A Level Mathematics: Pure Mathematics 2 (Cambridge 9709)

This guide covers Pure Mathematics 2, the second content strand of Cambridge International AS & A Level Mathematics 9709, 2026–2027 series. 9709 is modular: six papers across three content strands — Pure Mathematics (Papers 1–3), Mechanics (Paper 4), and Probability & Statistics (Papers 5–6). Paper 1 is compulsory for both the AS Level and the A Level. Pure Mathematics 2 is examined through Paper 2, which is offered only as part of the standalone AS Level award (Paper 1 plus Paper 2); it does not count towards the full A Level, for which candidates take Paper 3 (Pure Mathematics 3) instead, and candidates are not permitted to take both Paper 2 and Paper 3.

Where this fits in 9709

Pure Mathematics 2 builds directly on the algebraic and calculus foundations laid in Pure Mathematics 1 (Paper 1), extending differentiation and integration to exponential, logarithmic and trigonometric functions, and introducing techniques — the modulus function, radian measure in calculus, and numerical methods for equations that cannot be solved exactly. Paper 2 and Paper 3 are alternative routes built on the same Paper 1 foundation, not a sequence: a candidate takes Paper 1 plus Paper 2 for the AS Level, or Paper 1 plus Paper 3 for the A Level, never both Paper 2 and Paper 3. Pure Mathematics 2’s content is often where AS Level students meet, for the first time, functions and equations that genuinely cannot be handled by purely algebraic methods, which is part of why numerical methods appear here rather than in Pure Mathematics 1.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL MATHEMATICS 9709 — PURE MATHEMATICS 2

• 2.1 Algebra — the modulus function |x|, including its graph and the solution of equations and inequalities involving a modulus; division of polynomials, the factor theorem and the remainder theorem, and using these to simplify expressions and solve equations
• 2.2 Logarithmic and exponential functions — the laws of logarithms, the graphs of exponential and logarithmic functions, and solving equations involving indices and logarithms
• 2.3 Trigonometry — further trigonometric identities and equations building on Pure Mathematics 1, including the use of the secant, cosecant and cotangent functions and their identities
• 2.4 Differentiation — differentiating exponential, logarithmic and trigonometric functions, and applying the product, quotient and chain rules to combinations of these; differentiating functions defined parametrically and implicitly
• 2.5 Integration — integrating exponential, logarithmic and trigonometric functions, using integration to find areas and evaluate definite integrals involving these functions, and estimating the value of a definite integral by means of the trapezium rule, including recognising whether the rule gives an over- or under-estimate
• 2.6 Numerical solution of equations — locating roots of equations by sign changes, and using simple iterative methods to find approximate solutions where an equation cannot be solved algebraically

How to approach it

Pure Mathematics 2 rewards fluency with the factor theorem and polynomial division (2.1) before moving on, since later work across the paper assumes this is automatic rather than a fresh challenge each time it appears. Logarithmic and exponential functions (2.2) are best practised as a pair — converting confidently between index form and logarithmic form, and recognising when an equation calls for logarithms to solve an unknown exponent — because exam questions frequently mix the two representations within a single problem rather than testing either in isolation.

For trigonometry (2.3), the secant, cosecant and cotangent functions are often the least familiar part of this topic to students moving on from Pure Mathematics 1, so deliberate practice manipulating identities involving all six trigonometric functions — not just sine, cosine and tangent — closes a common gap early. Differentiation and integration of exponential, logarithmic and trigonometric functions (2.4–2.5) are the parts of this topic most heavily tested in combination: expect questions that require differentiating a product or quotient involving an exponential and a trigonometric function in the same expression, or integrating a function that first needs simplifying using a logarithm law. Building a habit of checking whether an expression can be simplified algebraically before attempting to differentiate or integrate it directly tends to save time and avoid errors.

Numerical solution of equations (2.6) is conceptually different from the rest of the topic — it is about approximating, not solving exactly — so the priority is understanding what a sign change across an interval actually demonstrates (that a root lies within it, not what its value is), and being comfortable setting up and iterating a formula of the form x_(n+1) = g(x_n) to converge on an approximate root. A frequent source of lost marks is presenting an iteration without enough decimal places carried through each step, so carry more figures than the final answer needs until the last stated iteration. Because Pure Mathematics 2 is assessed as part of the standalone AS Level award (not the full A Level), students moving from Paper 1 into Paper 2 should expect a step up in how far a single question can combine techniques from across this topic, rather than testing each sub-topic in isolation as Paper 1 more often does.

A further habit worth building across this whole topic is checking, before starting a calculation, which representation of a function is easiest to work with — logarithmic form often simplifies an equation that looks unmanageable in index form, and a trigonometric identity can turn an integral that looks intractable into one of the standard forms covered in 2.5. Examiners at this level increasingly reward candidates who can recognise which of several valid methods is most efficient for a given question, rather than simply applying the first technique that comes to mind. Past-paper practice specific to Paper 2 is the most reliable way to build this instinct, since the way this syllabus strand combines sub-topics within a single question is not always obvious from the syllabus content list alone.

Official syllabus

Cambridge International AS & A Level Mathematics 9709 syllabus for 2026 and 2027 — cambridgeinternational.org.

Related resources

• Pure Mathematics 1: Quadratics and beyond — the first content strand of 9709, a prerequisite for this topic