Skip to content

Home
Glossary
Chapter 1: Proof and mathematical argument
- Overview
- 1.1 Statements, assumptions and the structure of a proof
- 1.2 Proof by deduction
- 1.3 Proof by exhaustion
- 1.4 Disproof by counter example
- 1.5 Proof by contradiction
- Review
Chapter 2: Algebra and functions
- Overview
- 2.1 Laws of indices for rational exponents
- 2.2 Surds and rationalising the denominator
- 2.3 Quadratic functions, the discriminant and completing the square
- 2.4 Solving quadratic equations, including in a function of the unknown
- 2.5 Simultaneous equations by elimination and substitution
- 2.6 Linear and quadratic inequalities, and set notation
- 2.7 Polynomials, the factor theorem and algebraic division
- 2.8 Rational expressions and partial fractions
- 2.9 Graphs, the modulus function, asymptotes and intersections
- 2.10 Composite and inverse functions
- 2.11 Transformations of graphs
- 2.12 Functions in modelling: limitations and refinements
- Review
Chapter 3: Coordinate geometry
- Overview
- 3.1 Straight lines, gradients and straight-line models
- 3.2 Parallel and perpendicular lines
- 3.3 The equation of a circle and its properties
- 3.4 Parametric equations and Cartesian conversion
- 3.5 Parametric equations in modelling
- Review
Chapter 4: Sequences and series
- Overview
- 4.1 Binomial expansion for positive integer n; n! and nCr
- 4.2 Binomial expansion for rational n and approximation
- 4.3 Sequences from formulae and recurrences
- 4.4 Sigma notation for sums
- 4.5 Arithmetic sequences and series
- 4.6 Geometric sequences and series, and the sum to infinity
- 4.7 Sequences and series in modelling
- Review
Chapter 5: Trigonometry
- Overview
- 5.1 The sine and cosine rules and the area of a triangle
- 5.2 Radian measure, arc length and sector area
- 5.3 The sine, cosine and tangent functions: graphs and periodicity
- 5.4 Exact trigonometric values
- 5.5 Small-angle approximations
- 5.6 Secant, cosecant, cotangent and the inverse trigonometric functions
- 5.7 Trigonometric identities
- 5.8 Addition and double-angle formulae
- 5.9 The forms r cos(theta +/- alpha) and r sin(theta +/- alpha)
- 5.10 Solving trigonometric equations in an interval
- 5.11 Proofs with trigonometric functions and identities
- 5.12 Trigonometry in context: vectors, kinematics and forces
- Review
Chapter 6: Exponentials and logarithms
- Overview
- 6.1 The functions a^x and e^x
- 6.2 The gradient of e^kx and the exponential model
- 6.3 Logarithms and the laws of logarithms
- 6.4 Solving equations of the form a^x = b
- 6.5 Logarithmic graphs and estimating parameters
- 6.6 Exponential growth and decay in modelling
- Review
Chapter 7: Differentiation
- Overview
- 7.1 The derivative as a gradient and a rate of change
- 7.2 Differentiation from first principles
- 7.3 Differentiating x^n, e^kx, a^kx, sin kx, cos kx, tan kx and ln x
- 7.4 The second derivative, convexity and points of inflection
- 7.5 Tangents, normals, maxima, minima and stationary points
- 7.6 Increasing and decreasing functions
- 7.7 The product, quotient and chain rules
- 7.8 Connected rates of change and inverse functions
- 7.9 Implicit and parametric differentiation
- 7.10 Constructing differential equations
- Review
Chapter 8: Integration
- Overview
- 8.1 The Fundamental Theorem of Calculus and standard integrals
- 8.2 Definite integrals and the area under a curve
- 8.3 The area between two curves
- 8.4 Integration as the limit of a sum
- 8.5 Integration by substitution and by parts
- 8.6 Integrating with partial fractions
- 8.7 Separable first-order differential equations
- 8.8 Interpreting differential-equation solutions in context
- Review
Chapter 9: Numerical methods
- Overview
- 9.1 Locating roots by change of sign
- 9.2 Simple iterative methods, cobweb and staircase diagrams
- 9.3 Newton–Raphson and other recurrence relations
- 9.4 The trapezium rule and limits on the estimate
- 9.5 Numerical methods in context
- Review
Chapter 10: Vectors
- Overview
- 10.1 Vectors in two and three dimensions
- 10.2 Magnitude and direction; component and magnitude/direction form
- 10.3 Vector addition and scalar multiplication
- 10.4 Position vectors and the distance between two points
- 10.5 Vectors in pure mathematics and in context
- Review
Chapter 11: Statistical sampling and data presentation
- Overview
- 11.1 Population, sample and informal inference
- 11.2 Sampling techniques and their critique
- 11.3 Diagrams for single-variable data and the histogram
- 11.4 Scatter diagrams, regression lines and correlation
- 11.5 Measures of central tendency and variation, including standard deviation
- 11.6 Outliers, data cleaning and critiquing presentation
- 11.7 Working with the large data set
- Review
Chapter 12: Probability
- Overview
- 12.1 Mutually exclusive and independent events
- 12.2 Conditional probability and the conditional probability formula
- 12.3 Modelling with probability and critiquing assumptions
- Review
Chapter 13: Statistical distributions
- Overview
- 13.1 Discrete probability distributions and the binomial distribution
- 13.2 Calculating binomial probabilities
- 13.3 The Normal distribution as a model
- 13.4 Links to the binomial, histograms and points of inflection
- 13.5 Selecting an appropriate distribution
- Review
Chapter 14: Statistical hypothesis testing
- Overview
- 14.1 The language of hypothesis testing
- 14.2 Binomial tests for a proportion, one- and two-tailed
- 14.3 Normal tests for a mean and the significance level
- 14.4 Interpreting correlation coefficients with p-values and critical values
- Review
Chapter 15: Quantities, units and kinematics
- Overview
- 15.1 Fundamental and derived SI quantities and units
- 15.2 The language of kinematics
- 15.3 Displacement–time and velocity–time graphs
- 15.4 The constant-acceleration formulae
- 15.5 Calculus in kinematics
- 15.6 Motion under gravity and projectiles using vectors
- Review
Chapter 16: Forces, Newton's laws and moments
- Overview
- 16.1 Force and Newton's first law
- 16.2 Newton's second law and F = ma
- 16.3 Weight and motion under gravity
- 16.4 Newton's third law, equilibrium and connected particles
- 16.5 Resolving forces and coplanar equilibrium
- 16.6 Resultant forces and dynamics in a plane
- 16.7 Friction and the F <= mu R model
- 16.8 Moments in static contexts
- Review
About this book

