Proof by mathematical induction

Complex numbers and quadratic, cubic and quartic equations

Argand diagrams and modulus-argument form

Loci in the Argand diagram

De Moivre's theorem and exponential form

Roots of complex numbers and roots of unity

Matrix algebra and linear transformations

Invariant points and lines

Determinants and inverse matrices

Simultaneous equations and planes

Roots of polynomials

Sums of series

Method of differences

Maclaurin series

Volumes of revolution

Improper integrals and the mean value of a function

Integration using partial fractions

Inverse trigonometric functions: differentiation and integration

Vector and Cartesian equations of a line

Planes and the scalar product

Intersections and distances in three dimensions

Polar coordinates and polar curves

Area enclosed by a polar curve

Hyperbolic functions and their graphs

Calculus of hyperbolic functions

Inverse hyperbolic functions

First-order differential equations and integrating factors

Second-order linear differential equations

Simple harmonic motion and damped oscillations

Coupled first-order differential equations

The t-formulae

Taylor series and limits

Leibnitz's theorem and L'Hospital's rule

The Weierstrass substitution

Series solutions and reducible differential equations

Parabola and rectangular hyperbola

Ellipse and hyperbola

Vector product and scalar triple product

Applications of vectors to 3-D geometry

Numerical solution of differential equations

Simpson's rule

Algebraic inequalities

Group axioms and examples

Order, subgroups and Lagrange's theorem

Isomorphism

Reduction formulae

Arc length and area of a surface of revolution

Eigenvalues and eigenvectors

Diagonalisation and the Cayley-Hamilton theorem

Further loci and regions in the Argand diagram

Transformations from the z-plane to the w-plane

Division theorem, Euclidean algorithm and Bezout's identity

Modular arithmetic and divisibility tests

Fermat's little theorem and congruence equations

Combinatorics

First-order recurrence relations

Second-order recurrence relations

Mean and variance of discrete distributions

The Poisson distribution

Mean, variance and Poisson approximation to the binomial

The geometric distribution

The negative binomial distribution

Hypothesis test for the mean of a Poisson distribution

Hypothesis test for the parameter of a geometric distribution

The Central Limit Theorem

Goodness of fit tests and contingency tables

Probability generating functions

Type I and Type II errors and power

Least squares regression and residuals

Probability density and cumulative distribution functions

Mean and variance of continuous random variables

The continuous uniform distribution

Product moment and Spearman's rank correlation

Testing for zero correlation

Linear combinations of Normal random variables

Estimators, standard error and bias

Confidence intervals for a Normal mean

Tests for the difference between two Normal means

Variance of a Normal distribution and the F-test

t tests for a mean and the paired t-test

Difference between two means: pooled t-test

Momentum and impulse in one dimension

Momentum and impulse as vectors

Work, energy and power

Hooke's law

Elastic energy and the work-energy principle

Direct impact and Newton's law of restitution

Successive direct impacts

Oblique impact

Successive oblique impacts

Horizontal circular motion

Vertical circular motion

Moments and centres of mass of discrete masses

Centres of mass of plane figures and equilibrium of a lamina

Centres of mass of rigid bodies

Equilibrium, toppling and sliding

Variable forces and motion in one dimension

Simple harmonic motion

Variable acceleration in one dimension

Algorithms, bin packing and sorting

Graph types and Eulerian graphs

The planarity algorithm and Hamiltonian cycles

Minimum spanning trees

Dijkstra's algorithm

Floyd's algorithm

The Route Inspection (Chinese Postman) problem

The Travelling Salesman problem

Activity networks and precedence tables

Critical paths and float

Resource histograms and scheduling

Formulating linear programs

Graphical solution of linear programs

The Simplex algorithm

Two-stage Simplex and big-M methods

Transportation problems

The Hungarian algorithm

Hungarian algorithm as a linear programme

Cuts, labelling procedure and max-flow min-cut

Multiple sources and sinks, restricted capacities and optimal flow

Dynamic programming

Zero-sum games, play-safe strategies and saddle points

Mixed strategies: the graphical method

Dominance and mixed strategies by the Simplex algorithm

First-order recurrence relations

Second-order recurrence relations

Decision trees and expected monetary value