AQA A-Level Maths: Pure Mathematics Revision Guide

Pure Mathematics forms the backbone of AQA A-Level Maths. It accounts for two-thirds of your total marks, spanning Papers 1 and 2, and underpins everything you do in Statistics and Mechanics as well. If your pure maths is strong, the rest of the course becomes far more manageable. If it is weak, you will feel the effects across all three papers.

This guide covers every major pure maths topic in the AQA specification, explains what examiners expect, and gives you a clear revision strategy for Papers 1 and 2. Whether you are starting your revision early or looking for a focused plan in the final weeks, this is your roadmap.

Overview of the AQA A-Level Maths Papers

AQA A-Level Maths is assessed through three written papers, each lasting 2 hours and worth 100 marks:

• Paper 1 -- Pure Mathematics (100 marks, 2 hours)
• Paper 2 -- Pure Mathematics (100 marks, 2 hours)
• Paper 3 -- Statistics and Mechanics (100 marks, 2 hours)

Pure content is assessed entirely on Papers 1 and 2, which together account for 200 marks out of 300 -- two-thirds of the entire qualification. Paper 3 covers Statistics and Mechanics only. Any topic from the pure specification can appear on either Paper 1 or Paper 2, so you need to be confident across the full range of material.

There is no choice of questions and no coursework. Every mark comes from the exam, which means fluency, accuracy, and exam technique are everything.

Algebra and Functions

Algebra is the language of A-Level Maths. It runs through every other topic, so weaknesses here will follow you everywhere.

Core Skills

You need to be completely fluent with surds, indices, and the laws of logarithms. These are not just standalone topics -- they appear inside calculus problems, trigonometric equations, and sequences questions. The ability to simplify expressions confidently and manipulate equations accurately is non-negotiable at this level.

Quadratics remain important. You must be able to complete the square, use the discriminant to determine the nature of roots, and solve quadratic equations efficiently. These skills extend into coordinate geometry (finding intersections with circles) and calculus (optimisation problems).

Polynomial division and the factor theorem are used to factorise cubics and higher-order polynomials. You should be confident dividing polynomials by linear and quadratic factors, and using the factor theorem to test for roots systematically.

Partial fractions are essential for integration later in the course. You need to decompose rational expressions into simpler fractions, including cases with repeated factors and improper fractions.

Functions

Understand the modulus function and be able to solve equations and inequalities involving |f(x)|. Know how to sketch modulus graphs and find intersections.

Composite and inverse functions come up frequently. You should be able to find fg(x), state the domain and range of composite functions, and determine whether an inverse exists. The relationship between a function and its inverse -- both algebraically and graphically (reflection in y = x) -- is a common exam topic.

Transformations of graphs -- translations, stretches, and reflections -- must be automatic. Know the effect of y = f(x + a), y = f(x) + a, y = af(x), y = f(ax), y = -f(x), and y = f(-x). Be able to apply these to unfamiliar functions and state the new coordinates of given points.

For focused practice on algebra and functions, try LearningBro's Pure Maths 1 course and the Advanced Algebra course.

Coordinate Geometry

Coordinate geometry connects algebra to visual representations and appears in a wide range of exam questions.

Straight Lines

You need fluency with the equation of a line in all its forms: y = mx + c, y - y1 = m(x - x1), and ax + by + c = 0. Be confident finding gradients, midpoints, and distances. Know the conditions for parallel lines (equal gradients) and perpendicular lines (product of gradients equals -1).

Circles

The equation of a circle with centre (a, b) and radius r is (x - a)^2 + (y - b)^2 = r^2. You must be able to find the equation of a circle from given information, determine whether a point lies inside, on, or outside a circle, and find the equation of a tangent at a given point. Remember that the tangent to a circle is perpendicular to the radius at the point of contact -- this is a frequently tested relationship.

Parametric Equations

Parametric equations define x and y separately in terms of a parameter, usually t. You need to be able to convert between parametric and Cartesian forms, find the gradient of a parametric curve using dy/dx = (dy/dt) / (dx/dt), and find the equation of a tangent or normal at a given point.

Parametric questions often combine with calculus -- for example, finding the area under a parametric curve using integration, or locating stationary points.

For structured revision on these topics, see the Coordinate Geometry in Depth course.

Trigonometry

Trigonometry at A-Level goes well beyond the basic ratios you learned at GCSE. It becomes a powerful tool for solving equations, proving identities, and modelling periodic behaviour.

Foundations

You must know the exact values of sin, cos, and tan for 0, 30, 45, 60, and 90 degrees (and their radian equivalents: 0, pi/6, pi/4, pi/3, and pi/2). These come up constantly and you cannot afford to derive them every time.

The fundamental identity sin^2(theta) + cos^2(theta) = 1 and the relationship tan(theta) = sin(theta)/cos(theta) are starting points for many proofs and equation-solving problems. You should also know the derived identities: 1 + tan^2(theta) = sec^2(theta) and 1 + cot^2(theta) = cosec^2(theta).

