WJEC A2 Unit 3 Pure Mathematics B: a complete overview of advanced proof, functions, series, trigonometry, calculus and numerical methods

What Unit 3 actually demands

A2 Unit 3 Pure Mathematics B is the largest single unit of the qualification, worth 35 per cent of the A level. It deepens every strand of the AS pure work: proof becomes proof by contradiction, functions gain the modulus, composites, inverses and partial fractions, calculus gains the chain, product, quotient and implicit rules and three integration techniques, and a new numerical-methods topic handles equations and integrals with no exact answer. The AS pure content is assumed throughout, so the whole pure course is in play.

This guide walks through the eight topics of the unit, then sets out the exam structure. Each topic has a matching dot-point page with worked exam questions; this overview ties them together.

Proof by contradiction

The unit opens with proof by contradiction: assume the negation of the statement, reason to a logical impossibility, and conclude the statement is true. The set-piece examples are the irrationality of $\sqrt{2}$ and the infinitude of the primes, both of which you should be able to reproduce.

Functions: modulus, composite, inverse and partial fractions

This topic covers the modulus function $|x|$ (graphs, equations and inequalities), composite functions $fg(x)$ and inverse functions $f^{-1}(x)$ (domains, ranges, and the reflection in $y = x$), and resolving rational expressions into partial fractions, a skill reused in integration.

Sequences and series

You work with arithmetic and geometric series, sigma notation, and the sum to infinity $\dfrac{a}{1 - r}$ of a convergent geometric series ($|r| < 1$). The big extension is the binomial expansion for any rational index, an infinite series valid for $|x| < 1$.

Trigonometry

Advanced trigonometry covers radians (with arc length and sector area), the reciprocal functions $\sec$, $\csc$ and $\cot$ and their identities, the compound and double-angle formulae, and the harmonic form $R\sin(\theta + \alpha)$ for $a\sin\theta + b\cos\theta$.

Parametric equations

You describe curves with parametric equations, convert between parametric and Cartesian form by eliminating the parameter, and use parametric differentiation $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$ to find gradients and tangents.

Differentiation

Calculus extends to the chain, product and quotient rules, implicit differentiation, and the derivatives of the standard functions ($\mathrm{e}^x$, $\ln x$, $\sin x$, $\cos x$, $\tan x$), with the second derivative for concavity and connected rates of change.

Integration

Integration covers the standard functions and the three core techniques: substitution (reversing the chain rule), by parts (reversing the product rule), and partial fractions (each giving a logarithm). Definite integrals give areas.

Numerical methods

The unit closes with numerical methods: locating roots by change of sign, iterative formulae $x_{n+1} = g(x_n)$, the fast-converging Newton-Raphson method, and the trapezium rule for estimating definite integrals.

How Unit 3 is examined

WJEC Unit 3 Pure Mathematics B is a written paper of 2 hours 30 minutes carrying 120 marks, worth 35 per cent of the full A level. It uses structured and unstructured questions that integrate several topics, and a calculator is allowed. Because the AS pure content is assumed, revise the whole pure syllabus, not just the A2 additions, and practise multi-topic questions.

The eight topics, dot point by dot point

Each topic has a dot-point answer page with worked exam questions and cross-links. Browse them from this unit overview and the subject hub.

For the official specification

WJEC publishes the full specification, past papers and mark schemes at wjec.co.uk. Always revise from the current specification and WJEC's own past papers, because question style is board-specific.