A-Level Maths revision guide: what to revise and how to actually do it
A-Level Maths is examined in three two-hour papers at the end of Year 13, all calculator-allowed, covering pure maths, statistics and mechanics. Effective revision means working through questions — past papers first, topic questions for weak spots — and marking them against the real mark schemes, because the marks are for method as much as answers. Rereading notes produces familiarity; producing solutions produces marks.
A-Level Maths has a reputation for being the hardest popular A-Level, which is mostly unfair. It is not hard because the ideas are impossible. It is hard because it is cumulative: every topic assumes the one before it, so a gap in Year 12 algebra becomes a hole in Year 13 calculus. Revise it as a connected course, not a list of chapters, and it gets much less frightening.
What you're actually sitting
Whichever board your school uses, the shape is the same: three written papers at the end of Year 13, a calculator allowed in all of them, and no coursework. What differs is how the applied content is packaged.
| Board | Paper 1 | Paper 2 | Paper 3 |
|---|---|---|---|
| Edexcel (9MA0) | Pure, 2h, 100 marks | Pure, 2h, 100 marks | Statistics + Mechanics, 2h, 100 marks |
| AQA (7357) | Pure, 2h, 100 marks | Pure + Mechanics, 2h, 100 marks | Pure + Statistics, 2h, 100 marks |
| OCR A (H240) | Pure, 2h, 100 marks | Pure + Statistics, 2h, 100 marks | Pure + Mechanics, 2h, 100 marks |
| OCR MEI (H640) | Pure + Mechanics, 2h, 120 marks | Pure + Statistics, 2h, 120 marks | Pure + Comprehension, 2h, 120 marks |
The important consequence: on AQA and OCR, pure maths turns up in every single paper. On Edexcel, the applied content is quarantined into Paper 3, which means students routinely under-revise it and then lose a third of their grade in one afternoon. Whichever board you are on, statistics and mechanics are a third of your marks, not an afterthought.
The topic breakdown
Pure maths — about two-thirds of the course
- Proof — including proof by contradiction and disproof by counterexample.
- Algebra and functions — indices, surds, quadratics, simultaneous equations, inequalities, modulus, transformations, composite and inverse functions.
- Coordinate geometry — straight lines, circles, and parametric equations.
- Sequences and series — arithmetic and geometric, plus the binomial expansion for any rational n.
- Trigonometry — radians, all six trig functions, identities, equations, and R cos(x − α) form.
- Exponentials and logarithms — eˣ, ln x, and modelling growth and decay.
- Differentiation — first principles, chain/product/quotient rules, parametric and implicit differentiation, rates of change.
- Integration — substitution, parts, partial fractions, areas, and solving differential equations.
- Numerical methods — iteration and Newton–Raphson; the least-loved topic that comes up every year.
- Vectors — 2D and 3D, including the vector equation of a line.
Statistics
- Sampling methods and why some are biased.
- Presenting and interpreting data — histograms, box plots, outliers, measures of location and spread.
- Probability — Venn diagrams, tree diagrams, conditional probability.
- Statistical distributions — the binomial and the normal, including using your calculator for cumulative probabilities.
- Hypothesis testing — one-tailed and two-tailed, significance levels and critical regions.
Mechanics
- Kinematics — the SUVAT equations, displacement-time and velocity-time graphs, and motion handled with calculus.
- Projectiles — splitting motion into horizontal and vertical components.
- Forces and Newton's laws — resolving, friction (F ≤ μR), and F = ma.
- Moments — rigid bodies, ladders, and the dreaded tilting problem.
- Connected particles — pulleys, slopes and towing.
How the marking actually works
A-Level Maths mark schemes are built from method marks, accuracy marks and — on the longer questions — marks for the shape of your argument. A wrong final answer with correct method typically loses only the last mark or two. A correct answer with no working, on a question that says "show that" or "hence", can score almost nothing. This is the single most misunderstood thing about the exam: the working is the answer.
Worked questions, the way examiners set them
Find ∫ x·e^(2x) dx.
- Recognise a product of x (which simplifies when differentiated) and e^(2x) (which is easy to integrate) — integration by parts: ∫u dv = uv − ∫v du.
- Set u = x, so du/dx = 1. Set dv/dx = e^(2x), so v = ½e^(2x).
- Substitute: ∫ x·e^(2x) dx = x·½e^(2x) − ∫ ½e^(2x) dx.
- The remaining integral is standard: ∫ ½e^(2x) dx = ¼e^(2x).
∫ x·e^(2x) dx = ½x·e^(2x) − ¼e^(2x) + c. Check by differentiating back — the product rule recreates x·e^(2x).