Addition and Double Angle Formulae

The addition formulae -- sin(A +/- B), cos(A +/- B), and tan(A +/- B) -- are provided in the formula booklet. However, you must know how to use them fluently, and you should be able to derive the double angle formulae from them. The double angle formulae for cos(2A) have three forms, and knowing which one to use in a given context is a skill that comes from practice.

The Rsin(theta + alpha) Form

This technique rewrites a linear combination of sin and cos as a single trigonometric function. It is essential for solving equations of the form a sin(theta) + b cos(theta) = c and for finding maximum and minimum values of trigonometric expressions. The method is systematic: expand Rsin(theta + alpha) using the addition formula, compare coefficients, and solve for R and alpha.

Solving Trigonometric Equations

You need a reliable method for finding all solutions in a given interval. Use the CAST diagram or the unit circle to identify which quadrants give valid solutions, then apply transformations to account for the argument of the function. Always check that your solutions are within the specified range.

Small Angle Approximations

For small values of x (in radians): sin(x) is approximately x, cos(x) is approximately 1 - x^2/2, and tan(x) is approximately x. These approximations are used to simplify expressions and find limits, and they appear in both pure and applied contexts.

Practise trigonometry thoroughly with the Trigonometry in Depth course.

Calculus

Calculus is the single largest topic in A-Level Pure Maths. It splits into differentiation and integration, and both are tested extensively across Papers 1 and 2.

Differentiation

Differentiation from first principles is the formal definition: the limit of (f(x + h) - f(x)) / h as h tends to 0. You must be able to use this to differentiate simple functions (x^2, x^3, sin(x)) and explain the process. Exam questions on first principles are typically worth several marks and require careful algebraic manipulation.

Rules of differentiation are the tools you use day to day:

• The power rule for x^n (including fractional and negative powers).
• The chain rule for differentiating composite functions: dy/dx = dy/du times du/dx. This is probably the most frequently used rule in the entire course.
• The product rule for differentiating the product of two functions: d/dx(uv) = u(dv/dx) + v(du/dx).
• The quotient rule for differentiating a fraction: d/dx(u/v) = (v(du/dx) - u(dv/dx)) / v^2.

You must be able to differentiate trigonometric, exponential, and logarithmic functions. The standard results -- d/dx(sin x) = cos x, d/dx(cos x) = -sin x, d/dx(e^x) = e^x, d/dx(ln x) = 1/x -- should be memorised. Combined with the chain rule, these allow you to differentiate a wide range of functions.

Applications of differentiation include finding stationary points, determining the nature of turning points (using the second derivative or sign changes), finding equations of tangents and normals, and solving optimisation problems.

Integration

Integration is the reverse of differentiation, but it is also much more. At A-Level you need to master several techniques:

• Basic integration -- reversing the power rule, integrating standard functions (sin, cos, e^x, 1/x).
• Definite integrals -- evaluating integrals between limits to find exact values. Remember that a definite integral gives a number, not a function.
• Finding areas -- using integration to calculate the area between a curve and the x-axis, or between two curves. Be careful with regions below the x-axis, where the integral gives a negative value and you need to take the absolute value.
• Integration by substitution -- a technique for simplifying integrals by changing the variable. You must be able to choose an appropriate substitution, change the limits (for definite integrals), and convert the entire integral into the new variable.
• Integration by parts -- used when the integrand is a product of two functions. The formula is: integral of u(dv/dx) dx = uv - integral of v(du/dx) dx. Know when to apply it (typically when one factor becomes simpler on differentiation) and be prepared to use it more than once for some integrals.

Differential equations connect calculus to real-world modelling. You need to be able to solve first-order differential equations by separating variables, find particular solutions using initial conditions, and interpret the results in context.

Explore these techniques with the Calculus Applications course.

Sequences and Series

Arithmetic Sequences and Series

An arithmetic sequence has a constant common difference, d. The nth term is a + (n - 1)d, and the sum of the first n terms is S_n = n/2(2a + (n - 1)d) or equivalently n/2(a + l), where l is the last term. You must be able to find any of these quantities given the others, and solve problems where two or more pieces of information are given simultaneously (often leading to simultaneous equations).

Geometric Sequences and Series

A geometric sequence has a constant common ratio, r. The nth term is ar^(n - 1), and the sum of n terms is S_n = a(1 - r^n) / (1 - r). When |r| < 1, the sum to infinity exists and equals a / (1 - r). Questions on geometric series frequently involve setting up and solving equations, proving convergence, and finding the number of terms required for a sum to exceed a given value.

The Binomial Expansion

You need to know two forms of the binomial expansion:

• The expansion of (a + b)^n for positive integer n, using binomial coefficients (nCr). This is the version you learned at GCSE or early in the course.
• The general binomial theorem for (1 + x)^n where n is rational (not a positive integer). This produces an infinite series valid for |x| < 1, and is provided in the formula booklet. You must be able to apply it to expressions of the form (1 + bx)^n and (a + bx)^n, including finding the range of validity.

