WJEC AS Unit 1 Pure Mathematics A: a complete overview of proof, algebra, coordinate geometry, trigonometry, calculus and vectors

What Unit 1 actually demands

AS Unit 1 Pure Mathematics A is the foundation of the whole WJEC A-level. It builds the core pure toolkit that every later unit relies on: rigorous proof, fluent algebra, coordinate geometry, trigonometry, the start of calculus, and vectors. The paper rewards accuracy and clear method, and many questions deliberately blend topics, so the techniques must be automatic rather than memorised in isolation.

This guide walks through the nine 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

The unit opens with mathematical proof: proof by deduction (reasoning from general representations such as $2n+1$ for an odd number), proof by exhaustion (checking a finite, complete set of cases), and disproof by counterexample (a single value that breaks a "for all" claim). The same logical discipline is expected wherever you justify a result later in the paper.

Algebra and functions

This is the workhorse topic: surds and the laws of indices, quadratics (the formula, completing the square and the discriminant), simultaneous equations and inequalities, polynomial division and the factor theorem, and graph transformations ($f(x)+a$, $f(x+a)$, $af(x)$, $f(ax)$). Accuracy here protects marks across every other topic.

Coordinate geometry

Coordinate geometry covers straight lines (gradient, the forms of the equation, parallel and perpendicular conditions $m_1 m_2 = -1$) and circles (the equation $(x-a)^2+(y-b)^2=r^2$, finding the centre and radius by completing the square, and the radius-tangent property). Intersections come from solving equations simultaneously.

Sequences and series

At AS, this topic is the binomial expansion of $(a+b)^n$ for a positive integer $n$, using binomial coefficients $\binom{n}{r}$ and Pascal's triangle. You expand brackets and pick out a specified term or coefficient with the general term $\binom{n}{r}a^{n-r}b^r$. The expansion for fractional or negative powers belongs to A2 Unit 3.

Trigonometry

Trigonometry covers the graphs of $\sin$, $\cos$ and $\tan$, the identities $\sin^2\theta + \cos^2\theta = 1$ and $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, solving equations in a given interval using the quadrants, and solving non-right-angled triangles with the sine rule, cosine rule and the area formula $\tfrac{1}{2}ab\sin C$.

Exponentials and logarithms

This topic introduces the exponential function including $\mathrm{e}^x$, the laws of logarithms, solving equations with the unknown in the exponent (by taking logs), and modelling exponential growth and decay with a log-linear graph whose gradient and intercept give the constants.

Differentiation

Calculus begins with differentiation: the meaning of the derivative as a gradient, differentiation from first principles, the power rule $\dfrac{d}{dx}(x^n) = nx^{n-1}$, tangents and normals, increasing and decreasing functions, and finding and classifying stationary points with the second derivative.

Integration

Integration is treated as the reverse of differentiation: indefinite integrals with the constant of integration, finding a curve from its gradient function and a point, definite integrals evaluated between limits, and the area under a curve when it lies above the $x$-axis.

Vectors

The unit closes with two-dimensional vectors: representing them in $\mathbf{i}$, $\mathbf{j}$ form, finding magnitude and direction, adding and scaling them, working with position vectors ($\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$), and dividing a segment in a given ratio. This sets up the forces and velocities of the mechanics unit and the three-dimensional vectors of A2.

How Unit 1 is examined

WJEC Unit 1 Pure Mathematics A is a written paper of 2 hours 30 minutes carrying 120 marks, worth 25 per cent of the full A level. It uses a mixture of structured and unstructured questions, a calculator is allowed, and questions frequently integrate several pure topics in one. Revise topic by topic, but practise mixed questions so the connections (for example the discriminant inside a coordinate-geometry problem) are second nature.

The nine 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.