating)
    −NOX2, −COOH, −COR the 3-position (and 5) less reactive (deactivating)
    Key factsChanging the number of carbon atoms
    Step Effect Reagent and conditions
    halogenoalkane → nitrile +1 C KCN in ethanol, heat under reflux
    aldehyde or ketone → hydroxynitrile +1 C HCN with KCN catalyst, heat
    arene → alkylarene +1 C or more (the carbons of the alkyl group) CHX3Cl (or another halogenoalkane) with AlClX3, heat
    arene → aryl ketone +2 C (for CHX3COCl) CHX3COCl with AlClX3, heat
    alkylbenzene → benzoic acid the side chain is cut to one C (the COOH carbon) hot alkaline KMnOX4, then dilute acid
    alkene with C=C in the chain → two shorter molecules fewer C hot, concentrated, acidified KMnOX4
    Key factsRacemic products from planar groups

    When a reagent attacks a planar part of a molecule, it can approach from either face with equal probability. If the product has a chiral centre, both enantiomers form in equal amounts: a racemic mixture, which does not rotate the plane of plane-polarised light. The planar groups are the trigonal-planar carbon of a C=O group (aldehyde or ketone) and the carbocation of an SN1 reaction.

    Key factsWhat each spectrum tells you
    • Mass spectrum: the molecular ion peak gives Mr. A peak at M+2 with one-third the height of M shows one Cl; a 1:1 pair shows one Br. The [M+1]⁺ peak gives the number of carbon atoms, n=100×a(M+1)1.1×a(M), where a is the abundance (peak height).
    • Infrared: C=O (carbonyl, carboxyl) 1670–1740, C=O of an amide 1640–1690, ester 1710–1750, O–H of a carboxylic acid 2500–3000, O–H of an alcohol or phenol 3200–3600, N–H (amine, amide) 3300–3500, C≡N 2200–2250 cm⁻¹. The ring (C=C aromatic) is at 1500–1680.
    • ¹³C NMR: the number of peaks is the number of different carbon environments. Their shifts (Data section) show the type: 160–185 carboxyl; 190–220 aldehyde or ketone; 110–160 ring or C=C; 100–125 nitrile; 0–50 alkyl. See Chromatography and NMR spectroscopy and Infrared spectroscopy and mass spectrometry.
    Key factsTypical by-products
    Step By-products and side reactions
    nitration of an arene the other isomers (e.g. 2- as well as 4-), and dinitro compounds; water
    Friedel–Crafts alkylation more than one alkyl group on the ring; HCl
    halogenoalkane + NHX3 secondary and tertiary amines and quaternary ammonium salts; NHX4X
    halogenoalkane + NaOH an alkene by elimination (if the solvent is ethanol); NaX
    oxidation of a primary alcohol a carboxylic acid, if the aldehyde is not distilled out
    esterification water; an equilibrium, so unreacted acid and alcohol remain
    acid + PClX5 / SOClX2 / PClX3 POClX3 + HCl / SOX2 + HCl / HX3POX3
    acyl chloride + nucleophile HCl (or NHX4Cl with excess NHX3)
    diazotisation above 10 °C phenol and nitrogen
    reduction with Sn and HCl tin salts; the amine is present as its salt until NaOH is added

    Revise: Organic synthesis (A Level)

    Chromatography and NMR spectroscopy

    Key factsInterpreting a chromatogram
    • The number of spots is at least the number of substances in the mixture. Two compounds with the same Rf would overlap as a single spot.
    • A spot in the mixture’s lane at the same height as a standard may be that standard. Rf values only compare when the same plate material, solvent and temperature are used, and the safest comparison is on the same plate, as in the figure.
    • Matching Rf values suggest, but do not prove, that two compounds are the same. Repeating with a different solvent, where two different compounds are unlikely to match twice, is stronger evidence.
    FormulaPercentage composition from a chromatogram

    percentage of a component=area of its peaktotal area of all the peaks×100 Use areas, not heights: a tall thin peak can hold less than a short wide one. For a narrow peak, area ≈12×base×height.

    Key factsHow to count ¹³C environments
    1. Draw the structure with every carbon visible.
    2. Look for symmetry: carbon atoms that you could swap by a mirror plane or rotation, such as the two CHX3 groups of propan-2-ol, are equivalent.
    3. Count the groups of equivalent carbon atoms. That number is the number of peaks.
    Key factsThe n + 1 rule

    A signal from hydrogens that have n equivalent hydrogens on the adjacent carbon atoms is split into n + 1 lines.

    Revise: Chromatography and NMR spectroscopy

    Practical skills and qualitative analysis

    FormulaPercentage uncertainty

    percentage uncertainty=uncertainty in the measurementmeasured value×100 A quantity found from two readings (a titre from two burette readings, a temperature change from two thermometer readings) has twice the uncertainty of one reading.

    Revise: Practical skills and qualitative analysis

    Planning, analysis and evaluation

    Key factsDrawing and using a graph
    • Dependent variable (or processed value) on the y-axis, independent variable on the x-axis, each labelled with quantity / unit.
    • Scale: 1, 2 or 5 units (or ×10, ÷10 of these) to a 20 mm square, with the points filling at least half the grid in both directions. The origin need not be shown.
    • Plot each point as × or ⊙ and draw one straight line of best fit with about the same number of points above and below it.
    • Mark any anomalous point and ignore it when drawing the line.
    • Gradient: use two points on the line (not data points unless they are on it) more than half the length of the line apart, and show them on the graph. Gradient =Δy/Δx, with its unit.
    • y-intercept: the value of y where the line meets x=0; extend the line if the axis does not start at zero.

    Revise: Planning, analysis and evaluation

    PreviousPhysics formula and data sheetNextBiology key equations and statistics
    Toolbox/Formula sheets

    Biology key equations and statistics

    Reference

    Every key formula from the course in one place, each linked back to the chapter that teaches it.

    Cambridge 9700 papers give you the statistical formulae you need, but you must know which test to choose and how to read the result. This page collects every key box from the Biology chapters.

    Biology: the course map

    Key factsWhat is printed, and what is not
    • A calculator is allowed in every paper. You should show all your working and use appropriate units.
    • There is no data booklet, no formula sheet and no periodic table in any paper. You must remember every definition, equation and unit yourself. For the few equations you need (magnification, percentage change, rate), this course teaches them early and uses them often.
    • Statistics formulae (A Level only) are printed inside the question that needs them, and a table of critical values (the numbers a test result is compared with to decide whether it is significant) is given when needed. You do not have to remember the formulae. Degrees of freedom are never given, so you must be able to work them out (degrees of freedom is a number worked out from the sample sizes or the number of categories; see Statistics for biologists).
    • Paper 1 gives one mark per correct answer and takes nothing away for a wrong answer. Never leave a multiple-choice question blank: a guess costs nothing.
    • Papers 2 to 5 are answered on the question paper in black or dark blue pen, with HB pencil for diagrams and graphs. The marks for each part are shown in brackets, and they tell you how many separate points to make.
    Key factsFive habits of A* biology answers
    1. Write in mark-scheme language. Most marks are single creditworthy points. Use the exact terms (“active site”, “complementary shape”, “water potential gradient”, “partially permeable membrane”) and build cause-and-effect chains: change, so effect, so result.
    2. Obey the command word. “Describe” means say what happens (for a graph, the trend with figures and units). “Explain” means give the biological reason. “Suggest” means apply your knowledge to something unfamiliar. Count the marks: three marks means at least three separate points.
    3. Quote the data. Use the numbers and the units from the graph or table, name the turning point, and do any calculation with working shown. “Increases” alone is never enough.
    4. Know the practical skills as lists. Variables (independent, dependent, standardised), control, repeats and mean, sources of error, improvements and risk assessment appear in every practical question. Learn them as checklists and practise them.
    5. Practise with real past papers. Use the chapters to learn, then answer real questions under time. Mark yourself against the mark scheme and write down every point you missed.

    Revise: Biology: the course map

    Biology from zero

    Key factsCondensation and hydrolysis
    • Condensation: monomer + monomer → bigger molecule + water. Building up. Water is a product.
    • Hydrolysis: bigger molecule + water → monomer + monomer. Breaking down. Water is a reactant.
    • The word lysis means splitting, and hydro means water: splitting using water.
    Key factsThe length ladder

    1 mm=1000 μm1 μm=1000 nm1 mm=1 000 000 nm

    • Going to a smaller unit (mm → µm → nm): the number gets bigger, so multiply by 1000 for each step.
    • Going to a larger unit (nm → µm → mm): the number gets smaller, so divide by 1000 for each step.
    Key factsStandard form
    • Move the decimal point until you have one non-zero digit in front of it. The number of places you moved it is n.
    • Moved the point left (a big number): n is positive. 4 500 000=4.5×106.
    • Moved the point right (a small number): n is negative. 0.000 003=3×10−6.
    • On a calculator use the key marked EXP or ×10x; 4.5×106 is entered as 4.5 EXP 6.
    Key factsReading a graph
    1. Axes: what is on each axis, and in what units? Cambridge writes them as quantity / unit, for example time / s. The independent variable (the one you change) goes on the x-axis; the dependent variable (the one you measure) goes on the y-axis.
    2. Scale: how much does one small square or one gridline represent? Work it out before you read a value.
    3. Read values by ruling a line up or across to the curve and then to the axis. Quote the number with its unit.
    4. Describe the shape: is it rising, falling, flat, or does it peak? Where are the turning points? Use figures from the graph.
    5. Gradient gives a rate: gradient =change in ychange in x, with units “y unit per x unit”.

    Revise: Biology from zero

    Microscopy and magnification

    Key factsTotal magnification

    Total magnification = magnification of the eyepiece lens × magnification of the objective lens.

    With a ×10 eyepiece: the ×10 objective gives ×100 and the ×40 objective gives ×400.

    FormulaMagnification

    M=IA M = magnification (a number, written ×500, with no units); I = size of the image (drawing or micrograph); A = actual size of the specimen. I and A must be in the same unit before you divide.

    Key factsSteps for any magnification calculation
    1. Measure the image with a ruler, in mm, along its longest dimension (or read the scale bar).
    2. Convert so that image and actual size use the same unit. Most often convert to µm (mm × 1000).
    3. Substitute into the correct form of M=I/A and calculate.
    4. Check the size is sensible, give the unit for a length (µm, not “×”) and write a magnification as ×, with no unit.
    Key factsCalibrating the graticule (one objective at a time)
    1. Put the stage micrometer on the stage and focus on its scale with the chosen objective.
    2. Turn the eyepiece (and move the slide) until the zero marks of the two scales line up.
    3. Look along the scales for another place where a mark on the graticule lines up exactly with a mark on the stage micrometer. Read how many eyepiece units match how many stage divisions.
    4. Length of the stage divisions =divisions×10 μm.
    5. length of one eyepiece unit=length of the stage divisionsnumber of eyepiece units.
    6. Take the stage micrometer away, put the specimen on, and measure it in eyepiece units. Actual length = units × length of one eyepiece unit.
    Key factsWhy electron microscopes resolve more

    Electrons behave as waves with a much shorter wavelength than light. A microscope that uses a beam of electrons can therefore distinguish points that are much closer together: its resolution is higher. In practice a transmission electron microscope (TEM) can resolve points about 0.5 nm apart, compared with about 200 nm for a light microscope.

    Revise: Microscopy and magnification

    Eukaryotic cells and organelles

    Key factsThe secretory pathway (the marking points)
    1. The protein is made on ribosomes on the rough endoplasmic reticulum.
    2. It passes into the RER, where it is folded (and processed).
    3. A vesicle buds off the RER and carries it to the Golgi body.
    4. The Golgi body modifies and packages the protein into secretory vesicles.
    5. The secretory vesicles move to the cell surface membrane and fuse with it, releasing the protein outside the cell. This is exocytosis.
    Key factsATP and the cell

    Cells use ATP (adenosine triphosphate), made in respiration, as the immediate source of energy for energy-requiring processes. ATP is made from the energy released when nutrients such as glucose are broken down, mostly in the mitochondria; it is broken down when a cell needs energy, and the energy released is used for processes such as active transport, protein synthesis, movement and cell division.

    Revise: Eukaryotic cells and organelles

    Prokaryotes and viruses

    Key factsStructure of a typical bacterium
    • Unicellular: each bacterium is one cell that lives and reproduces on its own.
    • Small: generally 1–5 µm in diameter, roughly ten times smaller than most eukaryotic cells.
    • Cell wall made of peptidoglycan, a polymer of sugar chains cross-linked by short chains of amino acids. It is rigid and strong, and it is not cellulose.
    • Cell surface membrane inside the wall, which controls what enters and leaves, as in eukaryotic cells.
    • Circular DNA: one loop of DNA in the cytoplasm, not surrounded by a nuclear envelope and not wound around histone proteins (the proteins that DNA is wound around in eukaryotes) as eukaryotic DNA is. The region where it lies is the nucleoid.
    • 70S ribosomes in the cytoplasm, which make proteins. They are smaller than the 80S ribosomes of eukaryotic cells.
    • No organelles surrounded by double membranes: no nucleus, mitochondria or chloroplasts. Bacteria also have no endoplasmic reticulum or Golgi body.
    Key factsWhat every virus has, and what it lacks
    • A nucleic acid core: DNA or RNA (never both in the same virus).
    • A capsid: a coat of protein subunits around the nucleic acid.
    • Some viruses have an envelope, a layer of phospholipid (from the host cell) with proteins in it. Viruses with no envelope are called non-enveloped.
    • A virus is not a cell. It has no cytoplasm, no cell surface membrane (the envelope, where present, is not one), no ribosomes and no way of releasing energy for itself.
    • A virus is very small: about 20–300 nm, much smaller than a bacterium, and below the resolution of a light microscope (200 nm) for many viruses.

    Revise: Prokaryotes and viruses

    Carbohydrates

    Key factsCarbohydrates in one box
    • Carbohydrates contain C, H and O only.
    • Monosaccharides have the formula (CHX2O)n: triose n=3, pentose n=5, hexose n=6.
    • Glucose, fructose and galactose are all hexoses with the formula CX6HX12OX6.
    Key factsHow to draw the glucose ring (the order that works)
    1. Draw a six-sided ring with the oxygen at the top right. The thick edge is at the front (bottom).
    2. Write CHX2OH above the carbon at the top left (carbon 5). That is carbon 6.
    3. Carbon 1 is on the right. Write OH below the ring for α, and OH above the ring for β. The H goes on the other side.
    4. Carbon 2: OH below. Carbon 3: OH above. Carbon 4: OH below. The H goes on the opposite side each time.
    5. Check: α = OH on carbon 1 down, β = OH on carbon 1 up. Carbons 2, 3 and 4 are identical in both.
    Key factsThe role of covalent bonds

    Covalent bonds join monomers into polymers (here, monosaccharides into disaccharides and polysaccharides). They are strong, so the polymer is stable; they can be broken, one at a time, by hydrolysis.

    Key factsReducing or non-reducing?
    • Reducing: glucose, fructose, galactose, maltose (and lactose).
    • Non-reducing: sucrose, and the polysaccharides starch, glycogen and cellulose.
    • Benedict’s test: blue stays blue (negative) for a non-reducing sugar; brick-red precipitate (positive) for a reducing sugar.
    Key factsWhy starch and glycogen are good stores
    • Compact (coiled or branched): a lot of glucose in a small space.
    • Insoluble and large: they do not dissolve in the cytoplasm and cannot diffuse out of the cell, and they have no osmotic effect (they do not lower the water potential of the cell, unlike many separate glucose molecules).
    • A polymer of α-glucose: enzymes hydrolyse the glycosidic bonds to release glucose for respiration.
    • Branched (amylopectin, glycogen): many free ends, so glucose is released rapidly.

    Revise: Carbohydrates

    Lipids and water

    Key factsWater in one box
    • Hydrogen bonds form between the δ+ hydrogen of one water molecule and the δ− oxygen of another.
    • Solvent: polar molecules and ions dissolve, so water transports substances in blood and xylem and is the medium for reactions in the cell.
    • High specific heat capacity: much energy (breaking hydrogen bonds) is needed to warm water, so temperatures in cells and in water habitats stay steady.
    • High latent heat of vaporisation: much energy is needed to evaporate water, so evaporation (sweating, transpiration) cools the surface.
    Key factsTriglyceride in one box
    • One glycerol + three fatty acids, joined by three ester bonds; three molecules of water are released (condensation).
    • Non-polar and hydrophobic: the tails are hydrocarbon chains, so triglycerides do not dissolve in water.
    • Saturated (no C=C, straight tails, solid fats) or unsaturated (C=C, kinked tails, liquid oils).

    Revise: Lipids and water

    Proteins

    Key factsLevels of structure and what holds them
    • Primary: order of amino acids; peptide bonds.
    • Secondary: α-helix or β-pleated sheet; hydrogen bonds between C=O and N–H of the backbone.
    • Tertiary: 3D folding of one polypeptide; R-group interactions: hydrophobic interactions, hydrogen bonds, ionic bonds, disulfide bonds.
    • Quaternary: two or more polypeptides together (same interactions).
    • The primary structure decides all the others: a different sequence means different R groups, so different bonds and a different fold.

    Revise: Proteins

    Testing for biological molecules

    Key factsThe four tests in one table
    Test Positive Negative Heat?
    Benedict’s (reducing sugar) green, yellow, orange or brick-red precipitate stays blue yes
    Benedict’s after acid (non-reducing sugar) green to brick-red after acid, blue before blue at both stages yes, with acid
    Iodine (starch) blue-black stays orange-brown no
    Emulsion (lipid) cloudy white emulsion stays clear no
    Biuret (protein) purple (lilac) stays blue no

    Revise: Testing for biological molecules

    Enzymes

    Key factsEnzymes in one box
    • Enzymes are globular proteins that act as biological catalysts.
    • Intracellular enzymes work inside the cell that made them; extracellular enzymes are secreted and work outside it.
    • An enzyme is not used up: one molecule catalyses the same reaction many times.
    Key factsHow an enzyme works (the marking points)
    1. The substrate has a shape complementary to the active site.
    2. Substrate and enzyme collide; the substrate binds to the active site, forming an enzyme–substrate complex.
    3. (Induced fit) the active site changes shape slightly to fit the substrate closely, straining bonds.
    4. The activation energy is lowered, so the reaction happens faster.
    5. Products form, leave the active site, and the enzyme is unchanged and reused.
    Key factsFinding the initial rate from a progress curve
    1. Draw a straight tangent that touches the curve at t=0 and follows its initial slope.
    2. Make a large triangle under the tangent (at least half the grid).
    3. Initial rate =change in ychange in x, with units, for example cm3 s−1.
    Key factsUsing a colorimeter to follow an enzyme reaction
    1. Filter: choose the filter of the colour the solution absorbs most, which is the complementary colour. A solution looks blue because it absorbs red light, so a blue or blue-black solution needs a red filter; a purple or red solution absorbs green light, so it needs a green filter.
    2. Blank: fill a cuvette with everything except the coloured substance (for starch tests, iodine solution in water) and set the reading to zero absorbance (100% transmission).
    3. Calibration curve: measure the absorbance of solutions of known concentration, plot absorbance against concentration and draw the line of best fit.
    4. Samples: take samples of the reaction mixture at set times, treat them in the same way (for example, add the same volume of iodine) and read the absorbance. Read each concentration from the calibration curve.
    5. Keep the cuvettes clean and use the same cuvette, the same way round, so only the sample changes.

