Cambridge AS and A Level Mathematics 9709 exam guide: Pure Mathematics 1, 2 and 3
Every pure mathematics topic in the 2026–2027 syllabus, from 1.1 to 3.9, mapped to a free lesson and a book chapter, with the papers, routes, calculator rules, how answers are marked and a 10-week revision plan.
The papers at a glance
Exams in 2027 follow the 2026–2027 syllabus. Exams are available in the June and November series, and also in the March series in India. All papers are written, externally assessed, and every question is compulsory.
| Paper | What it is | Time | Marks | Weighting |
|---|---|---|---|---|
| Paper 1 | Pure Mathematics 1: 10 to 12 structured questions | 1 hour 50 minutes | 75 | 60% of AS, 30% of A Level |
| Paper 2 | Pure Mathematics 2: 6 to 8 structured questions | 1 hour 15 minutes | 50 | 40% of AS; AS only |
| Paper 3 | Pure Mathematics 3: 9 to 11 structured questions | 1 hour 50 minutes | 75 | 30% of A Level |
| Paper 4 | Mechanics: 6 to 8 structured questions | 1 hour 15 minutes | 50 | 40% of AS, 20% of A Level |
| Paper 5 | Probability & Statistics 1: 6 to 8 questions | 1 hour 15 minutes | 50 | 40% of AS, 20% of A Level |
| Paper 6 | Probability & Statistics 2: 6 to 8 questions | 1 hour 15 minutes | 50 | 20% of A Level |
Which papers you sit Permitted routes
| Route | Papers | Notes |
|---|---|---|
| AS Level | P1 + P2, P1 + P4 or P1 + P5 | Two papers in the same series. P1 + P2 is pure only and cannot count towards A Level. Graded a–e. |
| A Level | P1 + P3 + P4 + P5 or P1 + P3 + P5 + P6 | Linear: all four in one series. Graded A*–E. |
| Staged A Level | P1 + P4 then P3 + P5; P1 + P5 then P3 + P4; P1 + P5 then P3 + P6 | AS in one series, then two more papers in a later series, with the AS result carried forward. |
| Not allowed | P4 with P6; P2 and P3 in the same series | Paper 6 builds on Paper 5. |
Paper 1 is compulsory for both AS and A Level, and Papers 3 and 5 are compulsory for A Level. Paper 2 content is largely a subset of Paper 3, which is why the topic tables below for Papers 2 and 3 look alike. Cambridge gives about 180 guided learning hours for AS and about 360 for the full A Level.
Does anything change from 2028?No significant changes. The 2028–2030 syllabus keeps the same papers, durations, marks and weightings, and Cambridge says there are no significant changes which affect teaching. Version 4 of the 2026–2027 syllabus (December 2025) only fixed a weblink and typos.
Pure Mathematics topics (1.1–3.9)
Cambridge gives weightings by paper, not by topic, so each heading shows the paper’s weighting. Each topic links to a free step of our Mathematics guide and to the matching chapter of Mathematics Made Simple.
