Proof by deduction, exhaustion and counter-example

Proof by contradiction

Laws of indices and surds

Quadratic functions and the discriminant

Simultaneous equations

Linear and quadratic inequalities

Polynomials, factor theorem and algebraic division

Rational expressions

Graphs of functions, asymptotes and proportional relationships

The modulus function

Composite and inverse functions

Graph transformations

Partial fractions

Modelling with functions

Equations of straight lines

Circles

Parametric equations

Parametric equations in modelling

Binomial expansion for positive integer powers

Binomial expansion for rational powers

Sequences, recurrence relations and sigma notation

Arithmetic sequences and series

Geometric sequences and series

Sequences and series in modelling

Trigonometric ratios, sine and cosine rules and area

Radian measure, arc length and exact values

Small angle approximations

Trigonometric graphs and identities

Reciprocal and inverse trigonometric functions

Compound and double angle formulae

Harmonic form and trigonometric proof

Solving trigonometric equations

Trigonometry in context

Exponential functions and e^x

Logarithms and their laws

Exponential and logarithmic equations

Logarithmic graphs and modelling growth and decay

Differentiation from first principles

Convex and concave curves and points of inflection

Differentiating powers of x and stationary points

Differentiating standard functions

Product, quotient and chain rules

Implicit and parametric differentiation

Constructing differential equations

Integration as the reverse of differentiation

Integrating standard functions

Definite integrals and area under a curve

Area between curves and integration as the limit of a sum

Integration by substitution

Integration by parts

Integration using partial fractions and reverse chain rule

Separable differential equations

Locating roots and iteration

Newton-Raphson method

Numerical integration with the trapezium rule

Numerical methods in context

Vectors in two dimensions

Vectors in three dimensions

Position vectors and distance

Vectors in problem solving

Population, sample and sampling techniques

Single-variable data: diagrams and histograms

Bivariate data, correlation and regression

Measures of location and spread

Outliers and cleaning data

Mutually exclusive and independent events

Conditional probability and tree and Venn diagrams

Modelling with probability

Discrete probability distributions

The binomial distribution

The normal distribution

Choosing a distribution and approximations

Language of hypothesis testing

Hypothesis test for correlation

Hypothesis tests for a binomial proportion

Hypothesis tests for a Normal mean

Quantities and S.I. units in mechanics

Language of kinematics and motion graphs

Constant acceleration (suvat) equations

Kinematics with vectors in two dimensions

Calculus in kinematics

Projectile motion

Forces and Newton's first law

Newton's second law in a straight line

Weight and motion under gravity

Newton's third law and connected particles

Resolving forces and equilibrium of a particle

Resultant forces and dynamics in a plane

Friction

Moments and equilibrium of rigid bodies