    Revise: Enzymes

    Enzyme kinetics, inhibitors and immobilised enzymes

    Key factsReading Vmax and Km from a graph
    1. Find Vmax: the rate the curve levels off towards (draw a dashed horizontal line through the plateau).
    2. Calculate half of Vmax: divide Vmax by two. This is a rate on the y-axis.
    3. Draw a horizontal line across from half Vmax to the curve, then a vertical line down to the x-axis.
    4. Read the substrate concentration where it meets the axis: that is Km. Give it units of concentration (for example mmol dm⁻³).
    Key factsEffect of a competitive inhibitor on the graph
    • Vmax is unchanged: with enough substrate, the substrate wins the competition and every active site is eventually filled with substrate.
    • Km increases: more substrate is needed to reach half Vmax, so the apparent affinity of the enzyme for its substrate is lower.
    • The rate is reduced most at low substrate concentrations, and hardly at all at very high ones.
    Key factsEffect of a non-competitive inhibitor on the graph
    • Vmax decreases: some enzyme molecules cannot work however much substrate is present.
    • Km is unchanged: the enzyme molecules that still work bind the substrate as well as before, so half of the (lower) Vmax is still reached at the same substrate concentration.
    • Raising the substrate concentration does not bring the rate back up to the uninhibited curve.
    Key factsCompetitive and non-competitive inhibitors compared
    feature Competitive Non-competitive
    Shape similar to the substrate need not resemble it
    Binds to the active site elsewhere (allosteric site)
    More substrate reduces its effect does not help
    Vmax unchanged decreases
    Km increases unchanged

    Revise: Enzyme kinetics, inhibitors and immobilised enzymes

    Membrane structure and cell signalling

    Key factsWhy the bilayer is a barrier
    1. The tails in the middle are hydrophobic, so ions and polar molecules (glucose, amino acids) cannot dissolve in them.
    2. Small, non-polar (lipid-soluble) molecules such as oxygen, carbon dioxide and fatty acids dissolve in the tails and cross freely.
    3. Everything else needs a transport protein.
    Key factsRoles of the components (one line each)
    • Phospholipids: form the bilayer; permeability (barrier to ions and polar molecules); fluidity.
    • Cholesterol: regulates fluidity (less at high temperature, stays fluid at low); reduces permeability; stability.
    • Glycolipids and glycoproteins: carbohydrate chains on the outer surface: cell recognition (antigens), receptors, cell adhesion.
    • Channel and carrier proteins: transport of ions and polar molecules.
    • Receptors: cell signalling.

    Revise: Membrane structure and cell signalling

    Movement into and out of cells

    Key factsFactors that change the rate of diffusion
    1. Concentration gradient: a steeper gradient gives a faster rate.
    2. Distance: a shorter diffusion distance (a thinner membrane or layer) gives a faster rate.
    3. Surface area: a larger surface area (for example folded membranes) gives a faster rate.
    4. Temperature: higher temperature gives molecules more kinetic energy, so they move faster and the rate is higher.
    5. Size of the molecule: smaller molecules diffuse faster than larger ones.
    6. Lipid solubility and charge: lipid-soluble, uncharged molecules cross a bilayer faster than charged or polar ones.
    Key factsPredicting the direction of osmosis
    1. Water moves from the less negative (higher) water potential to the more negative (lower) water potential.
    2. It crosses a partially permeable membrane. The solutes stay where they are: only water moves.
    3. Movement continues until the water potentials are equal, when there is no net movement.

    Revise: Movement into and out of cells

    Chromosomes and the mitotic cell cycle

    Key factsChromosome structure: the five terms
    • DNA: one long molecule in each chromosome (or in each chromatid, once copied).
    • Histone proteins: the DNA is wound round them and coiled up to make the chromosome short and thick.
    • Sister chromatids: the two identical copies of a replicated chromosome.
    • Centromere: joins the sister chromatids and is where spindle fibres attach.
    • Telomeres: the ends of the chromosome; repeated non-coding sequences.
    Key factsCounting rule
    • A chromosome is one DNA molecule (with its proteins) when it has not been copied.
    • After DNA replication a chromosome has two chromatids and contains two DNA molecules. It is still one chromosome.
    • When the sister chromatids separate in anaphase, each chromatid is now a chromosome.
    FormulaMitotic index

    mitotic index=number of cells in mitosistotal number of cells Give it as a decimal or multiply by 100 for a percentage. Count the cells in all four stages, and count the total number of cells including those in interphase.

    Key factsTime spent in a stage

    time in a stage=cells in that stagetotal cells×time for the whole cell cycle This works because the cells on a slide are at random points in their cycle, so the proportion of cells in a stage matches the proportion of the cycle that the stage takes.

    Key factsWhat telomeres do
    1. The end of a chromosome is not fully copied at each DNA replication, so the chromosome gets shorter.
    2. The telomere is non-coding DNA, so what is lost is part of the telomere, not genes.
    3. Telomeres therefore prevent the loss of genes from the ends of chromosomes.
    4. Telomeres get a little shorter at every division. When they are very short, the cell stops dividing, which limits the number of times a cell can divide.

    Revise: Chromosomes and the mitotic cell cycle

    Nucleic acids and DNA replication

    Key factsNucleotides in one box
    • A nucleotide = pentose sugar + phosphate group + nitrogenous base.
    • DNA sugar: deoxyribose. RNA sugar: ribose.
    • Purines (A, G): double ring. Pyrimidines (C, T, U): single ring.
    • DNA bases: A, G, C, T. RNA bases: A, G, C, U.
    Key factsThe structure of DNA
    1. Two polynucleotide strands wound into a double helix.
    2. Each strand has a sugar–phosphate backbone, with nucleotides joined by phosphodiester bonds.
    3. The strands are antiparallel (one runs 5′ to 3′, the other 3′ to 5′).
    4. Complementary base pairing: A–T (two hydrogen bonds) and C–G (three hydrogen bonds); a purine always pairs with a pyrimidine, so the width is constant.
    5. The sequence of bases carries the genetic information.
    Key factsBase ratios in double-stranded DNA
    • The number of A = the number of T, and the number of C = the number of G.
    • The total of purines (A + G) = the total of pyrimidines (T + C), each 50% of the bases.
    • If you know the percentage of one base, you can find all four.

    Revise: Nucleic acids and DNA replication

    Protein synthesis and gene mutation

    Key factsFeatures of the genetic code
    • Triplet: three bases code for one amino acid.
    • Degenerate: most amino acids are coded for by more than one triplet (64 triplets, 20 amino acids).
    • Non-overlapping: each base is part of only one triplet, and the bases are read in groups of three, one group after another.
    • Universal: the same triplets code for the same amino acids in almost all organisms (bacteria, plants, animals and viruses).
    • Start and stop: the codon AUG codes for methionine and is where translation starts. The codons UAA, UAG and UGA are stop codons: they code for no amino acid, and translation stops there.
    Key factsFrom DNA to polypeptide in one chain

    DNA template triplet → mRNA codon → tRNA anticodon → amino acid. For example: template TAC → codon AUG → anticodon UAC → methionine.

    Key factsThree kinds of gene mutation
    1. Substitution: one nucleotide is replaced by a different one.
    2. Insertion: one or more extra nucleotides are added to the sequence.
    3. Deletion: one or more nucleotides are removed from the sequence.
    Key factsInsertions and deletions
    • Insert or delete 1 or 2 bases (not a multiple of 3): frameshift. Every amino acid after the change is affected.
    • Insert or delete 3 bases (or a multiple of 3): the frame is kept. One amino acid (or a few) is added or lost and the rest is unchanged.
    • A substitution never causes a frameshift, because the number of bases is the same.

    Revise: Protein synthesis and gene mutation

    Plant tissues and water transport

    Key factsWhat is transported

    Some mineral ions and organic compounds are transported in plants dissolved in water. Water is the solvent in both tissues. Xylem moves water and mineral ions; phloem moves sucrose and amino acids dissolved in water.

    Key factsRules for a plan diagram
    1. Use a sharp pencil and draw clear single lines. No shading, no colouring, no cells.
    2. Draw only tissue boundaries: the outline of the section, and the regions of xylem, phloem and other tissues.
    3. Keep proportions correct: if the xylem fills a quarter of the radius on the slide, it fills a quarter of the radius in your drawing.
    4. Make the drawing large (fill at least half the space), with a title that says what the section is (for example “TS of a dicotyledonous stem”) and the magnification or a scale bar.
    5. Label with ruled lines that touch the tissue, do not cross, and are written horizontally to the side of the drawing.
    Key factsWhere the tissues are in a dicotyledonous stem
    • The vascular bundles form a ring near the outside of the stem.
    • In each bundle, xylem is on the inside (towards the centre) and phloem on the outside. A layer of cambium, which can divide to make new xylem and phloem, lies between them.
    • The cortex lies outside the ring of bundles, under the epidermis; the pith fills the centre.
    Key factsWhere the tissues are in a dicotyledonous root
    • The vascular tissue is in the centre of the root. The xylem forms a star or cross shape, and the phloem lies in patches between the arms of the xylem.
    • Around the vascular tissue is a ring of cells called the pericycle and, outside it, the endodermis, then the wide cortex.
    • The outer layer is the epidermis, with root hairs (extensions of single epidermal cells).
    Key factsWhere the tissues are in a dicotyledonous leaf
    • The midrib holds the main vascular bundle, with xylem on the upper side and phloem on the lower side. Smaller veins branch through the spongy mesophyll, with the same arrangement.
    • Below the upper epidermis lies a layer of palisade mesophyll, with the spongy mesophyll below it, then the lower epidermis, which has most of the stomata.

    Revise: Plant tissues and water transport

    Blood vessels, blood and tissue fluid

    Key factsWhy a double circulation?
    • The blood loses pressure as it squeezes through the narrow capillaries of the lungs, so it returns to the heart at low pressure.
    • The heart pumps it again, so the blood enters the systemic circulation at high pressure. High pressure carries blood quickly to the whole body and drives tissue fluid out of the capillaries (see below).
    • The lungs, which have delicate thin-walled capillaries, receive blood at a lower pressure than the body, which avoids damaging them.
    • A double circulation also keeps oxygenated and deoxygenated blood apart, so tissues receive fully oxygenated blood.
    Key factsStructure and function of the vessels
    Vessel Wall Lumen Special feature Why it suits the job
    elastic artery very thick; much elastic tissue and collagen narrow stretches and recoils withstands high pressure; evens out the pulse
    muscular artery thick; more smooth muscle narrow muscle contracts or relaxes controls the blood flow to an organ
    capillary one layer of endothelium about 7 µm gaps between cells short diffusion distance for exchange
    vein thin; little muscle and elastic tissue wide valves low pressure, no stretch needed; valves stop backflow
    Key factsTelling the white blood cells apart
    1. Look at the nucleus first: several lobes means a neutrophil; kidney shape means a monocyte; a round nucleus filling almost the whole cell means a lymphocyte.
    2. Then size: a lymphocyte is about the size of a red cell (a little larger); a monocyte is the biggest.
    3. A red blood cell has no nucleus and a pale middle where it is thin (the biconcave shape), and it is far more numerous than any white cell.
    Key factsThe functions of tissue fluid
    • It supplies the cells with oxygen, glucose, amino acids and other nutrients, which diffuse (or are carried by facilitated diffusion or active transport) from the tissue fluid into the cell.
    • It removes carbon dioxide and wastes such as urea, which diffuse out of the cells into the tissue fluid and then into the capillaries.
    • It bathes the cells, so that they stay in a watery environment of stable composition, and their surroundings stay steady.

    Revise: Blood vessels, blood and tissue fluid

    Transport of oxygen and carbon dioxide

    Key factsUsing the shape of the curve
    • In the lungs (pO₂ about 13 kPa) the curve is on its flat top: haemoglobin is about 97% saturated, so it loads nearly all the oxygen it can. Even if the pO₂ falls a little, the saturation hardly changes, which gives a safety margin.
    • In respiring tissues (low pO₂, in a resting muscle about 4 kPa, in a very active one lower) the curve is on its steep part: a small fall in pO₂ releases a large amount of oxygen. The tissues that use oxygen fastest, and so have the lowest pO₂, get the most.
    • The haemoglobin does not give up all of its oxygen: in a resting muscle it is still about 61% saturated when it leaves, which is a reserve for exercise.
    Key factsWhy the chloride shift matters
    • It keeps the charge balanced across the membrane, so hydrogen carbonate ions can keep leaving the cell.
    • This keeps the concentration of HCOX3X− inside the red cell low, so the carbonic anhydrase reaction keeps going to the right and the cell can take up more carbon dioxide.
    • So more carbon dioxide is carried (as HCOX3X− in the plasma) than the red cell alone could hold.
    Key factsThe Bohr shift and why it matters
    • Active tissues (respiring fast) have a high pCO₂ and a low pO₂. The curve shifts right: haemoglobin has a lower affinity, so at the same pO₂ it unloads more oxygen (in the example, 20 percentage points more at 4 kPa). The tissues that most need oxygen receive more.
    • In the lungs the pCO₂ is low, so the curve is further to the left: haemoglobin has a higher affinity and loads oxygen fully, even though the pO₂ is only 13 kPa.
    • The shift is useful: it matches the oxygen delivery to the oxygen use, without the need for any change in the blood flow.

    Revise: Transport of oxygen and carbon dioxide

    The heart and the cardiac cycle

    Key factsThe heart in one box
    • Four chambers: two thin-walled atria receive blood from the veins; two thick-walled ventricles pump blood out into the arteries.
    • Septum keeps oxygenated and deoxygenated blood apart.
    • AV valves (tricuspid, bicuspid) lie between atrium and ventricle, held by tendinous cords. Semilunar valves lie at the base of the aorta and the pulmonary artery.
    • Coronary arteries supply the cardiac muscle.
    Key factsWhy the walls differ
    • Atria (thin walls): an atrium pumps blood only a short distance, into the ventricle just below it. Most of the filling happens by itself, because the blood is flowing in under the low pressure of the veins. A small, low-pressure squeeze is enough.
    • Ventricles (thick walls): a ventricle must pump the blood out of the heart and along an artery, so it must generate a much higher pressure.
    • Left ventricle (thickest): it pumps blood to the whole body (every organ except the lungs), a long path with a great resistance to flow (resistance: how hard it is for blood to flow through the narrow vessels), so it must produce a high pressure, about 16 kPa at its peak.
    • Right ventricle (thinner): it pumps blood only to the lungs, which are close by, at a much lower pressure. Low pressure in the lungs avoids damaging the delicate walls of the lung capillaries and means that less tissue fluid is forced out into the alveoli (the tiny air sacs at the ends of the airways in the lungs).
    FormulaHeart rate from the cardiac cycle

    heart rate / beats per minute=60time for one cycle / s

    If you are given the time for several cycles, divide by the number of cycles first.

    Key factsThe rule for every valve
    • Blood flows from higher pressure to lower pressure.
    • A valve opens when the pressure behind it (upstream) is higher than the pressure in front of it.
    • A valve closes when the pressure in front of it (downstream) is higher than the pressure behind it, and so it stops the blood flowing backwards.
    • In an answer always say which chambers you compare: “the pressure in the left ventricle is higher than the pressure in the left atrium, so the AV valve closes”.
    Key factsRoles, in the order they act
    • SAN: sets the rhythm (pacemaker) and starts the wave of excitation across the atria.
    • Fibrous tissue: stops the wave crossing straight to the ventricles.
    • AVN: delays the wave (about 0.1 s), so atrial systole finishes before ventricular systole starts.
    • Bundle of His and Purkyne tissue: carry the wave quickly down the septum to the apex and up the ventricle walls, so the ventricles contract together from the bottom upwards.

    Revise: The heart and the cardiac cycle

    The gas exchange system

    Key factsWhat a good exchange surface needs
    1. A large surface area, so many molecules can cross at once.
    2. A short diffusion distance: a thin surface, one or two cells thick.
    3. A steep concentration gradient, kept steep by moving the air (ventilation) on one side and the blood on the other.
    4. A moist surface, so that gases can dissolve before they diffuse.
    Key factsThe airway tree in one box
    • Path of air: trachea → bronchi → bronchioles → alveoli.
    • Each branch is narrower than the one before, but together the branches give a huge number of alveoli.
    • The lungs are made of the bronchi and bronchioles, the alveoli and the capillary networks around them.
    • Gas exchange happens only in the alveoli.
    Key factsWhich tissue does which job
    • Cilia + goblet cells + mucous glands: trap and remove dust and pathogens.
    • Cartilage: keeps the larger airways open.
    • Smooth muscle: narrows or widens the airway.
    • Elastic fibres: stretch on breathing in, recoil on breathing out.
    • Squamous epithelium: a very thin wall for gas exchange.
    Key factsHow the alveoli are adapted for gas exchange
    Feature Why it helps
    hundreds of millions of tiny alveoli a very large total surface area for gas exchange
    wall of squamous epithelium, one cell thick a very short diffusion distance
    capillary wall of squamous endothelium, one cell thick, with the capillary very close to the alveolus a second very short distance; red blood cells are pressed close to the air
    a dense network of capillaries around each alveolus a large blood supply; the blood carries oxygen away and brings carbon dioxide, which keeps the gradient steep
    ventilation brings fresh air into the alveoli and removes air rich in carbon dioxide the gradient for oxygen and for carbon dioxide stays steep
    a thin film of water on the inside of the wall oxygen dissolves in it before it diffuses into the cells
    elastic fibres in the wall the alveolus stretches as it fills and recoils to push air out

    Revise: The gas exchange system

    Infectious diseases

    Key factsThe four diseases at a glance
    Disease Pathogen Type How it is transmitted
    cholera Vibrio cholerae bacterium water or food contaminated with faeces
    malaria Plasmodium falciparum, P. malariae, P. ovale, P. vivax protoctists bite of an infected female Anopheles mosquito; also blood transfusion and mother to fetus
    tuberculosis (TB) Mycobacterium tuberculosis and M. bovis bacteria droplets from coughs and sneezes; M. bovis also in unpasteurised milk and meat from infected cattle
    HIV/AIDS human immunodeficiency virus (HIV) virus body fluids: unprotected sex, shared needles, blood products, mother to child

    Revise: Infectious diseases

    Antibiotics and resistance

    Key factsAntibiotics in one box
    • An antibiotic kills bacteria or stops them growing, with little or no harm to human cells.
    • It is selective because it attacks a target that bacteria have and human cells do not (such as the peptidoglycan wall).
    • Antibiotics act on bacteria only: not on viruses, and not on the human cells that the viruses live in.
    • A broad-spectrum antibiotic affects many types of bacteria; a narrow-spectrum one affects only a few.