A coin is tossed 40 times and lands heads 27 times. Test at the 5% level whether the coin is biased towards heads.
- Define hypotheses: H₀: p = 0.5, H₁: p > 0.5, where p is the probability of heads.
- Under H₀, X ~ B(40, 0.5). Find P(X ≥ 27) — a tail probability, not P(X = 27).
- Using the cumulative binomial: P(X ≥ 27) = 1 − P(X ≤ 26) ≈ 1 − 0.9597 = 0.0403.
- Compare with the significance level: 0.0403 < 0.05, so the result is significant.
- Conclude in context, not in symbols: there is evidence at the 5% level to suggest the coin is biased towards heads.
Reject H₀. P(X ≥ 27) ≈ 0.040 < 0.05 — significant evidence the coin favours heads.
A particle of mass 2 kg rests on a rough horizontal table, connected by a light inextensible string over a smooth pulley to a particle of mass 3 kg hanging freely. The coefficient of friction is 0.4. Find the acceleration when the system is released.
- For the hanging mass, the only equation is Newton's second law downwards: 3g − T = 3a.
- For the table mass, resolve vertically first: R = 2g, so friction F = μR = 0.4 × 2g = 0.8g.
- Horizontally: T − F = 2a, so T − 0.8g = 2a.
- Add the two equations to eliminate T: 3g − 0.8g = 5a, so 2.2g = 5a.
- a = 2.2 × 9.8 ÷ 5 = 4.312 m s⁻².
a ≈ 4.31 m s⁻². The classic error is writing F = ma for the whole system with the wrong mass — each particle gets its own equation.
Where marks actually get dropped
- The minus sign. Most lost marks in the whole qualification are dropped negatives — in expanding, in integrating e^(−x), in SUVAT with downwards motion.
- Degree mode. Trig questions are in radians at A-Level. A calculator left in degrees silently wrecks every answer.
- Ignoring "show that". The answer is printed; the marks are for the route. Skipping a step costs the step's mark.
- Premature rounding. Rounding mid-calculation compounds; keep full precision until the final line.
- Statistics treated as the easy third. Hypothesis-test conclusions must be in context — "reject H₀" alone scores nothing for the final mark.
- Vectors in 3D abandoned. The z-coordinate adds almost no difficulty; students still leave whole questions blank.
The revision checklist
- I can do every Year 1 topic without notes — Year 2 content assumes it completely.
- I have done at least six full timed papers and marked them against the real mark scheme.
- For every mark I dropped, I have a one-line note on why — and I have re-done those question types.
- I know which formulae are in my board's booklet and which I must recall.
- I can state the conditions for a binomial model and set up a hypothesis test without a template.
- My calculator is in radians, I can find cumulative binomial and normal probabilities on it, and I know its iterative-solve function.
- I have read at least one examiner report for my board.
Where Lumi fits
The sticking point in maths revision is rarely effort — it is the moment a mark scheme says a line of working and you cannot see why that line is allowed. That is exactly what Lumi is for: it draws the step on a board, works back to the piece of algebra or the rule you actually lost, and then gives you a similar question to check it landed. Upload the paper, point at the line, interrupt it when the explanation races ahead.
What to take from this
- Three papers, six hours, all calculator. Every board splits into pure plus applied — the applied half is where underprepared students lose easy marks.
- Pure maths is roughly two-thirds of the course and underpins the rest, so it repays the most revision time.
- Past papers are diagnosis, not a finishing ritual. Start them months before the exam.
- Method marks mean a wrong answer with correct working can score most of the marks. Show every line.
- The formula booklet is a safety net, not a plan — the students who rely on it lose minutes every question.
Questions people also ask
Every paper your board has set since the 2017 spec change is worth doing — that is eight or more full sets. Do them timed, mark them honestly, and redo the topics that cost you marks rather than just accumulating papers.
Yes — Edexcel, AQA and OCR all provide a formula booklet in every paper. It covers standard results but not the methods, and not the applied formulae you use constantly. Knowing where things are in it saves real time.
It moves every series, but as rough guidance an A has recently sat around 65–75% overall and an A* around 75–85%. Treat those as targets, not guarantees — boundaries are set after each paper is marked.
Yes — they are more learnable than pure because the question types repeat. A hypothesis test and a connected-particles question follow the same skeleton almost every series. Learn the skeleton and the marks are unusually reliable.
Both. The board's own site has the authoritative papers, mark schemes and examiner reports; topic-organised sites like PMT are better for drilling a single weak topic. Use topic questions to fix weaknesses, full papers for timing.