Paper 1: Pure Mathematics 1 60% of AS, 30% of A Level
| Topic | What you must know | Learn it here | Chapter in the book |
|---|---|---|---|
| 1.1 | Quadratics: completing the square, discriminant | Quadratics and Inequalities | Chapter 2 |
| 1.2 | Functions: domain, range, composite, inverse | Functions and Inverses | Chapter 3 |
| 1.3 | Coordinate geometry: lines and circles | Lines and Circles | Chapter 4 |
| 1.4 | Circular measure: radians, arcs and sectors | Trigonometry: Radians, Graphs and Equations | Chapter 7 |
| 1.5 | Trigonometry: graphs, exact values, equations | Trigonometry: Radians, Graphs and Equations | Chapter 7 |
| 1.6 | Series: binomial (positive n), AP and GP | Sequences, Series and the Binomial Theorem | Chapter 6 |
| 1.7 | Differentiation: gradient as a limit, \( x^n \), chain rule | The Derivative from First Principles, Power, Product and Quotient Rules, The Chain Rule | Chapter 10 |
| 1.7 | Tangents, normals, rates of change, stationary points | The Chain Rule, Rates of Change, Motion and Related Rates, Stationary Points, Concavity and Curve Sketching, Optimisation | Chapters 10, 11 and 12 |
| 1.8 | Integration of \( (ax+b)^n \); definite and simple improper integrals | The Fundamental Theorem of Calculus | Chapter 13 |
| 1.8 | Area under and between curves; volumes of revolution | Area Between Curves and Average Value, Volumes of Revolution and Arc Length | Chapter 15 |
Paper 2: Pure Mathematics 2 40% of AS (AS only)
| Topic | What you must know | Learn it here | Chapter in the book |
|---|---|---|---|
| 2.1 | Modulus, polynomial division, factor and remainder theorems | Indices, Surds and Polynomials, Quadratics and Inequalities | Chapters 1 and 2 |
| 2.2 | Logarithms and exponentials, \( e^x \) and ln, linearising data | Exponentials and Logarithms | Chapter 5 |
| 2.3 | sec, cosec, cot; compound and double angles; \( R\sin(\theta \pm \alpha) \) | Trigonometric Identities and the Harmonic Form | Chapter 8 |
| 2.4 | Differentiating exp, ln, trig, products, quotients, composites | Power, Product and Quotient Rules, The Chain Rule | Chapter 10 |
| 2.4 | First derivative of parametric and implicit functions | Implicit Differentiation and Inverse Functions, Parametric, Polar and Vector-Valued Functions | Chapters 11 and 20 |
| 2.5 | Integrating \( e^{ax+b} \), \( \tfrac{1}{ax+b} \), sin, cos, \( \sec^2 \) | The Fundamental Theorem of Calculus | Chapters 13 and 14 |
| 2.5 | Trapezium rule, over- and under-estimates | Riemann Sums and the Definite Integral | Chapter 13 |
| 2.6 | Sign change and iteration \( x_{n+1} = F(x_n) \) | Numerical Methods: Roots and Iteration | Chapter 17 |
Paper 3: Pure Mathematics 3 30% of A Level
| Topic | What you must know | Learn it here | Chapter in the book |
|---|---|---|---|
| 3.1 | Paper 2 algebra plus partial fractions | Indices, Surds and Polynomials, Integration by Parts and Partial Fractions | Chapters 1 and 14 |
| 3.1 | Binomial expansion for rational n, with validity | Sequences, Series and the Binomial Theorem | Chapter 6 |
| 3.2 | Logarithms and exponentials | Exponentials and Logarithms | Chapter 5 |
| 3.3 | Trigonometry: reciprocal functions, identities, R form | Trigonometric Identities and the Harmonic Form | Chapter 8 |
| 3.4 | Differentiation, including \( \tan^{-1}x \), parametric and implicit | Power, Product and Quotient Rules, The Chain Rule, Implicit Differentiation and Inverse Functions, Parametric, Polar and Vector-Valued Functions | Chapters 10, 11 and 20 |
| 3.5 | Integration: \( \tfrac{1}{x^2+a^2} \), \( \tfrac{f'(x)}{f(x)} \), partial fractions | Integration by Substitution, Integration by Parts and Partial Fractions | Chapter 14 |
| 3.5 | Integration by parts and by a given substitution | Integration by Substitution, Integration by Parts and Partial Fractions | Chapter 14 |
| 3.6 | Numerical solution of equations | Numerical Methods: Roots and Iteration | Chapter 17 |
| 3.7 | Vectors: lines, intersections, scalar product, angles | Vectors and Lines in 3D | Chapter 18 |
| 3.8 | First-order separable differential equations in context | Separable Differential Equations and Slope Fields | Chapter 16 |
| 3.9 | Complex numbers: Argand diagram, modulus–argument, loci | Complex Numbers | Chapter 19 |
Skip these parts of the lessonsOur lessons serve AP and IB students too, so some steps go further than 9709. These are not in the 9709 syllabus: Newton–Raphson, slope fields, Euler’s method, Maclaurin series, polar curves, the exponential form of a complex number and De Moivre’s theorem. In 3.4 you need the derivative of \( \tan^{-1}x \) but not \( \sin^{-1}x \) or \( \cos^{-1}x \); in 2.2 the change-of-base law is not required for Paper 2.