    Revise: Antibiotics and resistance

    The immune response

    Key factsAntigens in one box
    • An antigen is a molecule (usually a protein or glycoprotein) that stimulates an immune response.
    • Self antigens are on the body’s own cells and are ignored. Non-self antigens are on pathogens, toxins and other people’s cells and are attacked.
    • The immune system recognises an antigen by its shape, which must be complementary to the shape of a receptor.
    Key factsPhagocytosis: the marking points
    1. Phagocytes move towards the pathogen (chemotaxis) in response to chemicals it releases.
    2. Receptors on the phagocyte bind to antigens on the pathogen.
    3. The pathogen is engulfed by endocytosis and held in a phagosome.
    4. Lysosomes fuse with the phagosome and release hydrolytic enzymes.
    5. The enzymes digest the pathogen.
    Key factsThe primary response: the marking points
    1. Macrophage engulfs the pathogen and presents the antigen.
    2. T-helper cell with a complementary receptor binds, is activated and releases cytokines.
    3. Cytokines stimulate B-lymphocytes (with complementary receptors) to divide by mitosis.
    4. B cells become plasma cells that secrete antibodies.
    5. T-killer cells are activated and destroy infected cells.
    6. Some cells become memory cells.

    Revise: The immune response

    Antibodies and vaccination

    Key factsAntibody structure in one box
    • Four polypeptides: two heavy, two light, joined by disulfide bonds: quaternary structure.
    • Y-shaped. The variable regions at the tips make two identical antigen-binding sites, complementary to one antigen.
    • The constant region is the same in every antibody (phagocytes recognise it). The hinge gives flexibility.
    Key factsThe hybridoma method: the marking points
    1. Mouse injected with the antigen; B-lymphocytes taken from its spleen.
    2. B cells fused with myeloma cells to make hybridoma cells.
    3. Hybridomas screened for the wanted antibody.
    4. Selected hybridoma cloned and cultured; antibody collected and purified.
    5. Hybridoma: antibody production from the B cell, indefinite division from the myeloma cell.
    Key factsHow a vaccine gives long-term immunity
    1. The vaccine contains antigens (not antibodies) and cannot cause the disease.
    2. It causes a primary response: antibodies and memory cells are made.
    3. Memory cells are long-lived, so on later infection there is a rapid secondary response.

    Revise: Antibodies and vaccination

    Maths skills for biology

    Key factsConverting units
    1. Write down the conversion that links the two units: 1 mm=1000 μm, 1 μm=1000 nm, 1 dm3=1000 cm3, 1 kg=1000 g, 1 g=1000 mg, 1 min=60 s.
    2. Going to a smaller unit, the number gets bigger, so multiply. Going to a larger unit, the number gets smaller, so divide.
    3. Check that the answer is sensible: 0.35 mm is a small number of mm, so it must be a few hundred µm.
    Key factsThe significant figures rule (Cambridge)
    1. The answer to a calculation has the same number of significant figures as the least precise data you used, or one more.
    2. Keep every digit on your calculator during the working. Round once, at the end.
    3. Record measurements to the precision of the instrument, with the same number of decimal places down a column: a ruler marked in mm gives 4.0 cm, not 4 cm; a balance reading 0.01 g gives 2.50 g, not 2.5 g.
    FormulaPercentage change

    percentage change=final value−initial valueinitial value×100 A positive answer is a percentage increase (or gain) and a negative answer is a percentage decrease (or loss). Always divide by the initial value, the one you started with.

    FormulaPercentage error

    percentage error=uncertaintymeasured value×100 For a single reading, the uncertainty is half the smallest division of the instrument (so a reading on a scale marked every 0.2 cm³ has an uncertainty of ±0.1 cm3). If a value is found by subtracting two readings (a start and an end), the two uncertainties add.

    FormulaShapes you must be able to calculate
    • Rectangle: area =l×w; perimeter =2(l+w).
    • Triangle: area =12×base×height.
    • Circle (radius r): area =πr2; circumference =2πr. The radius is half the diameter.
    • Cuboid: volume =l×w×h; surface area =2(lw+lh+wh).
    • Cylinder (radius r, length h): volume =πr2h; surface area of the curved side =2πrh; each flat end =πr2.
    Key factsDrawing a graph
    1. The independent variable (the one you change) goes on the x-axis; the dependent variable (the one you measure) on the y-axis.
    2. Label each axis with the quantity / unit, exactly as in the table heading.
    3. Choose a scale that uses more than half of the grid in each direction, with simple steps (1, 2, 5 or 10 per large square, never 3 or 7). Start at zero unless that would waste most of the grid.
    4. Plot every point as a small cross or a dot in a circle.
    5. If the data show a clear trend and the points scatter, draw a line of best fit: a smooth curve or a ruled straight line with points on both sides. If you cannot say what happens between points, join them with ruled straight lines.
    6. Do not extend a line beyond the data unless the question says you can.
    Key factsFinding the rate at a point on a curve
    1. Mark the point on the curve (for example at t=2 min).
    2. Lay a ruler against the curve at that point, with the curve on the same side of the ruler either side of the point. Check that the angles between the ruler and the curve look equal on both sides. Draw the line.
    3. Make a large right-angled triangle under the tangent. Read two points on the tangent, far apart. Use points on the tangent, not points on the curve.
    4. Rate =change in ychange in x, with units.

    Revise: Maths skills for biology

    Practical skills: investigations (Paper 3)

    FormulaDilution

    c1V1=c2V2 c1 and V1 are the concentration and volume of the stock solution used; c2 and V2 are the concentration and final volume wanted. So the volume of stock to use is V1=c2V2c1, and you add water to make up the final volume V2.

    Key factsRules for a results table
    1. A ruled table with a heading for every column (and every row, if the rows are labelled).
    2. Each heading says the quantity and the unit, separated by a solidus: “temperature / °C”, “time / s”. No units in the body of the table.
    3. The independent variable is in the left-hand column (or the top row), and the dependent variable to its right.
    4. Raw data are recorded to the precision of the instrument, with the same number of decimal places in a column.
    5. Replicates have their own columns (or rows), followed by the mean and any calculated value, with headings and units for these too.
    6. Qualitative results are recorded in words that are clear and specific.
    Key factsA graph that scores full marks
    1. Axes: the independent variable on the x-axis, the dependent variable on the y-axis, each labelled with the same quantity and unit as the table heading.
    2. Scale: simple steps (1, 2, 5 or 10 per large square), using more than half of the grid in each direction, so that each value can be read to within half a small square. Start at zero if that is sensible.
    3. Points: plotted accurately as small crosses (×) or dots in circles (⊙), never large blobs that hide the position.
    4. Line: a smooth curve of best fit or a ruled straight line of best fit if there is a clear trend and the points scatter around it; ruled straight lines joining the points if you cannot be sure what happens between the points, or if the data are not expected to follow a smooth curve.
    5. No extrapolation past the last point, unless the question says it can be justified.
    6. Sharp pencil, ruler for straight lines, a title if the question asks for one.

    Revise: Practical skills: investigations (Paper 3)

    Practical skills: microscopy and drawing (Paper 3)

    Key factsUsing the microscope
    1. Put the slide on the stage, section under the objective, and clip it down. Turn the objective to the lowest power (in Paper 3, the ×10 objective) before you look.
    2. Look through the eyepiece and use the coarse focus to bring the specimen into view. Move the slide so the part you want is at the centre of the field.
    3. Use the fine focus to make it sharp. Adjust the light so the image is bright enough without glare.
    4. To change to ×40, turn the objective so that the lens clicks into place and focus with the fine focus only; the coarse focus can crush the slide.
    5. Look for the structure you were asked about; check that you have chosen the correct part of the specimen before you start to draw.
    Key factsThe rules for a good drawing
    1. Sharp pencil. Lines are fine, clear and unbroken: one confident line, not several short or sketchy ones. A rubber is used to remove mistakes completely.
    2. Large. Use most of the space available. A drawing should fill at least half of the area given.
    3. No shading and no colouring. Show a dense or dark area with a label, not with shading.
    4. Draw what you see. Draw the observable features only, not what you expect to be there or a diagram from a book.
    5. Correct proportions. Check them by measuring (see the next section) before you draw.
    6. Labels with ruled lines that touch the structure, written horizontally, with no arrowheads and no crossing lines, placed beside the drawing.
    7. Title and, if asked, the magnification or a scale bar.
    Key factsDrawing cell walls
    • Cell walls have a thickness. Draw a wall with two lines: one for each edge of the wall.
    • Where two cells touch, draw the wall they share with three lines: one for the edge of each cell’s wall and one in the middle where the two meet.
    • Draw cells in their true shapes and in the same relative sizes as on the slide. Include several cells, as the question asks.
    Key factsA comparison table
    1. A heading for the feature, then one column for each specimen.
    2. One feature per row, with an observation for both specimens in the same row. A row with an observation of only one specimen does not compare.
    3. Use observable features only (what you can see: the number of layers, the thickness, the shape, the presence of a structure), not explanations (what it is for).
    4. Include similarities as well as differences, and use measurements where you can (a thickness in µm, a count).
    5. Be specific: “two layers of palisade cells” and not “more palisade”.
    FormulaDensity from fields of view

    density=mean number per fieldarea of one field The field of view is a circle of area πr2, with the diameter measured against a stage micrometer or graticule (the radius is half the diameter). The density is per unit area, such as per mm².

    Revise: Practical skills: microscopy and drawing (Paper 3)

    Energy, ATP and respiratory quotient

    Key factsATP in one box
    • ATP is an energy carrier (currency), not a long-term store. Fats and carbohydrates are the stores.
    • Hydrolysis of ATP to ADP and Pi releases about 30.5 kJ mol⁻¹, in small amounts that suit one reaction.
    • ATP is made from ADP and Pi using energy from respiration, and used up within a short time.
    Key factsWhy lipids have the highest energy value
    1. A gram of lipid contains more hydrogen atoms (more C–H bonds) than a gram of carbohydrate.
    2. So more reduced NAD and reduced FAD form per gram.
    3. So more hydrogen reaches the electron transport chain, more protons are pumped and more ATP is made by chemiosmosis.
    4. Lipids contain less oxygen, so more oxygen is used per gram.
    FormulaBalancing complete oxidation and finding RQ

    CxHyOz+(x+y4−z2)O2→x CO2+y2 H2O

    RQ=xx+y4−z2

    x, y and z are the numbers of carbon, hydrogen and oxygen atoms in one molecule of the substrate. You can check any answer by counting oxygen atoms on both sides.

    FormulaRespirometer calculations

    volume of oxygen taken in=πr2d1

    volume of carbon dioxide produced=πr2(d1−d2)

    RQ=d1−d2d1

    d1 is the movement towards the tube with the absorber, d2 the net movement towards the tube without it (use a negative d2 if the fluid moves away from the tube), r the radius of the capillary. Because πr2 is the same on the top and the bottom, it cancels in the RQ.

    Revise: Energy, ATP and respiratory quotient

    Aerobic respiration

    Key factsGlycolysis, per molecule of glucose
    • Location: cytoplasm. No oxygen needed.
    • 2 ATP used, 4 ATP made: a net gain of 2 ATP (by substrate-linked phosphorylation).
    • 2 reduced NAD are formed.
    • Products: 2 pyruvate (3C each).
    Key factsOne turn of the Krebs cycle
    • 2 CO₂ are released (two decarboxylations).
    • 3 reduced NAD and 1 reduced FAD are formed (dehydrogenations).
    • 1 ATP is made by substrate-linked phosphorylation.
    • Oxaloacetate (4C) is regenerated. There are two turns per glucose, because each glucose gives two acetyl CoA.
    Key factsAerobic v anaerobic yield
    1. Aerobic: the electron transport chain works because oxygen is the final electron acceptor, so chemiosmosis makes most of the ATP.
    2. Anaerobic: only glycolysis, so a net gain of 2 ATP per glucose.
    3. In anaerobic respiration the substrate is only partly oxidised, so much energy stays in the products.

    Revise: Aerobic respiration

    Photosynthesis

    Key factsThe two stages in one box
    1. The light-dependent stage (in the thylakoids) transfers energy from light to ATP and reduced NADP, and releases oxygen.
    2. The light-independent stage, the Calvin cycle (in the stroma), uses that ATP and reduced NADP to turn carbon dioxide into triose phosphate, from which complex organic molecules are built.
    3. ATP and reduced NADP used up in the Calvin cycle go back to the thylakoids as ADP, Pi and NADP, ready to be charged again.
    Key factsNon-cyclic photophosphorylation: the marking points
    1. Light absorbed by PSII; photoactivation of chlorophyll, electrons leave.
    2. Photolysis of water replaces the electrons; protons and oxygen released.
    3. Electrons pass along the electron transport chain and release energy.
    4. Energy used to pump protons from the stroma into the thylakoid space (proton gradient).
    5. Electrons re-energised in PSI and passed to NADP, forming reduced NADP.
    6. Protons flow back through ATP synthase, making ATP.
    Key factsThe Calvin cycle in one box
    1. Fixation: COX2 + RuBP (5C) → 2 GP (3C), catalysed by rubisco.
    2. Reduction: GP → TP (3C), using ATP and reduced NADP.
    3. Regeneration: TP → RuBP, using ATP; one TP in six leaves the cycle.
    4. Per COX2 fixed: 3 ATP and 2 reduced NADP.
    5. TP makes carbohydrates, lipids and amino acids; GP makes some amino acids.

    Revise: Photosynthesis

    Limiting factors in photosynthesis

    Key factsReading a rate graph
    1. Where the line is rising, the factor on the x-axis is the limiting factor.
    2. Where the line is flat, the factor on the x-axis is no longer limiting. Something else is. Say what, and why.
    3. Raising the right factor lifts the whole plateau; raising any other factor changes nothing.
    FormulaLight intensity and distance

    I∝1d2

    • I is the light intensity at the plant; d is the distance from the lamp to the plant.
    • Doubling the distance gives 14 of the intensity; tripling it gives 19.
    • To compare two distances: I2I1=(d1d2)2.

    Revise: Limiting factors in photosynthesis

    Homeostasis and the kidney

    Key factsWhy homeostasis matters

    Constant conditions mean that enzymes keep their shape and work at a steady rate, cells keep their volume because the water potential outside them is steady, and cells have a constant supply of glucose to respire. Without it, the body could work only when the weather suited it.

    Key factsFrom amino acid to urine
    1. Excess amino acids go to the liver.
    2. Deamination removes the amino group as ammonia, leaving a keto acid.
    3. Ammonia and carbon dioxide are converted to urea in the liver.
    4. Urea travels in the blood plasma to the kidney, where it is filtered out and excreted in urine.

    Revise: Homeostasis and the kidney

    Neurones and action potentials

    Key factsThe action potential in one table
    Phase Channels Ion movement Membrane potential
    Resting most voltage-gated channels closed; K⁺ leak channels open K⁺ leak out; pump keeps gradients about −70 mV
    Depolarisation voltage-gated Na⁺ channels open Na⁺ diffuse in rises past −50 mV (threshold) to about +35 mV
    Repolarisation Na⁺ channels close; voltage-gated K⁺ channels open K⁺ diffuse out falls back to about −70 mV
    Hyperpolarisation K⁺ channels slow to close too many K⁺ leave below −70 mV, to about −80 mV
    Recovery K⁺ channels closed pump and leak channels restore gradients back to −70 mV

    Revise: Neurones and action potentials

    Synapses and muscle contraction

    Key factsThe marking points for synaptic transmission
    • action potential arrives; calcium channels open; Ca²⁺ diffuse in
    • vesicles fuse with the presynaptic membrane; ACh released by exocytosis
    • ACh diffuses across the cleft and binds to receptors (not “enters the next neurone”)
    • sodium channels open; Na⁺ diffuse in; depolarisation and, if threshold is reached, an action potential
    • acetylcholinesterase breaks down ACh, so the response stops
    Key factsThe roles in the sliding filament model
    Component Role
    Calcium ions bind to troponin; released from the SR when the fibre is stimulated
    Troponin binds Ca²⁺ and changes shape, which moves tropomyosin
    Tropomyosin covers the myosin-binding sites on actin at rest; moved away when Ca²⁺ bind troponin
    Myosin head binds to actin (cross-bridge), tilts in the power stroke, and hydrolyses ATP
    ATP binding to the head makes it detach from actin; its hydrolysis gives the energy to recock the head; also used to pump Ca²⁺ back into the SR

    Revise: Synapses and muscle contraction

    Control and coordination in plants

    Key factsThe trade-off

    Stomata open: carbon dioxide diffuses in (for photosynthesis) but water vapour diffuses out (transpiration). Stomata closed: water is kept in, but no carbon dioxide gets in. A plant regulates the stomatal aperture, the width of the pore, to balance the need for carbon dioxide with the need to minimise water loss.

    Revise: Control and coordination in plants

    Meiosis and genetic variation

    Key factsWhy meiosis is needed
    • Fertilisation joins two haploid gametes to make a diploid zygote.
    • Meiosis is a reduction division that halves the chromosome number (2n to n).
    • So the diploid number stays the same in every generation.
    • It also makes the gametes genetically different from each other, which you will see below.
    Key factsThe two divisions in one line each
    • Meiosis I: homologous chromosomes pair up (bivalents), cross over, line up in pairs, and are pulled apart. Chromosome number: 2n to n.
    • Meiosis II: sister chromatids are pulled apart, as in mitosis. Chromosome number stays at n, and each chromosome becomes a single chromatid.
    Key factsCombinations from random orientation

    For a diploid cell with n homologous pairs, the number of different combinations of maternal and paternal chromosomes in the gametes is 2n (ignoring crossing over). In humans, n=23, so 223=8 388 608, or about 8.4 million.

    Key factsHow genetic variation arises
    • Crossing over (prophase I): non-sister chromatids exchange sections, making new combinations of alleles on a chromosome.
    • Random orientation / independent assortment (metaphase I, and metaphase II): the pairs of homologous chromosomes, and the chromatids, are arranged and separated at random, so gametes differ in which chromosomes they get.
    • Random fertilisation: any male gamete can fuse with any female gamete, so zygotes differ.

    Revise: Meiosis and genetic variation

    Monohybrid inheritance

    Key factsThe layout of a genetic diagram
    1. Key: which letter stands for which allele and phenotype.
    2. Parental phenotypes, linked by ×.
    3. Parental genotypes.
    4. Gametes: one allele per gamete, each written in a circle. (In this guide round brackets stand for the circles.)
    5. A Punnett square, with the male gametes along the top and the female gametes down the side (or a labelled set of lines joining gametes). Each box is the zygote made when two gametes fuse.
    6. Offspring genotypes.
    7. Offspring phenotypes.
    8. The ratio of phenotypes, using the names of the phenotypes (for example 3 tall : 1 dwarf).
    Key factsReading a pedigree
    1. Two unaffected parents have an affected child: the condition is caused by a recessive allele, and both parents are carriers (heterozygous).
    2. Two affected parents have an unaffected child: the condition is caused by a dominant allele, and both parents are heterozygous.
    3. An affected female whose father is unaffected: the condition cannot be X-linked recessive. (An affected daughter has the recessive allele on both X chromosomes, and one of them came from her father, so he must be affected too.) So it is autosomal.
    4. An affected father and an affected son: if the mother is not affected and is not a carrier, the son must have had the allele from his father, so it is not X-linked (a son gets his Y, not his X, from his father) and the condition is autosomal. If the mother could be a carrier, X-linked is still possible, because the son may have had the allele from her.
    5. If many more males than females are affected, and affected males are born to unaffected parents, X-linked recessive inheritance is likely.

    Revise: Monohybrid inheritance

    Dihybrid crosses and the chi-squared test

    Key factsDihybrid crosses with unlinked genes
    • Heterozygote × heterozygote (AaBb × AaBb): gametes (AB), (Ab), (aB), (ab) from each parent; 9 : 3 : 3 : 1 phenotypes (9 genotypes, 4 phenotypes).
    • Test cross (AaBb × aabb): the second parent makes only (ab); offspring in the ratio 1 : 1 : 1 : 1.
    • For unlinked genes, multiply the ratios (or probabilities) from the two monohybrid crosses. Use this to check a Punnett square or to answer a probability question without drawing 16 boxes.
    Key factsDoing a chi-squared test
    1. State the null hypothesis: there is no significant difference between the observed and expected results (the ratio fits).
    2. Work out the expected numbers: total number × the fraction for each class, from the ratio. The expected values add up to the same total as the observed values.
    3. Calculate (O−E)2÷E for each class and add them to get χ2.
    4. Find the degrees of freedom: the number of classes minus 1.
    5. Compare χ2 with the critical value at p=0.05 (given in a table in the exam; the degrees of freedom are not given).
    6. State the conclusion in words.

    Revise: Dihybrid crosses and the chi-squared test

    Genes, phenotype and gene control

    Key factsThe chain from gene to phenotype
    1. A gene (an allele) has a particular base sequence.
    2. Transcription copies it into mRNA, whose codons are read in translation.
    3. The codons decide the amino acid sequence (primary structure) of the polypeptide.
    4. The primary structure decides the tertiary structure (the shape) of the protein.
    5. The shape decides what the protein does (or whether it does anything).
    6. What the protein does (or fails to do) produces the phenotype.

    Revise: Genes, phenotype and gene control

    Variation and the t-test

    Key factsGenes and variation
    • Discontinuous: one gene or a few with large effects; distinct categories; little environmental effect.
    • Continuous: many genes (polygenic) each with a small, additive effect, plus environmental effects; a range of phenotypes in a bell-shaped distribution.
    FormulaStandard deviation (given in the exam when needed)

    s=∑(x−xˉ)2n−1 x is each value, xˉ the mean of the sample, n the number of values in the sample, and ∑ means “the sum of”. Note n−1 in the denominator for a sample.

    FormulaStandard error and 95% confidence interval

    SE=sn95% confidence interval≈xˉ±2×SE

    Formulat-test (given in the exam when needed)

    t=∣xˉ1−xˉ2∣s12n1+s22n2 xˉ1, xˉ2: the means of the two samples. s1, s2: their standard deviations. n1, n2: the numbers of values. The bars ∣ ∣ mean “ignore the sign”: always subtract the smaller mean from the larger.

    FormulaDegrees of freedom (not given in the exam)

    degrees of freedom=n1+n2−2

    Revise: Variation and the t-test

    Natural and artificial selection

    Key factsSelection acts on phenotypes and changes allele frequencies

    The environment selects phenotypes, but what changes from one generation to the next is the frequency of alleles in the gene pool. An individual does not change its alleles. A successful individual is one that survives and reproduces.

    Key factsDrift compared with natural selection
    • Natural selection is non-random: it favours alleles that increase survival and reproduction, so a population becomes better adapted.
    • Genetic drift is random: the alleles that increase or decrease are not better or worse. It has the largest effect in small populations.
    FormulaThe Hardy–Weinberg equations (given in the exam when needed)

    p+q=1 p2+2pq+q2=1 p = frequency of the dominant allele; q = frequency of the recessive allele; p2 = frequency of the homozygous dominant genotype (AA); 2pq = frequency of the heterozygous genotype (Aa, the carriers); q2 = frequency of the homozygous recessive genotype (aa).

    Key factsCalculating with Hardy–Weinberg
    1. Identify what you are given. Usually this is the frequency of the recessive phenotype (the only phenotype that shows the genotype aa), which is q2.
    2. Take the square root to find q.
    3. Find p=1−q.
    4. Calculate p2, 2pq and q2, and check that they add up to 1.
    5. For a number of individuals, multiply the frequency by the size of the population.

    Revise: Natural and artificial selection

    Evolution and speciation

    Key factsEvolution in one box
    • A population evolves; an individual does not. An individual keeps the alleles it was born with. What changes is the frequency of alleles in the gene pool of the population.
    • Evolution needs variation (from mutation, and from meiosis and sexual reproduction) that is inherited.
    • When two populations stop exchanging alleles, their gene pools can drift apart until they are separate species.
    Key factsThe idea behind sequence comparison

    The more differences there are between the DNA (or protein) sequences of two species, the longer ago they shared a common ancestor, and the less closely they are related. Species with few differences separated recently.

    Key factsDiscussing DNA sequence evidence

    Strengths

    • A sequence has thousands of characters, so a comparison has far more data than appearance and is objective.
    • It can compare species that look very different, and avoids being fooled by convergent evolution (unrelated species that evolved similar features).
    • It works where there are no fossils, and from tiny samples.
    • Results can be checked by comparing several genes and against the fossil record.

    Limitations

    • The mutation rate is not always constant: it differs between genes, between lineages and with generation time, so the clock is only approximate.
    • A site can mutate more than once (even back to the original base), so some changes are hidden and old divergences look younger.
    • The clock must be calibrated with fossils, and the fossil record is incomplete.
    • Silent mutations and changes in non-coding DNA do not change the protein, so protein sequences understate the differences.
    • Bacteria can pass genes to other species, which breaks the idea of one line of descent.

    Revise: Evolution and speciation

    Classification

    Key factsArchaea and Bacteria: what to write

    Both are prokaryotes. The three differences to quote are the membrane lipids (ether bonds and branched chains in Archaea; ester bonds and unbranched chains in Bacteria), the base sequences of rRNA, and the cell wall (peptidoglycan in Bacteria only). Always say which domain each feature belongs to.

    Key factsHow to place a eukaryote in a kingdom

    Ask: Is it multicellular? Does it have a cell wall, and of what? Does it have chloroplasts? How does it feed?

    • Cellulose wall + chloroplasts + photosynthesis → Plantae.
    • Chitin wall + hyphae + extracellular digestion and absorption + no chloroplasts → Fungi.
    • Multicellular, no wall, no chloroplasts, ingests food, nervous system → Animalia.
    • A eukaryote that is none of these, usually single-celled → Protoctista.

    Revise: Classification

    Biodiversity and sampling

    Key factsWhat makes a good sample
    • It is representative: it must reflect the whole area, not only the parts that look rich or are easy to reach.
    • It is large enough: more samples reduce the effect of chance.
    • It is unbiased: the person collecting must not choose where to sample.
    FormulaMeasures from quadrats
    • Frequency (%):

    frequency=quadrats containing the speciestotal quadrats×100

    • Density (per m²):

    density=total individualsquadrats×area of one quadrat

    • Percentage cover = the percentage of the quadrat covered by the species (count the small squares of the grid that are more than half covered).

    With a 100-square grid the number of squares counted is the percentage cover. With a grid of another size, divide the number of squares counted by the number of squares in the grid and multiply by 100: 18 squares counted on a 25-square grid is 18÷25×100=72%.

    FormulaLincoln index

    N=n1×n2m2

    N = estimated population size; n1 = number caught and marked in the first sample; n2 = number caught in the second sample; m2 = number of marked animals in the second sample. The formula is given in the exam.

    FormulaSimpson’s index of diversity

    D=1−∑(nN)2

    n = number of individuals of one species; N = total number of individuals of all species; ∑ = add up the values for every species. The formula is given in the exam.

    Key factsWhich test?

    Pearson’s linear correlation (r): both variables are continuous and normally distributed, and the scatter graph shows a straight-line relationship. Use at least 5 pairs, preferably 10 or more.

    Spearman’s rank correlation (rs): the data are not normally distributed, or can only be ranked, or the relationship is increasing or decreasing but not a straight line. Use more than 5 pairs, preferably 10 to 30. The points are independent and selected at random.

    FormulaSpearman’s rank correlation coefficient

    rs=1−6∑D2n3−n

    D = the difference between the two ranks of each pair; n = number of pairs. The formula is given in the exam.

    FormulaPearson’s correlation coefficient

    r=∑xy−nxˉyˉ(n−1) sxsy

    x and y = the two variables; xˉ and yˉ = their means; sx and sy = their standard deviations (using n−1); n = number of pairs. The formula is given in the exam.

    Revise: Biodiversity and sampling

    Conservation

    Key factsCauses of extinction: cause, effect, result
    1. Habitat loss and fragmentation: less living space and smaller, isolated patches, so smaller populations, no gene flow and more edge; the species cannot find mates or enough food.
    2. Climate change: the area with suitable conditions moves faster than the species can migrate or evolve, and life-cycle timings (breeding and food supply) fall out of step.
    3. Overexploitation (hunting): adults are removed faster than they are replaced, which matters most for large, slow-breeding species; the population falls below a viable size.
    4. Invasive (introduced) species: compete for food and space, prey on natives or bring disease, so native numbers fall.
    5. Pollution: poisons individuals or degrades the habitat, so fewer survive and reproduce.
    6. In every case the final stage is a small population: lost alleles, inbreeding, and one chance event that kills the rest.
    Key factsIn situ v ex situ in one breath
    • In situ protects the species where it lives, so the whole community and natural selection carry on; it is best for large numbers but is hard to protect from poaching, fire and climate change.
    • Ex situ protects it somewhere else (zoo, botanic garden, seed bank, frozen zoo); it is a safety net for species whose habitat is lost, but the gene pool is small and the cause of the decline is not removed.
    • The best plan uses both: ex situ stock for reintroduction once the habitat is safe.

    Revise: Conservation

    Tools of genetic technology

    Key factsGenetic engineering in one box
    • Recombinant DNA is DNA from two or more sources joined together.
    • Get the gene: cut it from donor DNA, make cDNA from mRNA (reverse transcriptase, then DNA polymerase), or build it from nucleotides.
    • Restriction endonuclease cuts at a specific recognition site, leaving sticky ends; the same enzyme cuts the plasmid and the gene.
    • DNA ligase forms phosphodiester bonds and joins them.
    • A plasmid is the vector: it has an origin of replication, cutting sites and marker gene(s).
    • A promoter that the host recognises is needed for expression.
    • A marker gene (for example GFP, inserted into the plasmid joined to the desired gene) shows which cells took up the recombinant plasmid and express the gene.
    Key factsCopies in PCR

    After n cycles, starting from N0 molecules of the target: N=N0×2n Starting from one molecule: 2 copies after 1 cycle, 4 after 2, 8 after 3, and 210=1024 after 10. Doubling is exponential: it is not 2n (20 copies after 10 cycles is the usual wrong answer).

    Revise: Tools of genetic technology

    Genetic technology in medicine and agriculture

    Key factsGenetic screening: what a result does and does not mean
    • Positive result: the person has the tested allele. For a dominant allele with full penetrance (everyone who has the allele develops the disease, as in Huntington’s disease) it means the disease will develop; for BRCA1 or BRCA2 it means a much higher risk, not certainty.
    • A positive result does not say when symptoms start, how severe they will be, or that treatment is possible. A negative result removes only the risk from the alleles tested.
    • Benefits: early treatment, preventive action and informed choices about children. Costs: anxiety, possible insurance or employment discrimination, privacy of the data, and the pressure on relatives who may share the allele.
    Key factsGene therapy in one box
    • Somatic: gene added to body cells; affects only the patient; not inherited. Germ-line: gene added to gametes or embryos; every cell and all later generations carry it; not allowed in many countries.
    • Vector: a modified virus (efficient entry, but an immune response is possible) or a liposome (a phospholipid sphere that fuses with the cell surface membrane).
    • Ex vivo: cells removed, treated in the laboratory and returned (blood cells, for example SCID from lack of ADA). In vivo: vector put into the tissue (the eye).
    • It adds a working copy; the faulty allele stays in every other cell, so the effect may need repeating.

    Revise: Genetic technology in medicine and agriculture

    Statistics for biologists

    Key factsTwo rules for combining probabilities
    1. “And” rule (independent events): multiply. The probability that two independent events both happen is P(A)×P(B).
    2. “Or” rule (events that cannot both happen): add. The probability of one outcome or the other is P(A)+P(B).
    FormulaStandard deviation (provided in the exam)

    s=∑(x−xˉ)2n−1 x = each value, xˉ = the sample mean, n = the number of values. The denominator is n−1 because the values are a sample, not the whole population.

    FormulaStandard error and 95% confidence interval (provided in the exam)

    SE=sn95% CI=xˉ±2×SE (The factor 2 is the syllabus’s approximation.)

    Key factsReading error bars (a rule of thumb)
    • If the 95% CI error bars of two means overlap, the difference between the means is probably not significant: it could be due to chance.
    • If they do not overlap, the difference is probably significant.
    • This is only a guide. A statistical test, such as the t-test, decides. Bars that show SD cannot be used in this way, because SD does not describe how reliable the mean is.
    Key factsWhat to write for each test (at p=0.05)
    • Chi-squared: if calculated χ2 is less than the critical value, the difference between observed and expected is not significant, it is due to chance, and the null hypothesis is accepted. If greater, the difference is significant, the null hypothesis is rejected, and the observed results do not fit the expected ones.
    • t-test: if calculated t is greater than the critical value, there is a significant difference between the means (the null hypothesis is rejected). If less, there is no significant difference.
    • Pearson or Spearman: if the size of r or rs (ignoring the sign) is greater than the critical value, there is a significant correlation: positive if the value is positive, negative if it is negative. If less, there is no significant correlation.
    Key factsChoosing a test
    question data test conditions for validity
    Do observed numbers fit the numbers expected? nominal (frequencies in classes) chi-squared one row or column of counts; expected values from a ratio or from “no preference”
    Do two sample means differ? continuous t-test two samples; each approximately normally distributed; similar standard deviations; suits small samples (under 30 values in each)
    Do two variables change together in a straight line? continuous Pearson’s correlation both normally distributed; scatter graph shows a possible linear relationship; at least 5 pairs, ideally 10 or more
    Do two variables change together (rising or falling, not necessarily straight)? ordinal, ranked, or continuous but not normal Spearman’s rank correlation more than 5 pairs, ideally 10 to 30; points independent; random sample
    FormulaChi-squared (provided in the exam)

    χ2=∑(O−E)2E O = observed number in a class, E = expected number in that class. Degrees of freedom = number of classes −1 (not given).

    Formulat-test (provided in the exam)

    t=∣xˉ1−xˉ2∣s12n1+s22n2 ∣ ∣ means ignore the sign. Degrees of freedom =n1+n2−2 (not given).

    FormulaPearson’s correlation coefficient (provided in the exam)

    r=∑xy−nxˉyˉ(n−1) sxsy n = number of pairs, xˉ, yˉ = the means, sx, sy = the standard deviations (with n−1). The critical value depends on n (given in a table).

    FormulaSpearman’s rank correlation coefficient (provided in the exam)

    rs=1−6∑D2n3−n D = the difference between the two ranks of a pair, n = number of pairs.

    Revise: Statistics for biologists

    Planning investigations (Paper 5)

    Key factsPreparing a solution
    • % (w/v): mass needed (g) =percentage×volume (cm3)100.
    • mol dm⁻³: moles =c×V with V in dm³ (1 dm³ = 1000 cm³); mass =moles×M, where M is the molar mass (given in the question).
    • Dissolve the solute in less water than the final volume, then make up to the final volume in a volumetric flask or measuring cylinder.
    Key factsWhich dilution method?
    • Proportional: equal steps in concentration (0.2, 0.4, 0.6 ...). Use when the effect is expected to change steadily over a narrow range.
    • Serial: equal factors (1, 0.1, 0.01, 0.001 ...). Use when you need a very wide range of concentrations.

    Revise: Planning investigations (Paper 5)

    Analysis, conclusions and evaluation (Paper 5)

    Key factsA conclusion in four parts
    1. State the finding with data: the trend or difference, quoting values with units from the table or graph (and the result of the test, if one was done).
    2. Say how far the hypothesis is supported: “supports”, “partly supports”, “does not support”. Never “proves”. Mention the strengths and weaknesses of the evidence if asked.
    3. Explain it biologically: the mechanism that would cause the result.
    4. Stay within the data: only the range of the independent variable that was tested, and no cause from a correlation alone. Suggest a further prediction or hypothesis if asked.

    Revise: Analysis, conclusions and evaluation (Paper 5)

    Mixed practice: A Level

    Key factsWhich test? Choosing a statistical test
    Question Data Test Degrees of freedom
    Do observed counts fit an expected ratio? frequencies in categories chi-squared number of classes − 1
    Do two sample means differ? continuous, roughly normal, similar spread t-test n1+n2−2
    Do two variables change together? continuous and normal, linear scatter Pearson – (use n pairs)
    Do two variables change together? not normal, or ranks, or ordinal Spearman – (use n pairs)

    Chi-squared: calculated value below the critical value means no significant difference. t, Pearson and Spearman: a calculated value (ignoring any minus sign) above the critical value means a significant difference or correlation. Correlation does not show cause.

    Revise: Mixed practice: A Level

    PreviousChemistry data and key equationsNextEngineering formula sheet
    Toolbox/Formula sheets

    Engineering formula sheet

    Reference

    Every key formula from the course in one place, each linked back to the chapter that teaches it.

    The key equations engineers use every day, built on A Level Maths and Physics. This page collects every key box from the Engineering chapters.

    The formulas appear here as the chapters are written.

    PreviousBiology key equations and statisticsNextBusiness, economics and accounting formulas
    Toolbox/Formula sheets

    Business, economics and accounting formulas

    Reference

    Every key formula from the course in one place, each linked back to the chapter that teaches it.

    Cambridge Business, Economics and Accounting exams do not give you a formula sheet, so you must learn these by heart. This page collects every key box from those three courses.

    The formulas appear here as the chapters are written.

    PreviousEngineering formula sheetNextMaths: learn more online
    Toolbox/Learn more online

    Maths: learn more online

    Reference

    Every “Learn more online” link from the Maths chapters in one place: videos, practice questions, explorations, reading and past papers, grouped by topic.

    These links leave the guide and need the internet. Each one was chosen for a chapter, and each chapter keeps its own short list at the end. Use them after you have worked through the chapter, not instead of it: watch a video when an explanation has not clicked, and use the practice and past-paper links to test yourself under exam conditions. Pick a kind of link below to see only that kind.

    Before you start

    Mathematics: the course map

    Watch
    • CAIE past papers with video solutionsExamSolutionsattempt a paper first, then watch the worked solutions
    Practise
    • CAIE A Level Maths mock examsSave My Examstimed practice papersSome content paid
    Explore
    • Desmos graphing calculatorDesmoscheck every sketch you draw while you study (not allowed in the exam)
    Read
    • 9709 syllabus 2026–2027 (PDF)Cambridge Internationalevery learning outcome, straight from the board
    Papers
    • Cambridge 9709 qualification pageCambridge Internationalthe official home of 9709: syllabus, specimen papers and the MF19 formula list
    • 9709 past papers, all sessionsPapaCambridgequestion papers, mark schemes and examiner reportsUnofficial archive
    • 9709 Paper 1 past papers (menu for the other papers)Physics & Maths Tutorpapers with mark schemes, one page per paperUnofficial archive

    Numbers, fractions and percentages

    Watch
    • Videos and worksheets by topicCorbettmathsa short video and a worksheet for every GCSE number skill
    Practise
    • Dr Frost Maths question bankDr Frost Mathsfree practice from GCSE upwards (free account)
    Explore
    • Thinking about NumbersUnderground Mathematicsrich problems that make number facts stick
    Read
    • The world of numbersMaths is Funfriendly pages on fractions, percentages and ratio
    • Prealgebra 2e (free textbook)OpenStaxfull chapters on fractions, decimals and percentages, with exercises and answers

    Algebra from scratch

    Watch
    • Expanding two bracketsExamSolutionsshort video with examples
    • Linear inequalitiesExamSolutionswhy the sign flips when you divide by a negative
    Practise
    • Algebra 1Khan Academymastery practice with instant feedback
    Explore
    • Area Model AlgebraPhETsee why brackets expand the way they do
    • Thinking about AlgebraUnderground Mathematicsproblems that build fluency and insight
    Read
    • Algebra notesPaul’s Online Notesclear notes with many worked examples

    Indices, surds and standard form

    Watch
    • Introduction to surdsExamSolutionssimplifying surds step by step
    • Fractional indicesExamSolutionswhat a power of one half or two thirds means
    Practise
    • Algebra 2Khan Academyunits on exponents and radicals
    Read
    • Algebra indexMaths is Funshort pages on exponents, surds and standard form
    • Algebra and Trigonometry 2e (free textbook)OpenStaxthe prerequisites chapter covers exponents, radicals and scientific notation

    Graphs and straight lines

    Watch
    • Equation of a line from a point and a gradientExamSolutionsthe y − y₁ = m(x − x₁) form in action
    Practise
    • Algebra 1Khan Academyunits on linear equations and graphs
    Explore
    • Graphing LinesPhETdrag a line and watch m and c change
    • Graphing QuadraticsPhETvertex, roots and axis of symmetry you can move
    • Desmos graphing calculatorDesmosplot any graph from this chapter and test your sketches
    • Geometry of EquationsUnderground Mathematicsconnect equations to the shapes they draw

    Geometry and trigonometry basics

    Watch
    • Exact trig ratios for 30°, 45° and 60°ExamSolutionsthe two triangles you need to remember
    Practise
    • High School GeometryKhan AcademyPythagoras and right-angled trigonometry with practice
    • TrigonometryKhan Academyincludes the sine and cosine rules
    Explore
    • Thinking about GeometryUnderground Mathematicsgeometry problems that need real thought
    Read
    • TrigonometryMaths is FunSOH CAH TOA and beyond, with interactive diagrams
    • Geometry indexMaths is Funangles, circles, area and volume

    Pure Mathematics 1

    Quadratics

    Watch
    • Completing the squareExamSolutionsthe method shown slowly, with examples
    • Exam questions: roots and the discriminantExamSolutionsexam-style questions solved on video
    Practise
    • CAIE Pure 1 quadratics exam questionsSave My Examsquestions sorted by difficulty, with mark schemesSome content paid
    Explore
    • Graphing QuadraticsPhETmove a, b and c and watch the vertex and roots
    • QuadraticsUnderground Mathematicsrich problems for stretch
    Papers
    • 9709 Paper 1 past papersPhysics & Maths Tutorfind the quadratics questions in real papersUnofficial archive

    Functions

    Watch
    • The inverse of a functionExamSolutionsfinding f⁻¹(x) and its domain
    • Exam questions: functionsExamSolutionscomposite and inverse functions in exam style
    Practise
    • CAIE Pure 1 functions exam questionsSave My Examswith mark schemesSome content paid
    • CAIE Pure 1 transformations exam questionsSave My Examstranslations, stretches and reflectionsSome content paid
    Explore
    • Combining FunctionsUnderground Mathematicscomposition and inverses explored deeply
    • Desmos graphing calculatorDesmosadd a slider to f(x + a) or af(x) and watch the graph move

    Coordinate geometry: lines and circles

    Watch
    • Equation of a straight lineExamSolutionsy = mx + c from two points
    • Perpendicular bisectorExamSolutionsa favourite step in circle questions
    Practise
    • CAIE Pure 1 straight-line exam questionsSave My Examswith mark schemesSome content paid
    • CAIE Pure 1 circles exam questionsSave My Examstangents, chords and intersectionsSome content paid
    Explore
    • CirclesUnderground Mathematicsproblems on circle geometry and equations
    • GeoGebra calculatorGeoGebraplot a line and a circle and find where they meet

    Circular measure

    Watch
    • Arcs, sectors and segmentsExamSolutionss = rθ and A = ½r²θ in use
    • Exam questions: arcs, sectors and segmentsExamSolutionsexam-style questions solved on video
    Practise
    • CAIE Pure 1 radians exam questionsSave My Examswith mark schemesSome content paid
    • TrigonometryKhan Academythe radians unit, with practice
    Papers
    • 9709 Paper 1 past papersPhysics & Maths Tutorcircular measure appears in Paper 1Unofficial archive

    Trigonometry

    Watch
    • Trig equations in different rangesExamSolutionsfinding every solution in an interval
    • Exam questions: trig identitiesExamSolutionsprove-and-solve questions on video
    Practise
    • CAIE Pure 1 trig equations exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Trig TourPhETthe unit circle and the graphs move together
    • Trigonometry: Triangles to FunctionsUnderground Mathematicsfrom triangles to graphs, with rich problems
    Read
    • How to learn trigonometry intuitivelyBetterExplaineda picture-first explanation of sin, cos and tan

    Series: binomial expansion and progressions

    Watch
    • Binomial expansion for positive integer powersExamSolutionsPascal’s triangle and ⁿCᵣ
    • Exam questions: geometric seriesExamSolutionssums and sums to infinity in exam style
    Practise
    • CAIE Pure 1 binomial expansion exam questionsSave My Examswith mark schemesSome content paid
    • CAIE Pure 1 geometric progressions exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • SequencesUnderground Mathematicsproblems on APs, GPs and beyond
    • Counting & BinomialsUnderground Mathematicswhy binomial coefficients count things

    Differentiation

    Watch
    • The Essence of Calculus3Blue1Brownthe whole idea of calculus, visually
    • The paradox of the derivative3Blue1Brownwhat “instantaneous rate of change” really means
    • Exam questions: connected rates of changeExamSolutionsthe chain rule in rates questions
    Practise
    • CAIE Pure 1 applications of differentiationSave My Examstangents, stationary points, ratesSome content paid
    • Calculus 1Khan Academyderivative skills with instant feedback
    Explore
    • Calculus GrapherPhETdraw a curve and see its derivative
    • Calculus of PowersUnderground Mathematicsrich problems on differentiating powers

    Integration

    Watch
    • Integration and the fundamental theorem of calculus3Blue1Brownwhy area and antiderivatives are linked
    • What does area have to do with slope?3Blue1Brownthe same link seen another way
    • Exam questions: volume of revolution about the x-axisExamSolutionsexam-style questions solved on video
    Practise
    • CAIE Pure 1 integration exam questionsSave My Examswith mark schemesSome content paid
    Read
    • Area between curvesPaul’s Online Notesworked examples, including with respect to y
    • Volumes of solids of revolution (rings)Paul’s Online Notesthe washer method with pictures

    Pure Mathematics 2 and 3

    Modulus and polynomials

    Watch
    • Modulus inequalitiesExamSolutionssquaring both sides and the graphical method
    • Exam questions: remainder theoremExamSolutionsfactor and remainder theorem questions on video
    Practise
    • CAIE Pure 3 modulus exam questionsSave My Examswith mark schemesSome content paid
    • CAIE Pure 3 polynomials exam questionsSave My Examsdivision, factor and remainder theoremsSome content paid
    Explore
    • Polynomials & Rational FunctionsUnderground Mathematicsrich problems on polynomials
    Papers
    • 9709 Paper 2 past papersPhysics & Maths TutorPaper 2 (AS) has this topic tooUnofficial archive

    Logarithms and exponentials

    Watch
    • Rules of logarithmsExamSolutionsthe three laws with examples
    • Converting curves to linear formExamSolutionsy = kxⁿ and y = k(aˣ) from straight lines
    Practise
    • CAIE Pure 3 laws of logarithms exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Exponentials & LogarithmsUnderground Mathematicsrich problems on exponentials and logs
    Read
    • An intuitive guide to exponential functions and eBetterExplainedwhere e comes from
    • Demystifying the natural logarithmBetterExplainedln x as “time to grow”

    Further trigonometry

    Watch
    • The R-form (harmonic identities)ExamSolutionsa sin θ + b cos θ as R sin(θ + α)
    • Exam questions: double anglesExamSolutionsexam-style questions solved on video
    Practise
    • CAIE Pure 3 compound and double angle exam questionsSave My Examswith mark schemesSome content paid
    • TrigonometryKhan Academyunits on identities and equations
    Explore
    • Trigonometry: Compound AnglesUnderground Mathematicswhere the addition formulae come from
    Papers
    • 9709 Paper 3 past papersPhysics & Maths Tutorpapers with mark schemesUnofficial archive

    Further differentiation

    Watch
    • Visualizing the chain rule and product rule3Blue1Brownwhy the rules are true
    • What’s so special about Euler’s number e?3Blue1Brownwhy eˣ is its own derivative
    • Implicit differentiation, what’s going on here?3Blue1Brownthe idea behind the method
    • Differentiating parametric functionsExamSolutionsdy/dx from dy/dt and dx/dt
    Practise
    • CAIE Pure 3 further differentiation exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Calculus of Trigonometry & LogarithmsUnderground Mathematicsrich problems on these derivatives

    Further integration

    Watch
    • The trapezium ruleExamSolutionsthe method and the table layout
    • Exam questions: trapezium ruleExamSolutionsincluding over- and underestimates
    Practise
    • CAIE Pure 3 further integration exam questionsSave My Examswith mark schemesSome content paid
    • Calculus 2Khan Academyintegration practice with instant feedback
    Papers
    • 9709 Paper 2 past papersPhysics & Maths Tutorthe trapezium rule is examined on Paper 2Unofficial archive

    Numerical solution of equations

    Watch
    • Solving equations by iterationExamSolutionsxₙ₊₁ = F(xₙ) step by step
    • Exam questions: iterationExamSolutionssign change, iteration and accuracy
    Practise
    • CAIE Pure 3 numerical methods exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Desmos graphing calculatorDesmosplot y = x and y = F(x) and follow the staircase
    Papers
    • 9709 Paper 3 past papersPhysics & Maths Tutorpapers with mark schemesUnofficial archive

    Partial fractions and the binomial series

    Watch
    • Partial fractionsExamSolutionslinear and repeated factors
    • Exam questions: binomial expansionExamSolutionsrational powers and validity
    Practise
    • CAIE Pure 3 partial fractions exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Power SeriesUnderground Mathematicsseries expansions explored deeply
    Read
    • Partial fractionsPaul’s Online Notesevery denominator type, worked

    Integration techniques: parts, substitution and partial fractions

    Watch
    • Integration by partsExamSolutionschoosing u and dv/dx
    • Integration by substitutionExamSolutionschanging the variable and the limits
    Practise
    • CAIE Pure 3 further integration exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Product Rule & Integration by PartsUnderground Mathematicsrich problems linking the two
    • Chain Rule & Integration by SubstitutionUnderground Mathematicsrich problems linking the two
    Read
    • Integration by partsPaul’s Online Notesmany worked examples

    Differential equations

    Watch
    • Differential equations, studying the unsolvable3Blue1Brownwhat a differential equation describes
    • General and particular solutionsExamSolutionsseparating variables and using a condition
    • Exam questions: forming differential equationsExamSolutionsturning a rate statement into an equation
    Practise
    • CAIE Pure 3 differential equations exam questionsSave My Examswith mark schemesSome content paid
    Read
    • Separable equationsPaul’s Online Notesworked examples

    Vectors

    Watch
    • Intersecting and skew linesExamSolutionsthe three-equation test
    • Exam questions: parallel, intersecting and skew linesExamSolutionsexam-style questions solved on video
    Practise
    • CAIE Pure 3 vector lines and scalar product exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • GeoGebra 3D calculatorGeoGebraplot two lines in 3D and rotate them to see if they meet
    • Vector GeometryUnderground Mathematicsrich problems on lines and angles
    Read
    • Understanding the dot productBetterExplainedwhat a·b measures

    Complex numbers

    Watch
    • Square roots of a complex numberExamSolutionsequating real and imaginary parts
    • Exam questions: loci in the complex planeExamSolutionscircles, bisectors and half-lines
    • e^(iπ) in 3.14 minutes, using dynamics3Blue1Brownwhy re^(iθ) is a point on a circle
    Practise
    • CAIE Pure 3 Argand diagram exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • GeoGebra calculatorGeoGebradraw loci and check your sketches
    Read
    • A visual, intuitive guide to imaginary numbersBetterExplainedmultiplying by i as a rotation

    Probability and Statistics

    Representing and summarising data

    Watch
    • HistogramsExamSolutionsfrequency density and class widths
    • Exam questions: mean and standard deviationExamSolutionsfrom raw data and from Σx, Σx²
    Practise
    • CAIE S1 representation of data exam questionsSave My Examswith mark schemesSome content paid
    • Statistics and ProbabilityKhan Academyunits on displaying and summarising data
    Read
    • Introductory Statistics 2e (free textbook)OpenStaxthe descriptive statistics chapter
    Papers
    • 9709 Paper 5 past papersPhysics & Maths Tutortheir page “cie-paper-6” holds Paper 5 (9709/5)Unofficial archive

    Permutations and combinations

    Watch
    • Permutations with items togetherExamSolutionsthe “glue them together” method
    • Probability using permutations and combinationsExamSolutionscounting to find probabilities
    Practise
    • CAIE S1 permutations and combinations exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Counting & BinomialsUnderground Mathematicsrich counting problems
    Papers
    • 9709 Paper 5 past papersPhysics & Maths Tutortheir page “cie-paper-6” holds Paper 5Unofficial archive

    Probability

    Watch
    • Conditional probability with tree diagramsExamSolutionsP(A given B) from a tree
    • Exam questions: tree diagramsExamSolutionsexam-style questions solved on video
    Practise
    • CAIE S1 further probability exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Basic probabilitySeeing Theorysimulate coins and dice and watch the long run
    • Compound probabilitySeeing TheoryVenn diagrams and conditional probability you can play with

    Discrete random variables: binomial and geometric

    Watch
    • The binomial distributionExamSolutionsB(n, p) probabilities step by step
    • Binomial distributions3Blue1Brownwhere the binomial formula comes from
    • Exam questions: discrete random variablesExamSolutionstables, E(X) and Var(X)
    Practise
    • CAIE S1 binomial and geometric exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Plinko ProbabilityPhETwatch a binomial distribution build up ball by ball
    • Probability distributionsSeeing Theoryinteractive discrete distributions

    The normal distribution

    Watch
    • Normal distribution probabilitiesExamSolutionsstandardising and using the table
    • Normal approximation to the binomialExamSolutionsthe continuity correction in action
    • Why π is in the normal distribution3Blue1Brownfor the curious: the shape of the bell curve
    Practise
    • CAIE S1 normal distribution exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Probability distributionsSeeing Theorychange μ and σ and watch the curve
    Papers
    • 9709 Paper 5 past papersPhysics & Maths Tutortheir page “cie-paper-6” holds Paper 5Unofficial archive

    The Poisson distribution

    Watch
    • The Poisson distributionExamSolutionsPo(λ) probabilities step by step
    • Exam questions: Poisson distributionExamSolutionsexam-style questions solved on video
    Practise
    • CAIE S2 Poisson exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Probability distributionsSeeing Theoryinteractive distributions
    Papers
    • 9709 Paper 6 past papersPhysics & Maths Tutortheir page “cie-paper-7” holds Paper 6 (9709/6)Unofficial archive

    Linear combinations of random variables

    Watch
    • Var(aX + bY) for independent X and YExamSolutionswhy the variances add
    Practise
    • CAIE S2 linear combinations exam questionsSave My Examswith mark schemesSome content paid
    • Statistics and ProbabilityKhan Academythe random variables unit
    Read
    • Introductory Statistics 2e (free textbook)OpenStaxrandom variables explained with exercises
    Papers
    • 9709 Paper 6 past papersPhysics & Maths Tutortheir page “cie-paper-7” holds Paper 6Unofficial archive

    Continuous random variables

    Watch
    • Finding k in a pdfExamSolutionstotal area equals 1
    • Median, quartiles and percentiles from a pdfExamSolutionsareas up to the median
    Practise
    • CAIE S2 continuous random variables exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Probability distributionsSeeing Theorycontinuous distributions as areas
    Papers
    • 9709 Paper 6 past papersPhysics & Maths Tutorpapers with mark schemesUnofficial archive

    Sampling and estimation

    Watch
    • But what is the Central Limit Theorem?3Blue1Brownwhy sample means become normal
    • The central limit theoremExamSolutionsthe exam use of X̄ ~ N(μ, σ²/n)
    • Confidence interval for a population meanExamSolutionsbuilding and reading an interval
    Practise
    • CAIE S2 sampling and estimation exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Frequentist inferenceSeeing Theorydraw samples and watch confidence intervals hit or miss

    Hypothesis testing

    Watch
    • Type I and Type II errorsExamSolutionswhat each error means and its probability
    • Exam questions: binomial hypothesis testsExamSolutionsexam-style questions solved on video
    Practise
    • CAIE S2 hypothesis testing exam questionsSave My Examswith mark schemesSome content paid
    • Statistics and ProbabilityKhan Academythe significance tests unit
    Explore
    • Frequentist inferenceSeeing Theoryp-values and intervals you can simulate
    Papers
    • 9709 Paper 6 past papersPhysics & Maths Tutorpapers with mark schemesUnofficial archive

    Correlation and regression (UK boards)

    Watch
    • RegressionExamSolutionsthe least squares line from summary statistics
    • Hypothesis testing for zero correlationExamSolutionsthe UK-board test
    Practise
    • Statistics and ProbabilityKhan Academyunits on scatter plots and correlation
    Explore
    • Least-Squares RegressionPhETplace points and watch the line and r change
    • Regression analysisSeeing Theoryinteractive least squares

    Mechanics

    Forces and equilibrium

    Watch
    • Resolving forcesExamSolutionscomponents in two directions
    • Friction and limiting equilibriumExamSolutionsF = μR and “about to slip”
    Practise
    • CAIE Mechanics: resolving, slopes and friction exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Forces and Motion: BasicsPhETpush, friction and net force
    • Vector AdditionPhETadd forces as vectors and read the components
    Papers
    • 9709 Paper 4 past papersPhysics & Maths Tutorpapers with mark schemesUnofficial archive

    Kinematics: describing motion

    Watch
    • Constant acceleration (suvat)ExamSolutionschoosing the right equation
    • Exam questions: velocity–time graphsExamSolutionsareas and gradients in exam style
    Practise
    • CAIE Mechanics kinematics graphs exam questionsSave My Examswith mark schemesSome content paid
    • Isaac ScienceIsaac Sciencefree problem-solving practice, including kinematics (free account)
    Read
    • University Physics Volume 1 (free textbook)OpenStaxthe chapter on motion along a straight line

    Newton’s laws and connected particles

    Watch
    • Connected particles on inclined planesExamSolutionsstring over a pulley on a slope
    • Exam questions: tow-barsExamSolutionscar and trailer, tension and thrust
    Practise
    • CAIE Mechanics Newton’s second law exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • The RampPhETforces on a slope, with friction
    • Forces and Motion: BasicsPhETnet force and acceleration
    Papers
    • 9709 Paper 4 past papersPhysics & Maths Tutorpapers with mark schemesUnofficial archive

    Momentum and collisions

    Practise
    • CAIE Mechanics momentum exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Collision LabPhETcollide two balls and check that momentum is conserved
    Read
    • Physics archiveKhan Academythe impacts and linear momentum unit
    • University Physics Volume 1 (free textbook)OpenStaxthe linear momentum and collisions chapter
    Papers
    • 9709 Paper 4 past papersPhysics & Maths Tutorpapers with mark schemesUnofficial archive

    Work, energy and power

    Watch
    • The work–energy principleExamSolutionsenergy in, energy out, work against resistance
    • Car accelerating on a hillExamSolutionsP = Fv and F = ma together
    Practise
    • CAIE Mechanics work and energy exam questionsSave My Examswith mark schemesSome content paid
    Explore
    • Energy Skate ParkPhETKE and GPE trade places on a curved track
    • The RampPhETwork done against friction on a slope
    Papers
    • 9709 Paper 4 past papersPhysics & Maths Tutorpapers with mark schemesUnofficial archive

    Projectiles and moments (UK boards)

    Watch
    • ProjectilesExamSolutionshorizontal and vertical motion separately
    • Equation of the trajectoryExamSolutionswhy the path is a parabola
    • Horizontal beams in equilibriumExamSolutionstaking moments about a point
    Practise
    • Isaac ScienceIsaac Sciencefree problem-solving practice (free account)
    Explore
    • Projectile MotionPhETchange angle and speed and compare paths

    Further Mathematics

    Proof: how mathematicians know

    Practise
    • Proof revision (A Level)Physics & Maths Tutornotes and questions on each proof methodUnofficial archive
    • NRICH for studentsNRICHproblems where you must justify every step
    Explore
    • Divisibility & InductionUnderground Mathematicsinduction for divisibility problems
    Read
    • Mathematical InductionMaths is Funthe domino idea in plain words with examples

    Further Mathematics: a guided tour

    Watch
    • e^(iπ) in 3.14 minutes, using dynamics3Blue1Browna taste of complex exponentials
    Practise
    • NRICH for studentsNRICHproblems for students who love maths
    Explore
    • Divisibility & InductionUnderground Mathematicsproof by induction, explored
    • Hyperbolic FunctionsUnderground Mathematicsa first look at cosh and sinh
    • Power SeriesUnderground Mathematicstowards Maclaurin series
    Papers
    • 9231 Further Pure 1 past papersPhysics & Maths Tutorsee what Further Maths questions look likeUnofficial archive

    Exam preparation

    Paper 1 survival guide

    Papers
    • 9709 Paper 1 past papers (menu for the other papers)Physics & Maths Tutorpapers with mark schemes, one page per paperUnofficial archive

    Papers 2 and 3 survival guide

    Papers
    • 9709 Paper 2 past papersPhysics & Maths TutorPaper 2 (AS) papers with mark schemesUnofficial archive
    • 9709 Paper 3 past papersPhysics & Maths Tutorpapers with mark schemesUnofficial archive

    Paper 4 survival guide

    Papers
    • 9709 Paper 4 past papersPhysics & Maths Tutorpapers with mark schemesUnofficial archive

    Paper 5 survival guide

    Papers
    • 9709 Paper 5 past papersPhysics & Maths Tutortheir page “cie-paper-6” holds Paper 5Unofficial archive

    Paper 6 survival guide

    Papers
    • 9709 Paper 6 past papersPhysics & Maths Tutortheir page “cie-paper-7” holds Paper 6Unofficial archive
    PreviousBusiness, economics and accounting formulasNextPhysics: learn more online
    Toolbox/Learn more online

    Physics: learn more online

    Reference

    Every “Learn more online” link from the Physics chapters in one place: videos, practice questions, explorations, reading and past papers, grouped by topic.

    These links leave the guide and need the internet. Each one was chosen for a chapter, and each chapter keeps its own short list at the end. Use them after you have worked through the chapter, not instead of it: watch a video when an explanation has not clicked, and use the practice and past-paper links to test yourself under exam conditions. Pick a kind of link below to see only that kind.

    Start here

    Physics: the course map

    Explore
    • PhET simulations: Forces and MotionPhETsee forces and acceleration before you meet the equations
    • PhET simulations: Circuit Construction KitPhETbuild circuits and use virtual meters
    Read
    • Cambridge 9702 syllabus pageCambridge Internationalthe official syllabus, past papers and specimen papers
    • CAIE Physics revision notesPhysics & Maths Tutorshort notes and topic questionsUnofficial archive
    Papers
    • 9702 Paper 1 past papersPhysics & Maths Tutormultiple-choice papers with mark schemesUnofficial archive
    • 9702 Paper 2 past papersPhysics & Maths TutorAS structured questionsUnofficial archive
    • 9702 Paper 4 past papersPhysics & Maths TutorA Level structured questionsUnofficial archive

    Physics from zero

    Explore
    • Forces and Motion: BasicsPhETpush objects and see force, mass and acceleration together
    • Ohm’s LawPhETchange V and R and watch the current change
    • Build an AtomPhETbuild atoms from protons, neutrons and electrons

    AS Level

    Physical quantities, units and uncertainty

    Read
    • CAIE Physics revision notesPhysics & Maths Tutorshort notes and topic questionsUnofficial archive

    Kinematics

    Explore
    • Projectile MotionPhETlaunch objects and see the path, range and velocity components
    Read
    • CAIE Physics revision notesPhysics & Maths Tutorshort notes and topic questionsUnofficial archive

    Dynamics and momentum

    Explore
    • Collision LabPhETset masses and speeds, and test elastic and inelastic collisions in one and two dimensions
    • Forces and Motion: BasicsPhETadd forces and friction to see the resultant force and acceleration
    Papers
    • 9702 Paper 2 past papersPhysics & Maths Tutorfind the dynamics questions in real papersUnofficial archive

    Forces, density and pressure

    Explore
    • Balancing ActPhETbalance masses on a plank and see the moments
    • DensityPhETcompare the densities of materials by floating and sinking blocks
    • BuoyancyPhETchange a block’s density and the fluid and watch the upthrust
    Papers
    • 9702 Paper 2 past papersPhysics & Maths Tutorfind the moments, equilibrium and pressure questions in real papersUnofficial archive

    Work, energy and power

    Explore
    • Energy Skate Park: BasicsPhETdrag a skater around a track and watch kinetic, potential and thermal energy trade places
    Papers
    • 9702 Paper 2 past papersPhysics & Maths Tutorfind the work, energy and power questions in real papersUnofficial archive

    Deformation of solids

    Explore
    • Hooke’s LawPhETstretch and compress springs, and see force, energy and the spring constant change together
    Papers
    • 9702 Paper 2 past papersPhysics & Maths Tutorfind the deformation of solids questions in real papersUnofficial archive

    Waves

    Explore
    • Wave on a StringPhETchange frequency, amplitude and damping and see a wave travel along a string
    • Waves IntroPhETcompare water, sound and light waves
    • Sound WavesPhETsee compressions and rarefactions and hear how frequency changes pitch
    Papers
    • 9702 Paper 2 past papersPhysics & Maths Tutorfind the waves questions in real papersUnofficial archive

    Superposition

    Explore
    • Wave InterferencePhETsee water, sound and light waves interfere from two sources and change the slit separation
    • Wave on a StringPhETreflect pulses from a fixed end and watch stationary waves form
    Papers
    • 9702 Paper 2 past papersPhysics & Maths Tutorfind the superposition questions in real papersUnofficial archive

    Electricity

    Explore
    • Ohm’s LawPhETchange the p.d. and resistance and watch the current
    • Resistance in a WirePhETsee how resistivity, length and area change the resistance
    Papers
    • 9702 Paper 2 past papersPhysics & Maths Tutorfind the electricity questions in real papersUnofficial archive

    D.C. circuits

    Explore
    • Circuit Construction Kit: DCPhETbuild circuits with real meters and see the current flow
    Read
    • CAIE Physics revision notesPhysics & Maths Tutorshort notes and topic questionsUnofficial archive
    Papers
    • 9702 Paper 2 past papersPhysics & Maths Tutorfind the d.c. circuits questions in real papersUnofficial archive

    Particle physics

    Explore
    • Rutherford ScatteringPhETfire α-particles at a nucleus and see the paths
    • Build an AtomPhETbuild isotopes and ions from protons, neutrons and electrons
    • Beta DecayPhETwatch a neutron change into a proton
    • Alpha DecayPhETsee a nucleus emit an α-particle

    A Level (A2)

    Motion in a circle

    Read
    • CAIE Physics revision notesPhysics & Maths Tutorshort notes and topic questionsUnofficial archive

    Gravitational fields

    Explore
    • Gravity and OrbitsPhETchange the masses and distances and see the orbits and the force vectors
    • Gravity Force Lab: BasicsPhETsee how the force changes with mass and distance
    • My Solar SystemPhETbuild a system of bodies and watch it evolve

    Temperature

    Explore
    • States of MatterPhETheat and cool a substance and watch the molecules change state
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the thermal physics questions in real papersUnofficial archive

    Ideal gases

    Explore
    • Gas PropertiesPhETpump in molecules, change the volume and temperature, and watch the pressure
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the ideal-gas questions in real papersUnofficial archive

    Thermodynamics

    Explore
    • Gas PropertiesPhETheat and compress a gas and watch the temperature and pressure change
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the thermodynamics questions in real papersUnofficial archive

    Oscillations

    Explore
    • Pendulum LabPhETchange the length, mass and angle and see the period and energy
    • Masses & SpringsPhEThang masses on springs and measure the period, with damping
    Read
    • CAIE Physics revision notesPhysics & Maths Tutorshort notes and topic questionsUnofficial archive

    Electric fields

    Explore
    • Charges and FieldsPhETplace charges, see the field and the equipotentials
    • Coulomb’s LawPhETsee how the force changes with the charges and their separation
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the electric fields questions in real papersUnofficial archive

    Capacitance

    Explore
    • Capacitor Lab: BasicsPhETchange plate area, separation and voltage and watch the charge and energy
    Read
    • CAIE Physics revision notesPhysics & Maths Tutorshort notes and topic questionsUnofficial archive
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the capacitance questions in real papersUnofficial archive

    Magnetic fields

    Explore
    • Magnet and CompassPhETmove a compass around a bar magnet and see the field direction
    • Faraday’s Electromagnetic LabPhETexplore the field of a magnet, a coil and an electromagnet
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the magnetic fields questions in real papersUnofficial archive

    Electromagnetic induction

    Explore
    • Faraday’s LawPhETmove a magnet through a coil and see the e.m.f. and the lamp respond
    • Faraday’s Electromagnetic LabPhETbuild a generator and a transformer
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the induction questions in real papersUnofficial archive

    Alternating currents

    Explore
    • Circuit Construction Kit: ACPhETbuild circuits with an a.c. source and see the voltage and current change with time
    Read
    • CAIE Physics revision notesPhysics & Maths Tutorshort notes and topic questionsUnofficial archive
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the a.c. and rectification questions in real papersUnofficial archive

    Quantum physics

    Explore
    • Photoelectric EffectPhETchange the metal, the wavelength and the intensity and watch the photoelectrons
    • Models of the Hydrogen AtomPhETfire light at hydrogen and see which transitions occur
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the quantum physics questions in real papersUnofficial archive

    Nuclear physics

    Explore
    • Alpha DecayPhETwatch single nuclei decay at random and see the half-life emerge
    • Beta DecayPhETa neutron turns into a proton, an electron and an antineutrino
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the nuclear physics questions in real papersUnofficial archive

    Medical physics

    Read
    • CAIE Physics revision notesPhysics & Maths Tutorshort notes and topic questions on medical physicsUnofficial archive
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the medical physics questions in real papersUnofficial archive

    Astronomy and cosmology

    Explore
    • Blackbody SpectrumPhETchange the temperature and watch the peak and the colour move
    Read
    • CAIE Physics revision notesPhysics & Maths Tutorshort notes and topic questionsUnofficial archive
    Papers
    • 9702 Paper 4 past papersPhysics & Maths Tutorfind the astronomy and cosmology questions in real papersUnofficial archive

    Practical physics

    Practical skills: measurement, tables, graphs and uncertainty

    Explore
    • Curve FittingPhETfit lines and curves to data and see how well they match
    Papers
    • 9702 Paper 3 past papersPhysics & Maths Tutorpractise whole practical questionsUnofficial archive

    Planning, analysis and evaluation

    Explore
    • Least-Squares RegressionPhETdrag points and see the best-fit line and its residuals change
    • Curve FittingPhETfit lines and curves to data and see how well they match
    Papers
    • 9702 Paper 5 past papersPhysics & Maths Tutorpractise whole planning and analysis questionsUnofficial archive

    Exam preparation

    Paper 1 survival guide: multiple choice

    Read
    • Cambridge International AS & A Level Physics 9702Cambridge Internationalthe official syllabus page with the data and formulae list
    Papers
    • Physics & Maths Tutor: Cambridge A Level Physics Paper 1 past papersPhysics & Maths Tutorpast Paper 1 questions to time yourself onUnofficial archive

    Papers 2 and 4 survival guide: structured questions

    Read
    • Cambridge International AS & A Level Physics 9702Cambridge Internationalthe official syllabus page with command words, the data and formulae list and past papers
    Papers
    • Physics & Maths Tutor: Cambridge A Level Physics Paper 2 past papersPhysics & Maths Tutorpast Paper 2 questions to practise in timeUnofficial archive
    • Physics & Maths Tutor: Cambridge A Level Physics Paper 4 past papersPhysics & Maths Tutorpast Paper 4 questions, including synoptic onesUnofficial archive
    PreviousMaths: learn more onlineNextChemistry: learn more online
    Toolbox/Learn more online

    Chemistry: learn more online

    Reference

    Every “Learn more online” link from the Chemistry chapters in one place: videos, practice questions, explorations, reading and past papers, grouped by topic.

    These links leave the guide and need the internet. Each one was chosen for a chapter, and each chapter keeps its own short list at the end. Use them after you have worked through the chapter, not instead of it: watch a video when an explanation has not clicked, and use the practice and past-paper links to test yourself under exam conditions. Pick a kind of link below to see only that kind.

