Induction for series and sequences

Induction for divisibility and matrix powers

Complex number arithmetic

Complex conjugates and polynomial roots

Argand diagrams and modulus-argument form

Loci in the Argand diagram

De Moivre's theorem

Exponential form of a complex number

Roots of complex numbers and roots of unity

Matrix arithmetic

Linear transformations

Invariant points and lines

2x2 determinants and inverse matrices

3x3 determinants and inverse matrices

Solving simultaneous equations with matrices

Factorising determinants

Eigenvalues and eigenvectors

Diagonalisation of matrices

Roots and coefficients of polynomials

Summation of series

Method of differences with partial fractions

Maclaurin series of a function

Standard Maclaurin series

Limits using Maclaurin series and L'Hopital's rule

Inequalities with polynomials and rational expressions

Modulus of functions and reciprocal graphs

Rational function graphs

Oblique asymptotes of rational functions

Conic sections

Single transformations of curves

Composite transformations of curves

Improper integrals

Volumes of revolution and mean value

Inverse trigonometric functions: differentiation and integration

Partial fractions and reduction formulae

Arc length and surface area of revolution

Vector and Cartesian equations of lines

Vector and Cartesian equations of planes

Scalar product and perpendicular vectors

Angles between planes and between lines and planes

Vector product

Intersection of lines and distances

Intersections and distances involving planes

Polar and Cartesian coordinates

Sketching polar curves

Area enclosed by a polar curve

Hyperbolic functions and their graphs

Domains, ranges and reciprocal hyperbolic functions

Calculus with hyperbolic functions

Inverse hyperbolic functions

Basic hyperbolic identities

Further hyperbolic identities and proofs

First order equations and integrating factors

Second order equations with constant coefficients

Particular integrals

Simple harmonic motion

Damped oscillations

Coupled first order equations

Mid-ordinate and Simpson's rules

Euler's methods for differential equations

Dimensions and consistency

Conservation of momentum

Momentum, impulse and collisions requiring resolving

Coefficient of restitution and collisions

Impulse

Work done and energy

Hooke's law and elastic energy

Variable forces and power

Work and power with forces at an angle

Angular speed and circular motion

Vectors in circular motion

Conical pendulums

Circular motion in a vertical plane

Centre of mass of particles and composite bodies

Centre of mass by integration

Equilibrium, sliding and toppling

Distributions and expectation of DRVs

Linear functions of a DRV

Functions of a DRV

Discrete uniform distribution

Poisson model and probabilities

Sums of Poisson variables and Poisson hypothesis tests

Type I and Type II errors

Type I errors for normal distribution tests

Type II errors and power

Probability density functions

Part-discrete part-continuous distributions

Mean and variance of a CRV

Cumulative distribution functions

Rectangular distribution

Sums of independent random variables

Contingency tables and the chi-squared test

Yates' correction

Exponential distribution model, pdf and CDF

Mean and variance, and link to Poisson processes

One-sample t-test

Confidence intervals for a mean

Confidence intervals from small samples

Language and properties of graphs

Euler's formula for planar graphs

Kuratowski's theorem

Special graphs

Isomorphism of graphs

Spanning trees and network optimisation

Route inspection and travelling salesperson

Flows, cuts and the max-flow min-cut theorem

Augmenting flows and refinements

Formulating and graphical solution of LPs

The simplex algorithm

Activity networks and critical paths

Gantt charts, resource histograms and levelling

Pay-off matrices and play-safe strategies

Dominance and mixed strategies

Converting games to linear programming problems

Binary operations and their properties

Identity and inverses

Groups

Lagrange's theorem and generators

Isomorphism of groups