This guide covers the pure papers. Mechanics (Paper 4) is not part of the Mathematics guide; for the probability background behind Papers 5 and 6, see our Statistics guide.
How answers are marked
Cambridge’s assessment objectives are AO1 Knowledge and understanding and AO2 Application and communication. Paper 3 leans more towards application (45% AO1, 55% AO2); every other paper is 55% AO1, 45% AO2.
- M marks reward a correct method. Write the method down even when you can do it in your head: substitute the limits, show the quadratic you are solving, state the identity you are using. A slip later on then costs one mark, not the whole question.
- A marks are for accuracy and depend on the method. A correct answer reached by a wrong or missing method usually earns nothing, so a right number alone is not safe.
- B marks stand on their own. A correct statement, value or sketch can earn a B mark without any method, so always write down facts you know, such as the coordinates of a stationary point or the validity range of an expansion.
- No marks for unsupported calculator answers. The syllabus says so directly. If you solve an equation or evaluate an integral, show the working that gets you there.
- Give non-exact answers to 3 significant figures. Use one decimal place for angles in degrees, unless the question says otherwise, and do not round until the final answer. Give exact answers (surds, \( \pi \), ln) where the question asks for them.
- “Show that” means every step. The answer is given, so the marks are all in the working. Do not skip lines.
Calculators and the formula list
- A scientific calculator is expected in every paper. Graphical calculators, computers and calculators that can do symbolic algebra, differentiation or integration are not permitted.
- MF19 is supplied. The list of formulae and statistical tables is given in the exam. It is shared with Further Mathematics 9231, so ignore the Further Pure, Further Mechanics and Further Probability & Statistics sections. A copy is in section 5 of the syllabus: learn what is on it, and what is not, before the exam.
- Equipment. Bring a ruler and your scientific calculator. A protractor and a pair of compasses are not required.
10-week revision plan
| Week | Focus | Do this |
|---|---|---|
| 1 | P1 1.1–1.3 | Quadratics, functions and coordinate geometry. |
| 2 | P1 1.4–1.6 | Radians, trigonometry and series. |
| 3 | P1 1.7–1.8 | Differentiation and integration, with areas and volumes. |
| 4 | Paper 1 | Two timed Paper 1s (1 hour 50 minutes each), marked against the mark scheme. |
| 5 | P2/P3 algebra, logs, trig | Topics 2.1–2.3 (3.1–3.3 for A Level): factor theorem, partial fractions, R form. |
| 6 | P2/P3 calculus | Topics 2.4–2.5 (3.4–3.5): every derivative and integral, by parts and substitution. |
| 7 | Numerical methods | Topic 2.6 (3.6): sign change and iteration. AS students: timed Paper 2s this week. |
| 8 | P3 vectors and DEs | Topics 3.7 and 3.8. Practise intersection and skew-line questions. |
| 9 | P3 complex numbers | Topic 3.9: arithmetic, Argand diagrams, modulus–argument form and loci. |
| 10 | Full papers | Timed Paper 3s (or Paper 2s for AS), plus a final Paper 1. |
Common mistakes that cost marks
Common mistakeRounding in the middle of a calculation, so the final answer is wrong at 3 significant figures.
Common mistakeUsing degrees when the question is in radians, especially in calculus and sector problems.
Common mistakeLeaving out the validity condition on a binomial expansion for rational n.
Common mistakeForgetting the arbitrary constant in a differential equation, or finding it before rearranging.
Common mistakeWriting only the final answer to a “show that” or calculator question.
Ready to practise at exam level?
Mathematics Made Simple covers algebra, functions and calculus for AP, Cambridge and IB in 20 chapters, with worked examples and full worked answers. The practice book adds exam-style questions for every chapter.
Further reading
More free material for this exam: