A-level Maths Exam Study Guide (2027)
Everything the A-level Mathematics Exam tests, topic by topic: what each section covers, the facts that come up most, and how to study it. Pair it with the free 15-question practice test.
The exam at a glance
| Questions | 3 written papers, 100 marks each (300 marks in total) |
|---|---|
| Time | 6 hours in total (3 × 2 hours) |
| Grades and pass mark | Graded A* to E; grade E is a pass. Boundaries are set after every exam series, so aim for 70% in practice |
| Format | Written exam, calculator allowed on all three papers. A formula booklet is provided. About two thirds pure maths and one third statistics and mechanics |
| Exam boards | AQA, Pearson Edexcel and OCR |
| Test provider | At your school or sixth form college, or as a private candidate at an exam centre that accepts them |
Mind map: the whole exam on one screen
Read the complete A-level Maths study notes for “Proof, Algebra and Functions”
Our first full set of comprehensive notes, about 20 minutes of reading: The four kinds of proof · Setting out a proof · Indices, surds and quadratics · Simultaneous equations and inequalities · Polynomials, the factor theorem and partial fractions · Functions: domain, range, composites and inverses · Graphs and transformations, a worked exam example, the traps examiners use and a one-minute recap. Create a free account to read it and see how premium notes work before you decide.
1. Proof, Algebra and Functions
Proof by deduction, exhaustion and contradiction; indices and surds; quadratics and the discriminant; inequalities; polynomials and the factor theorem; partial fractions; functions, graphs and transformations. This underpins every pure question.
Key facts
- Proof methods: deduction (logical steps from known facts), exhaustion (check every case), counter-example (disprove a statement with one case) and contradiction (assume the opposite and reach an impossibility, e.g. √2 is irrational; there are infinitely many primes).
- The discriminant b² - 4ac of ax² + bx + c tells you the number of real roots: > 0 two distinct, = 0 one repeated, < 0 none. Use it for 'the line is a tangent' or 'has no real roots' questions.
- Solve quadratic inequalities by finding the critical values and sketching the curve: for (x - 1)(x - 4) < 0 the answer is 1 < x < 4; for > 0 it is x < 1 or x > 4.
- Factor theorem: (x - a) is a factor of f(x) if and only if f(a) = 0. Remainder theorem: the remainder when f(x) is divided by (x - a) is f(a).
- Partial fractions: split (5x + 1)/((x + 1)(x - 2)) into A/(x + 1) + B/(x - 2); repeated factors need A/(x - a) + B/(x - a)².
- Functions: fg(x) means do g first then f. f⁻¹ reflects f in y = x; its domain is the range of f. Only one-to-one functions have inverses. Modulus graphs |f(x)| reflect the negative parts in the x-axis.
- Transformations: y = f(x) + a moves up a; y = f(x + a) moves left a; y = af(x) stretches vertically by factor a; y = f(ax) stretches horizontally by factor 1/a; y = -f(x) reflects in the x-axis; y = f(-x) reflects in the y-axis.
Flashcards
Exam tip: In proof questions, end with a clear concluding sentence that repeats the statement you were asked to prove. Examiners often reserve the last mark for it.
2. Coordinate Geometry, Sequences, Series and Vectors
Straight lines and circles, parametric equations, the binomial expansion, arithmetic and geometric sequences and series, sigma notation, recurrence relations and vectors in 2D and 3D.
Key facts
- Line through (x₁, y₁) with gradient m: y - y₁ = m(x - x₁). Perpendicular gradients multiply to -1.
- Circle: (x - a)² + (y - b)² = r² has centre (a, b) and radius r. Complete the square to find them. A tangent is perpendicular to the radius at the point of contact; the perpendicular bisector of a chord passes through the centre.
- Parametric equations x = f(t), y = g(t): eliminate t to get a Cartesian equation, e.g. using sin² t + cos² t = 1.
- Binomial expansion: (a + b)^n = Σ nCr a^(n-r) b^r for positive integer n. For any rational n, (1 + x)^n = 1 + nx + n(n - 1)x²/2! + ..., valid for |x| < 1; for (a + bx)^n take a^n out first.
- Arithmetic series: uₙ = a + (n - 1)d and Sₙ = n/2 (2a + (n - 1)d). Geometric series: uₙ = ar^(n-1), Sₙ = a(1 - r^n)/(1 - r), and S∞ = a/(1 - r) when |r| < 1.
