TMUA topics
A plain-English summary of the official content specification (codes in brackets match UAT-UK's document).
Paper 1 and Paper 2: AS-level pure maths
- Algebra and functions (MM1)
- Indices including rational exponents, surds, quadratics and the discriminant, simultaneous equations, inequalities, polynomials and the Factor and Remainder Theorems, one-to-one functions, square root and modulus.
- Sequences and series (MM2)
- nth-term and recurrence sequences, arithmetic series, finite and infinite geometric series, the binomial expansion for positive integer powers.
- Coordinate geometry (MM3)
- Straight lines, circles in centre-radius and general form, circle theorems.
- Trigonometry (MM4)
- Sine and cosine rules, radians, arc length and sector area, exact values, graphs, the two identities, solving equations in an interval.
- Exponentials and logarithms (MM5)
- Graphs of a to the power x, laws of logarithms, solving equations of the form a to the power x equals b. Change of base is not set.
- Differentiation (MM6)
- Gradient and rate of change, differentiating powers of x, tangents and normals, stationary points, increasing and decreasing functions. First principles is not set.
- Integration (MM7)
- Definite integrals and areas, integrating powers of x, the Fundamental Theorem of Calculus, the trapezium rule, solving dy/dx = f(x).
- Graphs of functions (MM8)
- Sketching standard graphs, transformations, composite functions, the number of real roots, graphical meaning of solutions.
Paper 1 and Paper 2: GCSE Higher content
- Units, number, ratio and proportion (M1 to M3)
- Compound units, primes and HCF/LCM, counting, standard form, recurring decimals, exact calculation with surds and pi, bounds, percentages, direct and inverse proportion, similarity, compound growth.
- Algebra (M4)
- Factorising, rational expressions, rearranging formulae, identities, lines, kinematic graphs, simultaneous equations, quadratic sequences.
- Geometry (M5)
- Angle facts, congruence, Pythagoras in 2D and 3D, circle theorems, bearings, areas and volumes, arc length, trigonometry, vector proofs.
- Statistics and probability (M6 and M7)
- Histograms, cumulative frequency, quartiles, scatter graphs, frequency trees, tree and Venn diagrams, conditional probability.
Paper 2 only: mathematical reasoning
- Logic and conditions (Arg1 to Arg4)
- True and false, and, or, not, if-then and equivalent phrasings, converse and contrapositive, necessary and sufficient conditions, for all and there exists, negating statements.
- Proof (Prf1 to Prf5)
- Direct deduction, proof by cases, proof by contradiction, disproof by counterexample, forming conjectures from small cases, ordering the steps of a proof, multi-step reasoning.
- Spotting errors (Err1 and Err2)
- Finding the flaw in a purported proof, such as cancelling a common factor that could be zero, or inferring equal angles from equal sines.
Paper 1 needs the first two groups. Paper 2 needs all three. Source: UAT-UK: TMUA Content Specification. For the logic and proof topics, UAT-UK also publishes UAT-UK: Notes on Logic and Proof.
Revise each topic
Short revision pages with worked no-calculator examples, common mistakes and quick answers.
Paper 1TMUA Binomial ExpansionRevise the binomial expansion for positive integer powers for the TMUA: nCr, coefficients and constant terms, with worked no-calculator examples.Paper 1TMUA Coordinate Geometry: Lines and CirclesRevise straight lines, parallel and perpendicular gradients, circle equations and tangents for the TMUA, with worked no-calculator examples.Paper 1TMUA Counting and ProbabilityRevise systematic counting, tree diagrams, conditional probability and the addition and multiplication rules for the TMUA, with worked examples.Paper 1TMUA Differentiation and Stationary PointsRevise differentiation for the TMUA: gradients, tangents and normals, stationary points and optimisation, with worked no-calculator examples.Paper 1TMUA Exponentials and LogarithmsRevise exponential equations and the laws of logarithms for the TMUA, including equations that reduce to quadratics, with worked examples.Paper 1TMUA Functions and Graph TransformationsRevise one-to-one and many-to-one functions, composite functions and graph transformations for the TMUA, with worked no-calculator examples.Paper 1TMUA Geometry and Circle TheoremsRevise angle facts, circle theorems, polygons, similarity and Pythagoras in 2D and 3D for the TMUA, with worked no-calculator examples.Paper 1TMUA Indices and SurdsRevise index laws, fractional and negative powers, simplifying and rationalising surds for the TMUA, with worked examples and no calculator.Paper 1TMUA Inequalities and ModulusRevise linear and quadratic inequalities and modulus (absolute value) for the TMUA, with worked no-calculator examples and common sign traps.Paper 1TMUA Integration and AreaRevise integration for the TMUA: definite integrals, area under and between curves, and the trap when a curve crosses the axis. Worked examples.Paper 1TMUA Number, Factors and DivisibilityRevise primes, prime factorisation, HCF, LCM, counting factors and divisibility for the TMUA, with worked no-calculator examples.Paper 1TMUA Polynomials and the Factor TheoremRevise polynomial division, the remainder theorem and the factor theorem for the TMUA, with worked examples that need no calculator.Paper 1TMUA Quadratics and the DiscriminantRevise quadratics for the TMUA: the discriminant, completing the square, the vertex and sum and product of roots, with worked no-calculator examples.Paper 1TMUA Ratio, Proportion, Percentages and UnitsRevise ratio, percentage change, direct and inverse proportion, similar shapes and units for the TMUA, with worked no-calculator examples.Paper 1TMUA Sequences and SeriesRevise arithmetic and geometric sequences, sums, sum to infinity and recurrence relations for the TMUA, with worked no-calculator examples.Paper 1TMUA Simultaneous EquationsSolve a linear and a quadratic equation together for the TMUA, and read the answer as line and curve intersections. Worked no-calculator examples.Paper 1TMUA Statistics: Averages, Spread and DataRevise mean, median, quartiles, frequency density and grouped data for the TMUA, with worked no-calculator examples.Paper 1TMUA Triangle Trigonometry and RadiansRevise the sine and cosine rules, triangle area, radians, arc length and sector area for the TMUA, with worked examples and no calculator.Paper 1TMUA Trigonometric Equations and IdentitiesRevise trig identities, exact values and solving sine, cosine and tangent equations in a given interval for the TMUA, with worked examples.Paper 2TMUA Logic: Implication, Converse and ContrapositiveRevise 'if', 'only if', converse, contrapositive, necessary and sufficient conditions for TMUA Paper 2, with worked examples.Paper 2TMUA Proof by Cases, Exhaustion and CounterexampleLearn proof by cases, disproof by counterexample and conjecturing from small cases for TMUA Paper 2, with worked examples verified by hand.Paper 2TMUA Proof by ContradictionLearn proof by contradiction for TMUA Paper 2: how to start, what the contradiction looks like, with worked examples you can check by hand.Paper 2TMUA Quantifiers and NegationRevise 'for all', 'there exists' and how to negate statements correctly for TMUA Paper 2, with worked examples you can check by hand.Paper 2TMUA Spotting Errors in ProofsLearn to find the faulty step in a purported proof for TMUA Paper 2: illegal cancelling, sign errors and invalid inferences, with worked examples.Paper 2TMUA Truth-Teller and Liar PuzzlesPractise truth-teller and liar puzzles to build the multi-step logical reasoning needed for TMUA Paper 2, with worked examples verified by hand.