AQA A-Level Mathematics Pure mathematics: a complete overview of proof, algebra, calculus, trigonometry and vectors

What the Pure mathematics content demands

Pure mathematics is the backbone of AQA A-Level Mathematics. It develops the algebra, calculus, trigonometry and reasoning that every other part of the course depends on. The examiners test two linked skills: fluent technique with standard methods, and the judgement to choose and combine those methods in unfamiliar multi-step problems.

This guide walks through all ten pure topics in specification order, then sets out the exam patterns AQA repeats. Each topic has a matching dot-point page with practice questions; this overview ties them together.

Proof and algebra

The content opens with proof: deduction, exhaustion, disproof by counter-example and contradiction, including the classic results that $\sqrt{2}$ is irrational and that there are infinitely many primes. Clear logical layout earns marks across the whole paper.

Algebra and functions is the most reused topic. You manipulate surds and indices, solve quadratics by factorising, the formula and completing the square, work with the discriminant, handle simultaneous equations and inequalities, sketch curves, transform graphs, and expand using the binomial theorem. Weak algebra leaks marks everywhere, so it is the first thing to master.

Coordinate geometry and sequences

Coordinate geometry covers straight lines, gradients, the equation of a circle, and the relationship between tangents, chords and radii. Sequences and series introduces arithmetic and geometric sequences, sigma notation, the sum formulae, the condition for a convergent geometric series, and the binomial expansion for any rational power.

Trigonometry, exponentials and logarithms

Trigonometry works in radians, with the arc length and sector area formulae, exact values, the Pythagorean and addition identities, the reciprocal and inverse functions, and solving equations over a given interval. Exponentials and logarithms introduces the number $e$, the natural logarithm, the laws of logarithms, solving exponential equations, and using logarithms to linearise data and model growth and decay.

Calculus

Differentiation runs from first principles to the chain, product and quotient rules, derivatives of standard functions, stationary points and their nature, and optimisation. Integration reverses this with indefinite and definite integrals, areas, standard integrals, substitution, integration by parts and partial fractions. Calculus carries the most marks in the pure content.

Numerical methods and vectors

Numerical methods locate roots by sign change, use iteration and the Newton-Raphson method, and apply the trapezium rule, with attention to when methods fail. Vectors cover two and three dimensions, magnitude and direction, components, position vectors and geometric applications.

How the Pure content is examined

A typical AQA profile for Pure mathematics:

• Short technique questions. Differentiating and integrating standard functions, solving quadratics and trigonometric equations, and manipulating logarithms.
• Multi-step problems. Combining calculus with coordinate geometry to find tangents and areas, or trigonometry with algebra to solve equations.
• Proof and reasoning. Constructing deductive and contradiction proofs and disproving statements by counter-example.
• Applied calculus. Optimisation, connected rates of change, and using numerical methods when exact answers are not available.

Check your knowledge

A mix of recall and technique questions covering the Pure content. Attempt them under timed conditions, then check against the solutions.

1. Differentiate $y = 3x^4 - 2x$. (2 marks)
2. Find $\int (6x^2 + 4)\,dx$. (2 marks)
3. Solve $\sin x = \frac{1}{2}$ for $0 \le x \le 2\pi$. (3 marks)
4. Solve $5^x = 30$, giving your answer to three significant figures. (2 marks)
5. Find the magnitude of the vector $3\mathbf{i} - 4\mathbf{j}$. (2 marks)
6. Use the trapezium rule with two strips to estimate $\int_0^2 x^2\,dx$. (3 marks)