    Start here

    Chemistry from zero

    Watch
    • Atoms, compounds, and ionsKhan Academyvideo lessons on atoms, electron configurations, the Periodic Table and ionic formulas (written for a US course, so the order differs)
    Practise
    • Balancing Chemical EquationsPhETbalance equations with instant feedback, from easy to harder
    Explore
    • Build an AtomPhETadd protons, neutrons and electrons and watch the element, charge and mass change
    Read
    • Ionic bondingChemguidehow atoms form ions and how the ions pack into a lattice, in plain English
    • Covalent bondingChemguideshared pairs and the number of bonds each atom makes
    Papers
    • CAIE IGCSE Chemistry revisionPhysics & Maths TutorIGCSE questions sorted by topic, for extra practice at this levelUnofficial archive

    AS physical chemistry

    Atomic structure

    Practise
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive
    Read
    • Atomic structure and bonding menuChemguideslow, clear notes on atomic orbitals, electron configurations and ionisation energy
    • Ionisation energyChemguidethe factors and the trends, with graphs and worked reasoning
    • Electronic structuresChemguidehow to write configurations from hydrogen onwards

    Moles and stoichiometry

    Practise
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topic, including mole calculationsUnofficial archive
    Read
    • Chemguide: CIE A level chemistry supportChemguidenotes written for the Cambridge syllabus, including the calculation topics
    • Chemguide: atomic structure and bondingChemguidebackground on relative atomic mass and isotopes

    Chemical bonding

    Read
    • Bonding menuChemguideevery type of bonding explained slowly, with diagrams
    • Covalent bondingChemguidedot-and-cross diagrams, double bonds and expanded octets
    • Dative covalent bondingChemguideNH₄⁺, Al₂Cl₆ and why the bond is dative
    • Electronegativity and bond typeChemguidepolar bonds and the ionic–covalent spectrum
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Shapes of molecules and intermolecular forces

    Read
    • Shapes of molecules and ionsChemguideVSEPR step by step, with lone pairs and the unusual shapes
    • Van der Waals’ forcesChemguideid–id and pd–pd forces explained slowly
    • Hydrogen bondingChemguidewhy water and ammonia behave unusually
    • Bonding menuChemguideevery topic on bonding in one place
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    States of matter

    Practise
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive
    Read
    • Ideal gasesChemguidethe ideal gas equation and the assumptions behind it, with worked examples
    • Giant covalent structuresChemguidediamond, graphite and silicon dioxide explained with diagrams
    • Molecular structuresChemguidesimple molecular substances, their melting points and why they do not conduct
    • Ionic structuresChemguidethe sodium chloride lattice and why ionic solids melt at high temperatures

    Chemical energetics

    Practise
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive
    Read
    • Energetics menuChemguideenthalpy changes, definitions, Hess’s law and bond enthalpies in order
    • Hess’s law and enthalpy cyclesChemguideworked examples of energy cycles with formation and combustion data
    • Bond enthalpiesChemguideusing average bond energies to estimate enthalpy changes

    Redox and oxidation numbers

    Read
    • Redox menuChemguidedefinitions of oxidation and reduction, writing half-equations and ionic equations, and working out oxidation states
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Chemical equilibria

    Read
    • Le Chatelier’s principleChemguideconcentration, pressure, temperature and catalysts, with the reasoning for each
    • Equilibrium constants, KcChemguidewriting Kc expressions and calculating with them
    • Equilibrium constants, KpChemguidemole fraction, partial pressure and Kp calculations
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Acids and bases

    Read
    • Strong and weak acidsChemguidewhat “strong” and “weak” mean, with the equations (the pH and equilibrium-constant parts are A Level)
    • pH (titration) curvesChemguidethe four curves, equivalence point and end point explained step by step
    • Acid-base indicatorsChemguidehow indicators work and how to choose one (assumes the curves above)
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Reaction kinetics

    Read
    • An introduction to collision theoryChemguideactivation energy and collisions, step by step
    • The effect of temperature on rates of reactionChemguidethe Maxwell–Boltzmann distribution and why a small rise in temperature has a big effect
    • The effect of catalysts on rates of reactionChemguidelower activation energy shown on a distribution and an energy profile
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    AS inorganic chemistry

    Periodicity: Period 3

    Read
    • Period 3 menuChemguideone page each for the physical trends, the reactions of the elements, the oxides, the chlorides and the hydroxides of Na to Ar
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Group 2

    Read
    • Group 2 menuChemguideseparate pages on atomic properties, the reactions with water, oxygen and acids, the solubility trends and the thermal stability of the nitrates and carbonates
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Group 17: the halogens

    Read
    • Group 7 (halogens) menuChemguideseparate pages on physical properties, halogens as oxidising agents, reactions of halogens with sodium hydroxide, and halide ions as reducing agents and their tests
    • Redox reactions of halide ions with concentrated sulfuric acidChemguidethe half-equations and the products for chloride, bromide and iodide
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Nitrogen and sulfur

    Read
    • The Haber processChemguidehow ammonia is made and why the conditions are a compromise
    • The Contact processChemguidethe manufacture of sulfuric acid and its conditions
    • Types of catalysisChemguideheterogeneous and homogeneous catalysts with examples
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    AS organic chemistry and analysis

    Introduction to organic chemistry

    Read
    • Understanding the names of organic compoundsChemguidehow to name chains, branches and functional groups, with many examples
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Isomerism

    Read
    • Structural isomerismChemguidechain, position and functional-group isomers with diagrams
    • Geometric (cis/trans) isomerismChemguidewhy rotation is restricted and what makes cis and trans forms
    • Optical isomerismChemguidechiral centres, enantiomers and how they differ in plane-polarised light
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Hydrocarbons: alkanes and alkenes

    Read
    • Alkanes menuChemguideburning alkanes, halogenation by free radicals and cracking, explained step by step
    • Alkenes menuChemguidemaking alkenes, hydrogenation, halogens, hydrogen halides, potassium manganate(VII), hydration and polymerisation
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Halogenoalkanes

    Read
    • Nucleophilic substitution menuChemguideeach nucleophile’s mechanism explained slowly, with primary, secondary and tertiary cases
    • HalogenoalkanesChemguidereactions, elimination versus substitution and the silver nitrate test
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Alcohols

    Read
    • Alcohols menuChemguidemanufacture, dehydration, sodium, halogen replacement, oxidation, esterification and the iodoform reaction, in short pages
    • The oxidation of alcoholsChemguideprimary, secondary and tertiary alcohols with dichromate(VI), including distillation and reflux
    • The triiodomethane (iodoform) reactionChemguidewhich compounds give the yellow precipitate, and why
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Aldehydes and ketones

    Read
    • Aldehydes and ketones menuChemguidebackground, preparation, addition of HCN, reduction and the oxidation tests, one page each
    • Oxidation of aldehydes and ketonesChemguideTollens’ and Fehling’s tests, with the equations explained slowly
    • Addition to aldehydes and ketonesChemguidethe HCN mechanism step by step
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Carboxylic acids, esters and nitriles

    Read
    • Carboxylic acids menuChemguidebonding, preparation, reactions as acids and esterification, with one page each
    • Hydrolysis of estersChemguideacid and alkaline hydrolysis, including soap-making
    • Hydrolysis of nitrilesChemguidethe acid and alkali routes and their products
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Organic synthesis (AS)

    Read
    • Properties of organic compounds (menu)Chemguideone page per functional group, covering every reaction in the route map
    • Oxidation of aldehydes and ketonesChemguideTollens’, Fehling’s and dichromate tests, with the equations
    • The triiodomethane (iodoform) reactionChemguidewhich compounds give the yellow precipitate, and why
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Infrared spectroscopy and mass spectrometry

    Read
    • Infra-red spectroscopy (menu)Chemguidewhy bonds absorb, and how to read the spectra of simple compounds
    • Interpreting an infra-red spectrumChemguideidentifying common bonds in the spectrum of a carboxylic acid, with the ranges
    • Mass spectrometry (menu)Chemguidethe molecular ion, fragmentation, and the M+1 and M+2 peaks
    • Mass spectra: the M+2 peakChemguidewhy chlorine and bromine give pairs of peaks, including two chlorine atoms
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    A Level physical chemistry

    Lattice energy, entropy and Gibbs free energy

    Read
    • Lattice enthalpy (lattice energy)ChemguideBorn–Haber cycles and the effect of ion size and charge, step by step
    • Enthalpies of solution and hydrationChemguidethe energy cycle for dissolving and how ion size and charge change the numbers
    • Entropy and free energy menuChemguideentropy changes, ΔG and feasibility from first principles
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Electrochemistry: electrolysis and electrode potentials

    Read
    • Redox equilibria menuChemguideelectrode potentials, the electrochemical series and using E⦵ to predict reactions
    • An introduction to redox equilibria and electrode potentialsChemguidehalf-cells, the standard hydrogen electrode and how cell voltages arise
    • Making predictions using redox (electrode) potentialsChemguideusing E⦵ to decide whether a reaction happens, and its limits
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    pH, buffers, solubility product and partition

    Read
    • Buffer solutionsChemguidehow a buffer works, with the calculations, from the acid-base equilibria menu
    • Strong and weak acidsChemguidepH, Ka and the weak-acid calculation
    • The ionic product for water: KwChemguidepH of strong alkalis step by step
    • Solubility products menuChemguideintroduction, calculations and the common ion effect
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Rate equations and catalysis

    Read
    • Orders of reaction and rate equationsChemguideorders, rate constants and how to find them from data
    • Orders of reaction and mechanismsChemguidehow the rate equation points to the rate-determining step
    • Types of catalysisChemguideheterogeneous and homogeneous catalysts, with the persulfate and iodide example
    • Catalysts in inorganic industrial processesChemguidecatalysts in the Haber and Contact processes
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    A Level inorganic chemistry

    Transition elements

    Read
    • Introducing transition metalsChemguidethe definition, variable oxidation states and catalysis explained step by step
    • Complex ions menuChemguideligands, shapes and ligand exchange reactions, with diagrams
    • VanadiumChemguidethe four oxidation states of vanadium and their colours
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Complexes: colour, isomerism and stability

    Read
    • Complex ions - colourChemguidewhy complex ions are coloured, with the splitting of the d orbitals
    • Substitution in complex ions - ligand exchangeChemguidethe ligand exchange reactions of copper and cobalt with their colours
    • Substitution in complex ions - stability constantsChemguidestability constants and the chelate effect
    • Complex ions - shapesChemguideshapes of complex ions and the isomers they show
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    A Level organic chemistry and analysis

    Arenes: benzene and its compounds

    Read
    • Arenes (aromatic hydrocarbons) menuChemguideintroduction to arenes, nitration, halogenation and Friedel–Crafts reactions
    • Bonding in benzene: sp² hybridisation and delocalisationChemguidethe orbital picture step by step
    • Electrophilic substitution mechanisms menuChemguidethe mechanisms of the reactions in this chapter
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Phenol and the acidity of organic compounds

    Read
    • Phenol menuChemguideall the pages on phenol in one place
    • An introduction to phenolChemguidestructure and properties
    • The acidity of phenolChemguidewhy phenol is acidic and how it compares with alcohols and carboxylic acids
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Acyl chlorides and A Level carboxylic acids

    Read
    • Acyl chlorides menuChemguidebackground, preparation and the reactions with water, alcohols, phenol, ammonia and amines
    • Amides menuChemguidepreparing amides from acyl chlorides, and the hydrolysis of amides
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Amines, amides and amino acids

    Read
    • Aliphatic amines menuChemguidepreparation, amines as bases and as nucleophiles, and reaction with nitrous acid
    • Phenylamine (aniline) menuChemguidemaking phenylamine and its reactions
    • Amides menuChemguidepreparation, hydrolysis and the lack of base character
    • Amino acids and proteins menuChemguideacid–base behaviour of amino acids and zwitterions, then the structure of proteins
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Polymerisation

    Read
    • Polyesters: Terylene and PETChemguidehow PET forms, its structure and its uses
    • Polyamides: nylon and KevlarChemguidestructures, formation, hydrolysis and uses of nylon and Kevlar
    • The polymerisation of alkenesChemguideaddition polymers and the properties of common ones
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Organic synthesis (A Level)

    Read
    • Electrophilic substitution mechanismsChemguidenitration, Friedel–Crafts and the other reactions of benzene, step by step
    • AminesChemguidepreparing amines from halogenoalkanes and nitriles, and how they behave as bases
    • AmidesChemguidepreparing amides, hydrolysis and reduction
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Chromatography and NMR spectroscopy

    Read
    • Thin layer chromatographyChemguidehow a TLC plate is run, why the spots separate, and Rf values
    • Gas-liquid chromatographyChemguidehow the column, carrier gas and detector work, and retention times
    • The background to NMR spectroscopyChemguidewhat a proton NMR spectrum is and what chemical shift means
    • High resolution NMR spectraChemguidethe n + 1 rule and splitting patterns explained
    • Interpreting C-13 NMR spectraChemguidecounting carbon environments and reading shifts
    Papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Practical skills

    Practical skills and qualitative analysis

    Read
    • Testing for halide ionsChemguidethe silver nitrate and ammonia tests, with the colours of the precipitates
    Papers
    • Cambridge International AS & A Level Chemistry 9701: past papersCambridge Internationalpast papers, examiner reports and specimen papers, including the Paper 3 practical papers
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Planning, analysis and evaluation

    Read
    • Orders of reaction and rate equationsChemguidehow rates and orders are found from graphs, useful for the data analysis questions
    • Chemguide: CIE A level chemistry supportChemguidenotes written for the Cambridge syllabus, including calculation topics
    Papers
    • Cambridge International 9701 past papersCambridge Internationalthe board’s own page for past papers and mark schemes; practise Paper 5 under timed conditions
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Exam practice

    Paper 1 survival guide: multiple choice

    Read
    • Le Chatelier’s PrincipleChemguidea clear explanation of the effect of changing temperature, pressure and concentration on a position of equilibrium, which is a common multiple-choice topic
    • An introduction to collision theory in rates of reactionChemguidethe collision theory and the Boltzmann distribution behind many Paper 1 kinetics questions
    Papers
    • Cambridge International 9701 past papersCambridge Internationalthe board’s own page for past papers and mark schemes; Paper 1 papers are the multiple-choice ones
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Papers 2 and 4 survival guide: structured questions

    Read
    • Nucleophilic substitution menuChemguideeach nucleophile’s mechanism explained slowly, with primary, secondary and tertiary cases
    Papers
    • Cambridge International 9701 past papersCambridge Internationalthe board’s own page for past papers and mark schemes; Papers 2 and 4 are the structured ones
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast paper questions sorted by topicUnofficial archive

    Mixed practice: AS Level

    Read
    • CIE A level chemistry supportChemguidenotes on every syllabus topic, to read when a Revise link is not enough
    Papers
    • Cambridge International AS & A Level Chemistry 9701: past papersCambridge Internationalpast papers, mark schemes and examiner reports: use the Paper 2 papers for more structured practice
    • CAIE A Level Chemistry 9701 revisionPhysics & Maths Tutorpast-paper questions sorted by topic, good for the topics you lose marks onUnofficial archive

    Mixed practice: A Level

    Practise
    • Chemistry revision for CIE A LevelPhysics & Maths Tutortopic-by-topic questions to drill any set you found hardUnofficial archive
    Read
    • CIE A level chemistry supportChemguideexplanations to fill a gap that a set exposed
    Papers
    • Past Papers and Examiner ReportsCambridge Internationalthe real Paper 4 question papers with mark schemes and examiner reports, for timed practice after these sets
    PreviousPhysics: learn more onlineNextBiology: learn more online
    Toolbox/Learn more online

    Biology: learn more online

    Reference

    Every “Learn more online” link from the Biology chapters in one place: videos, practice questions, explorations, reading and past papers, grouped by topic.

    These links leave the guide and need the internet. Each one was chosen for a chapter, and each chapter keeps its own short list at the end. Use them after you have worked through the chapter, not instead of it: watch a video when an explanation has not clicked, and use the practice and past-paper links to test yourself under exam conditions. Pick a kind of link below to see only that kind.

    Start here

    Biology: the course map

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of the big ideas
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • CAIE A Level Biology (9700, exams from 2025)Save My Examsnotes and topic questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology topic notesPhysics & Maths Tutorfree notes and topic questions, one page per topic
    Papers
    • Past papers by paperPhysics & Maths TutorPapers 1 to 5, change the last part of the address for each paperUnofficial archive

    Biology from zero

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of cells, molecules and diffusion
    Practise
    • Flashcards for 9700 BiologySave My Examsquick-fire revision cards for key termsSome content paid
    • Cell structure topic questionsPhysics & Maths Tutornotes and exam questions for the first AS topic
    Read
    • Biological macromoleculesOpenStax Biology 2efree textbook chapter on monomers, polymers, condensation and hydrolysis, beyond the syllabus in places
    • Khan Academy BiologyKhan Academyfree lessons from the basics of chemistry and cells upwards

    Cells and molecules

    Microscopy and magnification

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of microscopes and cells
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Cell structure (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Explore
    • Learn.Genetics: the cellUniversity of Utahinteractive tours of cell size and structure
    Read
    • CAIE A Level Biology: cell structure notes and topic questionsPhysics & Maths Tutorfree revision notes and questions sorted by topic
    Papers
    • Paper 3 past papersPhysics & Maths Tutorfind the microscope and photomicrograph questions in real papersUnofficial archive

    Eukaryotic cells and organelles

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of cells and organelles
    Practise
    • Cell structure (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Explore
    • Learn.Genetics: the cellUniversity of Utahinteractive tours of plant and animal cells
    Read
    • CAIE A Level Biology: cell structure notes and topic questionsPhysics & Maths Tutorfree revision notes and questions sorted by topic
    • Cell structureOpenStax Biology 2efree university textbook chapter on cells and organelles, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the cell structure questions in real papersUnofficial archive

    Prokaryotes and viruses

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of prokaryotes and viruses
    Practise
    • Cell structure (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology: cell structure notes and topic questionsPhysics & Maths Tutorfree revision notes and questions sorted by topic
    • ProkaryotesOpenStax Biology 2efree university textbook chapter on bacteria, beyond the syllabus in places
    • VirusesOpenStax Biology 2efree university textbook chapter on virus structure and reproduction, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the cell structure questions in real papersUnofficial archive

    Carbohydrates

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of carbohydrates and macromolecules
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Biological molecules (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology: biological molecules notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Biological macromoleculesOpenStax Biology 2efree university textbook chapter on carbohydrates, lipids, proteins and nucleic acids, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the carbohydrate questions in real papersUnofficial archive

    Lipids and water

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of water, lipids and biological molecules
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Biological molecules (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    • CAIE A Level Biology flashcardsSave My Examsflashcards for lipids and waterSome content paid
    Read
    • CAIE A Level Biology: biological molecules notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Biological macromoleculesOpenStax Biology 2efree university textbook chapter that includes lipids, beyond the syllabus in places

    Proteins

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of proteins and their structure
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Biological molecules (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Explore
    • Learn.Genetics: DNA, genes and proteinsUniversity of Utahshort interactive pages on how genes make proteins
    Read
    • CAIE A Level Biology: biological molecules notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Biological macromoleculesOpenStax Biology 2efree university textbook chapter that includes protein structure, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the protein questions in real papersUnofficial archive

    Testing for biological molecules

    Watch
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of biological molecules
    Practise
    • Biological molecules (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology: biological molecules notes and topic questionsPhysics & Maths Tutorfree revision notes and questions sorted by topic
    Papers
    • Paper 3 past papersPhysics & Maths Tutorpractical papers with food-test questionsUnofficial archive

    Enzymes

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of enzymes and denaturation
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Enzymes (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology: enzymes notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Metabolism and enzymesOpenStax Biology 2efree university textbook chapter on activation energy and enzymes, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the enzyme questions in real papersUnofficial archive

    Enzyme kinetics, inhibitors and immobilised enzymes

    Watch
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of enzymes and inhibition
    Practise
    • Enzymes (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology: enzymes notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Metabolism and enzymesOpenStax Biology 2efree university textbook chapter on enzyme regulation, beyond the syllabus in places
    Papers
    • Paper 4 past papersPhysics & Maths Tutorfind enzyme questions in real papersUnofficial archive

    Membrane structure and cell signalling

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of the cell membrane
    Practise
    • Cell membranes and transport (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Explore
    • Membrane Transport simulationPhETmove molecules through channels and see how the membrane behaves
    Read
    • CAIE A Level Biology: cell membranes and transport notes and questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Plasma membranesOpenStax Biology 2efree university textbook chapter on membrane structure, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the membrane questions in real papersUnofficial archive

