Maths, topic by topic

Deduction, exhaustion, counter-examples and proof by contradiction.

Polynomials, the factor theorem, composite/inverse functions and the modulus.

Straight lines, circles, tangents and intersection conditions.

Binomial expansion, arithmetic and geometric sequences, sums to infinity.

Radians, sectors, identities, R cos(θ + α) and trigonometric modelling.

Log laws, solving equations, e and ln, and log-linear modelling.

First principles, product/quotient/chain rules, tangents and rates of change.

Indices before you integrate, substitution, areas and the trapezium rule.

Sampling methods, the large data set, histograms and outliers.

Venn/tree problems, binomial and normal distributions.

Binomial and normal tests, correlation, and writing conclusions that score.

SUVAT, velocity–time graphs, variable acceleration and projectiles.

F = ma, connected particles, friction and inclined planes.

Taking moments, equilibrium of rods and tilting problems.

Complex arithmetic, loci, de Moivre's theorem, roots and polynomial relationships.

Matrix algebra, transformations in two and three dimensions, invariants and systems.

Improper integrals, polar curves, hyperbolic functions and advanced calculus methods.

Induction, series, three-dimensional vectors and differential-equation modelling.

Momentum, work and energy, elastic strings, and direct and oblique collisions.

Discrete distributions, hypothesis tests, chi-squared tests and probability generating functions.

Quadratic inequalities, simultaneous equations and rearranging formulae.

Reverse percentages, direct/inverse proportion and ratio problems.

Circle theorems, 3D trigonometry and angles with reasons.

Fractional/negative indices, surds and rationalising the denominator.

Tree diagrams, conditional probability and combined events with and without replacement.

Histograms, cumulative frequency, box plots, estimated mean and scatter graphs.