A-level Mathematics

Welcome. This is a complete, interactive A-level Mathematics book for England. It covers pure mathematics, statistics and mechanics, across both years of the course. Everything here is free to read online, and there is nothing to install.

The content is the common content set by the Department for Education. It therefore follows the AQA 7357, Pearson Edexcel 9MA0 and OCR H240 specifications at once. Where the boards differ, a page says so.

How to use this book

Each section has three pages:

Page	What it is for
Learn	The teaching: ideas, worked examples, and something to try
Practise with help	Questions with hints you reveal only if you need them
Practise	Independent questions, with answers at the end

Each chapter opens with a problem you will be able to solve by the end. It closes with a Review that mixes the chapter's methods together, and returns to earlier chapters.

How the book is organised

The book has three strands.

Pure mathematics covers proof, algebra, coordinate geometry, sequences and series, trigonometry, exponentials and logarithms, differentiation, integration, numerical methods and vectors.
Statistics covers sampling, data presentation, probability, distributions and hypothesis testing.
Mechanics covers quantities and units, kinematics, forces, Newton's laws and moments.

The chapters are numbered as one sequence, so the strands appear in the order you meet them. The early pure chapters are the ones the applied chapters build on.

Checked answers

The questions and the answers in each Practise set come from one set that a computer checks every time the book is built. So the two cannot disagree. The computer checks the algebra, not the argument, so read the reasoning in Learn as well.

Chapters

Learn more

Glossary — the key words, by strand
Pure mathematics · Statistics · Mechanics