The TMUA specification, topic by topic

By The STEM-Tutor team · Updated October 2026

The TMUA specification lists exactly what the test can ask. Knowing it well stops you wasting time on topics that never appear, and shows you where the real challenge lies: applying familiar maths in unfamiliar ways.

How the specification is organised

The TMUA content specification has two sections. Section 1 lists the mathematical topics, which underpin both papers. Section 2 covers logic and proof, which applies to Paper 2, Mathematical Reasoning. Both papers have 20 multiple-choice questions in 75 minutes, with no calculator. UAT-UK, which runs the test, publishes the specification together with free Notes on Mathematics and Notes on Logic and Proof. Always check you are using the latest version.

Section 1: the mathematics

The maths is drawn from GCSE (or equivalent) and the pure mathematics in the first year of A level Maths. No statistics or mechanics are needed. The main areas are:

• Algebra and functions: indices and surds, quadratics and the discriminant, simultaneous equations, inequalities, polynomials and the factor theorem, and graph transformations.
• Sequences and series: arithmetic and geometric sequences, sums of series including the sum to infinity, and the binomial expansion for positive whole-number powers.
• Coordinate geometry: straight lines, gradients, parallel and perpendicular lines, and circles.
• Trigonometry: sine, cosine and tangent graphs, exact values, identities such as sin²x + cos²x = 1, and solving equations in a range.
• Exponentials and logarithms: laws of logs and solving exponential equations.
• Differentiation: gradients, tangents and normals, increasing and decreasing functions, and stationary points.
• Integration: indefinite and definite integrals of polynomials, and areas under curves.
• Graphs of functions: sketching, and using graphs to count solutions.

Section 2: logic and proof (Paper 2)

The logic is intentionally basic, but it must be precise. You need to understand:

• true and false statements, and combining them with and, or and not;
• conditional statements: “if A then B”, “A only if B” and “A if and only if B”;
• the converse and the contrapositive of a statement;
• necessary and sufficient conditions;
• “for all” and “there exists”, and how to negate statements that use them.

Symbolic notation and formal truth tables are not required. On proof, you should be able to follow and construct direct proofs, proofs by cases and proofs by contradiction, disprove a statement with a counterexample, and spot the error in a flawed argument.

Example questions by topic

These are original questions in the style of the TMUA, written to illustrate the specification.

Algebra: counting solutions

How many real solutions does (x² − 4)(x² + x + 1) = 0 have?

• x² − 4 = 0 gives x = ±2.
• x² + x + 1 has discriminant 1 − 4 = −3 < 0, so no real roots. Answer: 2.

Sequences: a geometric sum

A geometric series has first term 6 and sum to infinity 9. Find the common ratio.

• 6/(1 − r) = 9, so 1 − r = 2/3 and r = 1/3.

Coordinate geometry: a circle

Find the centre and radius of x² + y² − 6x + 4y = 3.

• Complete the square: (x − 3)² + (y + 2)² = 3 + 9 + 4 = 16.
• Centre (3, −2), radius 4.

Logs: solving an equation

Solve 2 log₃x − log₃(x + 6) = 1.

• log₃(x²/(x + 6)) = 1, so x² = 3x + 18 and (x − 6)(x + 3) = 0.
• x must be positive, so x = 6.

Calculus: a stationary point

Find the minimum value of f(x) = x³ − 12x for x > 0.

• f'(x) = 3x² − 12 = 0 gives x = 2 (for x > 0); f''(2) = 12 > 0, so it is a minimum.
• f(2) = 8 − 24 = −16.

Logic: necessary or sufficient?

For a whole number n, is “n is divisible by 4” necessary, sufficient, both or neither for “n is even”?

• Every multiple of 4 is even, so it is sufficient. But 6 is even and not a multiple of 4, so it is not necessary.
• Sufficient but not necessary.

Proof: finding the flaw

Claim: for all real x, x² > x. ‘Proof’: divide both sides by x to get x > 1, which is true. What is wrong?

• There are two errors. Dividing by x keeps the inequality the same way round only when x is positive; for negative x it would flip. And the final step, “x > 1, which is true”, is simply false for many x.
• A single counterexample settles it: x = 1/2 gives x² = 1/4, which is not greater than 1/2. So the claim is false. On Paper 2, look for division by something that could be zero or negative, and for steps that quietly assume the conclusion.

What is not on the TMUA

The specification is as useful for what it leaves out. You will not need second-year A level topics such as the chain, product or quotient rules, integration by parts, radians-based calculus of trig functions, parametric equations or vectors. Statistics and mechanics are not tested. Spending revision time on these would be wasted for the TMUA, although they matter for your A level exams.

Turning the specification into a plan

1. Print the specification and rate each topic: secure, shaky or new.
2. Fix shaky Section 1 topics first, without a calculator, since both papers depend on them.
3. Work through the Notes on Logic and Proof slowly, doing every exercise; this is where most students are least prepared.
4. Move to official practice papers, first untimed, then timed.
5. Keep an error log linked to the specification, so you can see which topics cost you marks.

Common mistakes

• Revising second-year A level content that the TMUA does not test.
• Leaving logic and proof until the last few weeks.
• Forgetting domain restrictions, for example with logarithms.
• Treating “A only if B” as meaning “if B then A”; it means “if A then B”.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is the TMUA specification the same every year?

It changes rarely, but UAT-UK can update it. Always download the current version from the official TMUA pages before you start revising.

Do I need Further Maths for the TMUA?

No. Everything on the specification is in GCSE and first-year A level pure maths.

Key takeaways

• Section 1: GCSE and first-year A level pure maths, for both papers.
• Section 2: logic and proof, for Paper 2.
• No calculator, no statistics, no mechanics.
• Logic and proof need dedicated practice.
• Use the official notes and practice papers alongside the specification.

Related guides

• What is the TMUA?
• ESAT specification
• A level Maths topics