- Sequences can be increasing, decreasing or periodic. A recurrence relation such as u(n+1) = 2u(n) - 3 defines each term from the previous one.
- Vectors: magnitude of ai + bj + ck is √(a² + b² + c²); a unit vector is the vector divided by its magnitude; the position vector of the point dividing AB in the ratio 1 : 2 is a + ⅓(b - a).
Flashcards
Exam tip: Write down a, d (or r) and n before using any series formula. Most lost marks come from using n when the question counts from n = 0 or asks for terms 'between' two values.
3. Trigonometry
Radians, arc length and sector area, small-angle approximations, trig graphs and exact values, reciprocal and inverse functions, identities, compound and double angle formulae, R cos(x ± α) form and solving equations.
Key facts
- π radians = 180°. Arc length s = rθ and sector area A = ½r²θ, with θ in radians.
- Small-angle approximations (θ in radians): sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1 - θ²/2.
- Exact values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, sin 45° = cos 45° = √2/2, tan 45° = 1, sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
- Identities: sin² x + cos² x = 1, tan x = sin x / cos x, 1 + tan² x = sec² x, 1 + cot² x = cosec² x.
- Compound angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B. Double angles: sin 2A = 2 sin A cos A; cos 2A = cos² A - sin² A = 2cos² A - 1 = 1 - 2sin² A.
- a sin x + b cos x can be written R sin(x + α) with R = √(a² + b²) and tan α = b/a. Use it to find maxima and minima and to solve equations.
- When solving, find the principal value, use the symmetry of the graph (or CAST) to find others, and adjust the interval for multiples such as 2x before solving.
- Inverse functions arcsin, arccos and arctan have restricted domains and ranges: arcsin and arctan give values in [-π/2, π/2], arccos in [0, π].
Flashcards
Exam tip: Check your calculator is in the right mode before every trig question. If the interval is in terms of π, work in radians from the start.
Test yourself on these topics
Take the free 15-question A-level Maths test, then save your results to unlock your Readiness Score and the mistakes bank.
4. Exponentials and Logarithms
Exponential functions and e^x, logarithms and their laws, solving exponential equations, modelling growth and decay, and using logs to turn curves into straight lines.
Key facts
- y = a^x passes through (0, 1) and is always positive. y = e^x has gradient equal to its value: d/dx(e^(kx)) = ke^(kx).
- log_a x = n means a^n = x. ln x is log_e x, the inverse of e^x, so e^(ln x) = x and ln(e^x) = x.
Flashcards
Exam tip: Never write log(a + b) = log a + log b. If an equation has a sum inside a log, combine the other logs first or rewrite it in exponential form.
5. Differentiation
Differentiation from first principles, standard derivatives, the chain, product and quotient rules, implicit and parametric differentiation, stationary points, concavity, tangents, normals and connected rates of change.
Key facts
- First principles: f'(x) = limit as h → 0 of (f(x + h) - f(x)) / h. You may be asked to prove the derivative of x² or x³ this way.
- Standard results: d/dx x^n = nx^(n-1), d/dx e^(kx) = ke^(kx), d/dx ln x = 1/x, d/dx sin kx = k cos kx, d/dx cos kx = -k sin kx, d/dx tan kx = k sec² kx.
Flashcards
Exam tip: When a question says 'hence' after a differentiation part, the answer you just found is the intended route. Using a different method can lose the marks.
6. Integration and Numerical Methods
Indefinite and definite integration, standard integrals, areas under and between curves, integration by substitution, by parts and with partial fractions, the trapezium rule, differential equations, and numerical root-finding.
Key facts
- ∫x^n dx = x^(n+1)/(n + 1) + c (n ≠ -1); ∫1/x dx = ln|x| + c; ∫e^(kx) dx = (1/k)e^(kx) + c; ∫cos kx dx = (1/k) sin kx + c; ∫sin kx dx = -(1/k) cos kx + c.
- A definite integral gives the signed area: regions below the x-axis come out negative, so split the integral at the roots for total area. Area between curves = ∫(upper - lower) dx.
Flashcards
Exam tip: Always add + c to indefinite integrals and change the limits when you substitute. Both are one-mark losses that examiners see every year.
7. Statistics
Sampling, data presentation and interpretation, the large data set, probability, the binomial and normal distributions, and hypothesis testing for a binomial proportion, a normal mean and correlation.
Key facts
- Sampling methods: simple random, systematic, stratified, quota, opportunity (convenience) and cluster. Know one advantage and one disadvantage of each.