Proof

Proof is a thread that runs through the entire specification. It is not just a standalone topic -- it is a way of thinking that examiners test in many different contexts.

Types of Proof

Proof by deduction is the most common type. Start from known facts or given information and use logical steps to reach the required conclusion. Every step must follow from the previous one, and you must not assume the result you are trying to prove.

Proof by exhaustion involves checking every possible case. This is only practical when the number of cases is small (for example, proving a result for all single-digit prime numbers).

Proof by contradiction starts by assuming the opposite of what you want to prove, then showing that this assumption leads to a logical impossibility. Classic examples include proving that the square root of 2 is irrational and proving that there are infinitely many primes.

Disproof by counterexample is used to show that a statement is false. You only need one example where the statement fails. This is the simplest type of proof but requires you to think carefully about what might go wrong with a general claim.

Writing Proofs Clearly

Structure matters. Start by stating what you are assuming or given. Label each step. Use "therefore" and "hence" to connect your reasoning. End with a clear concluding statement. Marks are often lost not because the mathematics is wrong but because the argument is unclear or steps have been skipped.

For more practice with proof and structured problem-solving, try the Problem Solving course.

Vectors

2D and 3D Vectors

You need to work confidently with vectors in two and three dimensions. This includes writing vectors in column form and in terms of unit vectors (i, j, k), adding and subtracting vectors, and multiplying by scalars.

Key Skills

Magnitude of a vector is found using Pythagoras' theorem extended to three dimensions: |v| = sqrt(x^2 + y^2 + z^2).

Position vectors describe the location of a point relative to the origin. The vector from point A to point B is given by OB - OA, where O is the origin. This is a fundamental skill for geometric problems.

Using vectors to solve geometric problems is a common exam question type. You might be asked to prove that lines are parallel (one vector is a scalar multiple of the other), find the point that divides a line segment in a given ratio, or show that three points are collinear.

Exam Technique for Papers 1 and 2

Strong mathematical knowledge is not enough on its own -- you need to deliver it effectively under exam conditions.

Timing

Papers 1 and 2 are each 2 hours long and worth 100 marks. That gives you roughly 1.2 minutes per mark. A 5-mark question should take about 6 minutes. If you are spending significantly longer than this on any question, move on and come back to it later. Practise under timed conditions so that pacing becomes instinctive.

Show All Working

Examiners award marks for method, not just the final answer. Even if your answer is wrong, clear working can earn you most of the marks. Write down the formula you are using, show your substitution, and present your algebra step by step. Never skip straight to the answer.

"Show That" Questions

These are a particular type of question where you are given the result and must prove it is correct. The key rule is: start from the given information and work towards the result. Do not start from the result and work backwards. Do not skip steps -- every line of working must follow logically from the previous one. If the answer is given as a specific fraction or exact value, you must arrive at exactly that form.

Use the Formula Booklet Wisely

The AQA formula booklet is provided in the exam. Know what is in it and what is not. The compound angle formulas, the general binomial expansion, standard derivatives and integrals, and the trapezium rule are all provided. The quadratic formula, laws of logarithms, basic differentiation results, and integration by parts formula are not. See our Formula Booklet Guide for a detailed breakdown of what to memorise and what to look up.

Common Pitfalls

• Forgetting the constant of integration (+c) for indefinite integrals.
• Sign errors in the chain rule, especially with cos and sin.
• Losing solutions when dividing both sides of an equation by a variable (you might be dividing by zero).
• Not checking the range of validity for binomial expansions.
• Giving decimal answers when exact values (surds, fractions, multiples of pi) are required.

Prepare with LearningBro

LearningBro offers a suite of A-Level Maths courses designed to build the fluency and understanding you need for every pure maths topic:

• Pure Maths 1 -- covers the core algebra, functions, coordinate geometry, and introductory calculus content.
• Pure Maths 2 -- covers the more advanced pure topics including further calculus, sequences, and proof.
• Advanced Algebra -- focused practice on algebraic manipulation, partial fractions, and functions.
• Trigonometry in Depth -- from exact values to Rsin(theta + alpha) and solving equations.
• Calculus Applications -- differentiation, integration, and differential equations with worked examples.
• Coordinate Geometry in Depth -- straight lines, circles, and parametric equations.
• Problem Solving -- proof, mathematical reasoning, and multi-step problem strategies.
• AQA Exam Preparation -- full exam-style practice with timed assessments and detailed mark scheme feedback.

Each course includes lessons, practice questions, and built-in flashcards with spaced repetition to help you commit key results to long-term memory. Try a free lesson preview to see how it works.

Final Thoughts

Pure Mathematics is not a topic you can cram. It demands consistent practice, deep understanding, and the ability to connect ideas across different areas of the specification. The students who do best are the ones who work steadily through the year, practise problems daily, and use past papers to refine their exam technique.

Start with the areas where you are least confident, build your fluency with regular practice, and test yourself under timed conditions as the exams approach. Two-thirds of your A-Level Maths grade depends on pure maths -- give it the time and attention it deserves.

Good luck with your revision. You have got this.