    Movement into and out of cells

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of osmosis and active transport
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Cell membranes and transport (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Explore
    • Membrane Transport simulationPhETsee diffusion, channels and gradients at work
    Read
    • CAIE A Level Biology: cell membranes and transport notes and questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Plasma membranesOpenStax Biology 2efree university textbook chapter on transport across membranes, beyond the syllabus in places
    Papers
    • Paper 3 past papersPhysics & Maths Tutorosmosis and diffusion practicals come up in real papersUnofficial archive

    Genetics at AS

    Chromosomes and the mitotic cell cycle

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of the cell cycle and mitosis
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • The mitotic cell cycle (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology: the mitotic cell cycle notes and questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Cell reproductionOpenStax Biology 2efree university textbook chapter on the cell cycle, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the mitosis questions in real papersUnofficial archive

    Nucleic acids and DNA replication

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of DNA structure and replication
    Practise
    • Nucleic acids and protein synthesis (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Explore
    • Learn.Genetics: DNA, genes and proteinsUniversity of Utahshort interactive pages on DNA and how it is copied
    Read
    • CAIE A Level Biology: nucleic acids and protein synthesis notes and questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • DNA structure and functionOpenStax Biology 2efree university textbook chapter on DNA structure and replication, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the DNA questions in real papersUnofficial archive

    Protein synthesis and gene mutation

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of protein synthesis and mutations
    Practise
    • Nucleic acids and protein synthesis (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Explore
    • Learn.Genetics: transcribe and translate a geneUniversity of Utahan interactive tool that makes mRNA and a protein from a gene
    • Gene expression essentialsPhETa simulation of transcription and translation, with mutations
    Read
    • CAIE A Level Biology: nucleic acids and protein synthesis notes and questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Genes and proteinsOpenStax Biology 2efree university textbook chapter on the genetic code, transcription and translation, beyond the syllabus in places

    Transport and exchange

    Plant tissues and water transport

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of water transport in plants
    Practise
    • Transport in plants (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology: transport in plants notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Plant form and physiologyOpenStax Biology 2efree university textbook chapter on plant transport, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the transport questions in real papersUnofficial archive

    Translocation in phloem

    Practise
    • Transport in plants (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology: transport in plants notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Plant form and physiologyOpenStax Biology 2efree university textbook chapter that covers transport in the phloem, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the translocation questions in real papersUnofficial archive

    Blood vessels, blood and tissue fluid

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of the circulatory system
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Transport in mammals (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology: transport in mammals notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Circulatory systemOpenStax Biology 2efree university textbook chapter on blood, vessels and the heart, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the transport questions in real papersUnofficial archive

    Transport of oxygen and carbon dioxide

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of haemoglobin and gas transport
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Transport in mammals (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • CAIE A Level Biology: transport in mammals notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Circulatory and respiratory systemsOpenStax Biology 2efree university textbook chapter on gas exchange and transport, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the dissociation curve questions in real papersUnofficial archive

    The heart and the cardiac cycle

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of the heart
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Transport in mammals (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology: transport in mammals notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Circulatory systemOpenStax Biology 2efree university textbook chapter on the heart and circulation, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the heart questions in real papersUnofficial archive

    The gas exchange system

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of the respiratory system
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Gas exchange (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • CAIE A Level Biology: gas exchange notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • The respiratory systemOpenStax Biology 2efree university textbook chapter on the lungs and gas exchange, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the gas exchange questions in real papersUnofficial archive

    Infection and immunity

    Infectious diseases

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of pathogens and disease
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Infectious diseases (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • CAIE A Level Biology: infectious disease notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • VirusesOpenStax Biology 2efree university textbook chapter on viruses, including HIV, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the infectious disease questions in real papersUnofficial archive

    Antibiotics and resistance

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of bacteria, viruses and antibiotics
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Infectious diseases (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • CAIE A Level Biology: infectious disease notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • ProkaryotesOpenStax Biology 2efree university textbook chapter on bacteria, including antibiotic resistance, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind the antibiotic and resistance questions in real papersUnofficial archive

    The immune response

    Watch
    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of the immune system
    Practise
    • Immunity (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • CAIE A Level Biology: immunity notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • The immune systemOpenStax Biology 2efree university textbook chapter, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind immunity questions in real papersUnofficial archive

    Antibodies and vaccination

    Watch
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
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    • Immunity (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • CAIE A Level Biology: immunity notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • The immune systemOpenStax Biology 2efree university textbook chapter, beyond the syllabus in places
    Papers
    • Paper 2 past papersPhysics & Maths Tutorfind immunity questions in real papersUnofficial archive

    AS skills

    Maths skills for biology

    Watch
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Practical skills and maths: topic questionsPhysics & Maths Tutornotes and questions on handling data
    • Topic questions (9700, exams from 2025)Save My Examsexam-style questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology notesZNotesconcise AS notes for revision
    Papers
    • Paper 2 past papersPhysics & Maths Tutorlook for the calculation questions in real papersUnofficial archive

    Practical skills: investigations (Paper 3)

    Watch
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Practical skills: notes and topic questionsPhysics & Maths Tutorrevision notes and questions on Paper 3 and Paper 5 skills
    • Topic questions (9700, exams from 2025)Save My Examsexam-style questions with mark schemes, some freeSome content paid
    Read
    • CAIE A Level Biology notesZNotesconcise AS revision notes
    Papers
    • Paper 3 past papersPhysics & Maths Tutorpractise the real Paper 3 questionsUnofficial archive

    Practical skills: microscopy and drawing (Paper 3)

    Watch
    • CAIE A Level Biology videosCognitoshort videos and questions matched to the syllabus
    Practise
    • Practical skills: notes and topic questionsPhysics & Maths Tutornotes and questions on microscope work and drawing
    Explore
    • Cells (Learn.Genetics)University of Utahinteractive tours of cells and what is inside them
    Read
    • Cell structureOpenStax Biology 2efree textbook chapter on cells and microscopes, beyond the syllabus in places
    Papers
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    Energy

    Energy, ATP and respiratory quotient

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    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of ATP and cellular respiration
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    • Cellular respirationOpenStax Biology 2efree university textbook chapter on ATP and respiration, beyond the syllabus in places
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    Aerobic respiration

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    • Energy and respiration (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • Cellular respirationOpenStax Biology 2efree university textbook chapter on respiration, beyond the syllabus in places
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    Anaerobic respiration and investigating respiration

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    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of fermentation and cellular respiration
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    • CAIE A Level Biology: energy and respiration notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Cellular respirationOpenStax Biology 2efree university textbook chapter that includes fermentation, beyond the syllabus in places
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    Photosynthesis

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    • PhotosynthesisOpenStax Biology 2efree university textbook chapter on the light and Calvin reactions, beyond the syllabus in places
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    Limiting factors in photosynthesis

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    Control and coordination

    Homeostasis and the kidney

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    • Homeostasis (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • Osmotic regulation and excretionOpenStax Biology 2efree university textbook chapter on the kidney, beyond the syllabus in places
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    Blood glucose, cell signalling and biosensors

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    • Cell communicationOpenStax Biology 2efree university textbook chapter on signalling pathways, beyond the syllabus in places
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    Neurones and action potentials

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    • Neuron simulationPhETstimulate a neurone, watch ion channels and the action potential
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    • CAIE A Level Biology: control and coordination notes and topic questionsPhysics & Maths Tutorfree revision notes and questions
    • The nervous systemOpenStax Biology 2efree university textbook chapter on neurones and nerve impulses, beyond the syllabus in places

    Synapses and muscle contraction

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    • Neuron simulationPhETsee neurotransmitter and ion channels at work in a model neurone
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    • CAIE A Level Biology: control and coordination notes and topic questionsPhysics & Maths Tutorfree revision notes and questions
    • The musculoskeletal systemOpenStax Biology 2efree university textbook chapter on muscle contraction, beyond the syllabus in places
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    Control and coordination in plants

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    • Plant form and physiologyOpenStax Biology 2efree university textbook chapter on plant responses and hormones, beyond the syllabus in places
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    Inheritance and evolution

    Meiosis and genetic variation

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    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of meiosis and crossing over
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    • Inheritance (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • CAIE A Level Biology: inherited change notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Meiosis and sexual reproductionOpenStax Biology 2efree university textbook chapter on meiosis, beyond the syllabus in places
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    Monohybrid inheritance

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    • Inheritance (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • Learn.Genetics: the basicsUniversity of Utahinteractive pages on genes, alleles and inheritance
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    • CAIE A Level Biology: inherited change notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Mendel’s experiments and the laws of probabilityOpenStax Biology 2efree university textbook chapter on Mendel and genetic crosses, beyond the syllabus in places
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    Dihybrid crosses and the chi-squared test

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    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of dihybrid crosses
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    • Inheritance (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • Learn.Genetics: the basicsUniversity of Utahinteractive pages on genes, alleles and inheritance
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    • CAIE A Level Biology: inherited change notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Mendel’s experiments and the laws of probabilityOpenStax Biology 2efree university textbook chapter on Mendel and dihybrid crosses, beyond the syllabus in places
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    • Paper 5 past papersPhysics & Maths Tutorfind the chi-squared questions in real papersUnofficial archive

    Genes, phenotype and gene control

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    • Transcribe and translate a geneLearn.Genetics, University of Utahan interactive view of how a gene becomes a protein
    • Gene Expression EssentialsPhETswitch transcription factors on and off and see the mRNA and protein made
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    • CAIE A Level Biology: inherited change notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Gene expressionOpenStax Biology 2efree university textbook chapter on the lac operon and eukaryotic gene control, beyond the syllabus in places

    Variation and the t-test

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    • Selection and evolution (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
    • Practical skills notes and questionsPhysics & Maths Tutorstatistics and data-handling questions for Papers 3 and 5
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    • CAIE A Level Biology: selection and evolution notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Khan Academy BiologyKhan Academyfree lessons on inheritance and populations
    Papers
    • Paper 5 past papersPhysics & Maths Tutorfind the statistics questions in real papersUnofficial archive

    Natural and artificial selection

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    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of natural selection and genetic drift
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    • Selection and evolution (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • Natural SelectionPhETchange the environment and watch allele frequencies in a bunny population
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    • CAIE A Level Biology: selection and evolution notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Evolution of populationsOpenStax Biology 2efree university textbook chapter on allele frequencies, drift and Hardy–Weinberg, beyond the syllabus in places
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    Evolution and speciation

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    • Selection and evolution (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • Natural selection simulationPhETchange the environment and watch allele frequencies shift over generations
    • EvolutionLearn.Genetics (University of Utah)short interactive explanations of evolution and relationships between species
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    • CAIE A Level Biology: selection and evolution notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Evolution and the origin of speciesOpenStax Biology 2efree university textbook chapter on speciation, beyond the syllabus in places
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    • Paper 4 past papersPhysics & Maths Tutorfind the evolution and speciation questions in real papersUnofficial archive

    Ecology and genetic technology

    Classification

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    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of classification and the domains
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    • Classification, biodiversity and conservation (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • CAIE A Level Biology: classification and biodiversity notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Phylogenies and the history of lifeOpenStax Biology 2efree university textbook chapter on taxonomy and the tree of life, beyond the syllabus in places
    • A Level Biology notesZNotesshort CAIE notes to check your recall
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    • Paper 4 past papersPhysics & Maths Tutorfind the classification questions in real papersUnofficial archive

    Biodiversity and sampling

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    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of ecology and sampling
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    • CAIE A Level Biology: classification and biodiversity notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Population and community ecologyOpenStax Biology 2efree university textbook chapter on populations, sampling and communities, beyond the syllabus in places
    Papers
    • Paper 5 past papersPhysics & Maths Tutorpractise planning and analysis questions on sampling and statisticsUnofficial archive

    Conservation

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    • Amoeba Sisters videosAmoeba Sisters (YouTube)a friendly first explanation of ecology and biodiversity
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    • Classification, biodiversity and conservation (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • CAIE A Level Biology: classification, biodiversity and conservation notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Conservation biologyOpenStax Biology 2efree university textbook chapter on biodiversity loss and conservation, beyond the syllabus in places
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    • Paper 4 past papersPhysics & Maths Tutorfind the conservation questions in real papersUnofficial archive

    Tools of genetic technology

    Practise
    • Genetic technology (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • Gel electrophoresis labLearn.Genetics, University of Utahrun a virtual gel and see how bands form
    • PCR labLearn.Genetics, University of Utahstep through the cycle and see the DNA double
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    • CAIE A Level Biology: genetic technology notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Biotechnology and genomicsOpenStax Biology 2efree university textbook chapter on cloning, PCR and gel electrophoresis, beyond the syllabus in places
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    • Paper 4 past papersPhysics & Maths Tutorfind the genetic technology questions in real papersUnofficial archive

    Genetic technology in medicine and agriculture

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    • Genetic technology (9700, exams from 2025)Save My Examsnotes and exam questions with mark schemes, some freeSome content paid
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    • The Basics: DNA, genes and proteinsLearn.Genetics, University of Utahclear interactive background on genes and how a change in a gene affects a protein
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    • CAIE A Level Biology: genetic technology notes and topic questionsPhysics & Maths Tutorfree revision notes and past-paper questions sorted by topic
    • Biotechnology and genomicsOpenStax Biology 2efree university textbook chapter that includes gene therapy and genetically modified organisms, beyond the syllabus in places
    Papers
    • Paper 4 past papersPhysics & Maths Tutorfind the genetic technology questions in real papersUnofficial archive

    A Level skills

    Statistics for biologists

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    • CAIE A Level Biology: practical skills and statisticsPhysics & Maths Tutorfree notes and questions on data handling and statistical tests
    • Khan Academy BiologyKhan Academyfree lessons if you want another explanation of the ideas behind sampling and variation
    Papers
    • Paper 5 past papersPhysics & Maths Tutorreal questions that use these testsUnofficial archive

    Planning investigations (Paper 5)

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    • Topic questions (9700, exams from 2025)Save My Examsexam questions with mark schemes by topicSome content paid
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    • CAIE A Level Biology: practical skills notesPhysics & Maths Tutorfree notes on planning, variables and evaluation
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    • Paper 5 past papersPhysics & Maths Tutorreal planning questions to practise onUnofficial archive

    Analysis, conclusions and evaluation (Paper 5)

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    • Topic questions (9700, exams from 2025)Save My Examsexam questions with mark schemes by topicSome content paid
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    • CAIE A Level Biology: practical skills notesPhysics & Maths Tutorfree notes on analysis, conclusions and evaluation
    • ZNotes A2 Biology notesZNotesfree summary notes for the A Level syllabus
    Papers
    • Paper 5 past papersPhysics & Maths Tutorreal analysis and evaluation questions to practise onUnofficial archive

    Exam readiness

    Paper 1: multiple choice

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    • CAIE A Level Biology topic questionsSave My Examsquestions sorted by topic, with mark schemes, some freeSome content paid
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    • CAIE A Level Biology topic notesPhysics & Maths Tutorfree notes and topic questions, one page per topic
    Papers
    • Past papers by paper: Paper 1Physics & Maths Tutorreal Paper 1 question papers and mark schemes to sort by yearUnofficial archive

    Papers 2 and 4: structured questions

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    • ZNotes AS notesZNotesshort revision notes for the AS topics
    • ZNotes A Level notesZNotesshort revision notes for the A Level topics
    Papers
    • Past papers by paper: Paper 2Physics & Maths Tutorreal Paper 2 question papers and mark schemesUnofficial archive
    • Past papers by paper: Paper 4Physics & Maths Tutorreal Paper 4 question papers and mark schemesUnofficial archive

    Papers 3 and 5: the practical skills papers

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    • CAIE A Level Biology practical skills notesPhysics & Maths Tutorfree notes and questions on practical skills
    Papers
    • Past papers by paper: Paper 3Physics & Maths Tutorreal Paper 3 question papers and mark schemesUnofficial archive
    • Past papers by paper: Paper 5Physics & Maths Tutorreal Paper 5 question papers and mark schemesUnofficial archive

    Mixed practice: A Level

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    • CAIE A Level Biology topic questionsSave My Examsexam questions grouped by topic, some freeSome content paid
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    • ZNotes A Level notesZNotesshort revision notes for the A Level topics
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    • Past papers by paper: Paper 4Physics & Maths Tutorreal Paper 4 question papers and mark schemesUnofficial archive
    • Past papers by paper: Paper 5Physics & Maths Tutorreal Paper 5 question papers and mark schemesUnofficial archive

    Mock Paper 3

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    • CAIE A Level Biology topic questionsSave My Examspractical-skills questions to follow up the marks you lostSome content paid
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    • Past papers by paper: Paper 3Physics & Maths Tutorreal Paper 3 question papers and mark schemes to try under timed conditionsUnofficial archive

    Mock Paper 4

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    Mock Paper 5

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    • CAIE A Level Biology topic questionsSave My Examspractical-skills questions to follow up the marks you lostSome content paid
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    • Past papers by paper: Paper 5Physics & Maths Tutorreal Paper 5 question papers and mark schemes to try under timed conditionsUnofficial archive
    PreviousChemistry: learn more onlineNextPractice quizzes
    Practice/Practice

    Practice quizzes

    Interactive

    Mixed multiple-choice questions drawn from every chapter. Pick a subject and a length, answer, and get a list of chapters to revise.

    TipHow to use quizzes well

    Quizzes are retrieval practice: the act of pulling an answer out of memory is what makes it stick. Do a short quiz at the start of each study session on topics from last week, not just today’s topic. When you get one wrong, open the chapter it came from and redo the worked examples before trying again.

    PreviousBiology: learn more onlineNextWhat to revise next
    Practice/Practice

    What to revise next

    Interactive

    Your weakest chapters first, from the mock papers you have marked, your recent quiz answers and the checklists you have not ticked yet.

    How the list is made

    Everything here comes from this browser: nothing is sent anywhere.

    • Mock papers. When you finish a mock and type in your mark for each question, every mark you lost counts against the chapter that question revises. Only your latest marking of each mock counts, so redoing a mock replaces the old result.
    • Quizzes. The last 10 answers you gave for each chapter’s questions, in chapter quizzes and in Practice quizzes. A wrong answer counts as much as 2 lost marks.
    • Checklists. In a chapter you have marked as learned, each “I can” line you have not ticked counts as half a mark.

    The bar shows how far behind each chapter is compared with the top one.

    TipUse the list well

    Work from the top. For each chapter: reread its key boxes, redo its worked examples with the answers covered, then try the questions you lost marks on again. Don’t spend all week on one chapter: two or three short sessions on the top five beat one long session on the first.

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