- Measures: mean, median, mode, standard deviation σ = √(Σx²/n - x̄²), interquartile range. Outliers are often defined as more than 1.5 × IQR beyond a quartile, or more than 2 standard deviations from the mean; use the definition given.
Flashcards
Exam tip: Write hypothesis-test conclusions in two parts: a statistical statement ('reject H₀') and a contextual one ('there is evidence at the 5% level that the coin is biased'). Never say the test proves anything.
8. Mechanics
Units and modelling assumptions, kinematics with constant and variable acceleration, projectiles, forces and Newton's laws, connected particles, friction and moments.
Key facts
- Modelling assumptions: particle (no size, no rotation), light (no mass), inextensible string (same acceleration throughout), smooth (no friction), rigid rod. Say how each affects the answer.
- suvat equations for constant acceleration: v = u + at, s = ut + ½at², s = ½(u + v)t, v² = u² + 2as, s = vt - ½at².
Flashcards
Exam tip: Draw a large, clear force diagram for every mechanics question, with the direction of acceleration marked. Most errors come from a missing force or a resolving slip.
How to pass
- 100 marks in 120 minutes is 1.2 minutes per mark. A 10-mark question deserves about 12 minutes; if you are stuck, write down what you know, move on and come back.
- Show every step. 'Show that' and proof questions give no credit for a correct final line without the working, and method marks are available even when the answer is wrong.
- Learn what is NOT in the formula booklet: differentiation and integration of standard functions, the quadratic formula, sin² x + cos² x = 1, the laws of logarithms and the suvat equations.
- Know your board's large data set well. Questions assume you know its variables, units and quirks, and these are easy marks if you have explored it in lessons.
- In statistics and mechanics, finish with a sentence in context: state the conclusion of a hypothesis test in words, and give units and a sensible accuracy (usually 3 significant figures) for every answer.
Frequently asked questions
How long should I study for the A-level Maths exam?
Most students do best starting 10 to 12 weeks before the first paper, with 30 to 45 minutes a day per subject. You're ready once your Readiness Score has stayed at 80 or higher for about a week.
Can I take the A-level Maths exam online?
No. A-level Maths is a written exam sat in person at your school, college or an approved exam centre. Format: written exam, calculator allowed on all three papers. A formula booklet is provided. About two thirds pure maths and one third statistics and mechanics. Check your exam board's timetable for dates.
What score do I need to pass the A-level Maths exam?
Graded A* to E; grade E is a pass. Boundaries are set after every exam series, so aim for 70% in practice. The exam has 3 written papers, 100 marks each (300 marks in total) and a time limit of 6 hours in total (3 × 2 hours).
What are the most tested topics?
The outline covers proof, algebra and functions, coordinate geometry, sequences, series and vectors, trigonometry, exponentials and logarithms, differentiation, integration and numerical methods, statistics, mechanics. Use the free practice test to see which ones you need most.
The premium A-level Maths study path
Premium A-level Maths prep is studied one topic at a time. Each step starts with that topic's key notes and flashcard deck, then its own question sets of 25, so you master one topic before you move to the next. When every topic is strong, the 6 full mock exams show you're ready.
Your step-by-step study path
- 1Proof, Algebra and Functions8 key notes, 100 flashcards, then 375 practice questions in 15 sets of 25Notes free with an account
- 2Coordinate Geometry, Sequences, Series and Vectors8 key notes, 100 flashcards, then 375 practice questions in 15 sets of 25Premium
- 3Trigonometry8 key notes, 100 flashcards, then 375 practice questions in 15 sets of 25Premium
- 4Exponentials and Logarithms8 key notes, 100 flashcards, then 375 practice questions in 15 sets of 25Premium
- 5Differentiation7 key notes, 100 flashcards, then 375 practice questions in 15 sets of 25Premium
- 6Integration and Numerical Methods7 key notes, 100 flashcards, then 375 practice questions in 15 sets of 25Premium
- 7Statistics7 key notes, 100 flashcards, then 375 practice questions in 15 sets of 25Premium
- 8Mechanics7 key notes, 100 flashcards, then 375 practice questions in 15 sets of 25Premium
- Final step: 6 full mock exams50 questions each, timed like exam day. Hit your target on them before your first paper.Premium
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Summarised from the exam boards' published specifications and the rules cited, checked November 2026. Confirm current requirements with AQA, Pearson Edexcel and OCR.