TMUA topics A–Z

Polynomials, factorising, surds, inequalities and function notation.

Arithmetic and geometric progressions, sigma notation and binomial basics.

Straight lines, circles, parabolas and curve sketching on paper.

Identities, exact values, graphs and equations without a calculator.

Laws of indices and logs, exponential modelling and solving.

Gradients, stationary points, tangents and rates of change.

Definite and indefinite integrals, areas under curves, kinematics.

Transforming graphs, symmetry, asymptotes and interpreting plots.

Fractions, percentages, bounds, estimation and mental arithmetic.

Angles, area and volume, Pythagoras, vectors and constructions.

Averages, distributions, Venn diagrams, conditional probability.

Statements and connectives: conclusions, premises and argument structure in short passages.

Necessary vs sufficient conditions and if-and-only-if statements.

Quantifiers — for all / there exists — and what follows strictly from the premises.

Negation of statements, plus everyday quantitative reasoning with rates and proportions.

What counts as a proof: definitions, direct argument, examples vs proof.

Proof by cases: splitting into exhaustive cases and proving each, step by step.

Proof by contradiction and contrapositive: assuming the negation and deriving absurdity.

Disproof by counterexample: refuting a universal claim with a single counterexample.

Following proofs: what was proved, which structure was used, and where it breaks.

Finding the first invalid step: division by zero, converse errors, scope slips.

Common fallacies: generalising from one case, confusing correlation with cause, missing warrants.