Key Facts
| Specification edition | TMUA Content Specification for assessment in October 2026 and January 2027 |
|---|---|
| Section 1 | MM1–MM8 (AS-level pure) and M1–M7 (Higher GCSE); both papers |
| Section 2 | Logic of arguments, mathematical proof, identifying errors in proofs; Paper 2 |
| Rules | No calculator, no formulae booklet, no negative marking |
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The TMUA specification is UAT-UK's list of everything the test can examine, and it has two sections: mathematical knowledge for both papers, covering AS-level pure mathematics and Higher GCSE mathematics, and mathematical reasoning for Paper 2, covering logic, proof and finding errors in proofs. Paper 1 applies the knowledge in Section 1; Paper 2 uses Section 1 and Section 2 together. There is no formulae booklet and no calculator, so every formula in the specification has to be recalled. Below are the topics in each section and original example questions with worked answers.
How the TMUA specification is organised
The TMUA Content Specification sets out the structure of the test and then the content, topic by topic, with a reference code for each.
| Paper | Name | Questions and time | Content examined |
|---|---|---|---|
| Paper 1 | Applications of Mathematical Knowledge | 20 multiple-choice questions in 75 minutes | Section 1 |
| Paper 2 | Mathematical Reasoning | 20 multiple-choice questions in 75 minutes | Sections 1 and 2 |
Questions across both papers carry equal weight, there is no penalty for a wrong answer, and calculators are not allowed.
Section 1: mathematical knowledge
Section 1 has two parts. UAT-UK says Part 1 is almost all covered by the pure mathematics in an AS-level mathematics course, and Part 2 is almost all covered by Higher GCSE mathematics, with some overlap between them.
| Code | Topic | Includes |
|---|---|---|
| MM1 | Algebra and functions | Indices, surds, quadratics and the discriminant, simultaneous equations, inequalities, polynomials, factor and remainder theorems |
| MM2 | Sequences and series | Recurrence relations, arithmetic and geometric series, sum to infinity, binomial expansion |
| MM3 | Coordinate geometry | Straight lines, circles and circle properties |
| MM4 | Trigonometry | Sine and cosine rules, radians, exact values, graphs, identities, trigonometric equations |
| MM5 | Exponentials and logarithms | Exponential graphs, laws of logarithms, equations of the form a to the power x equals b |
| MM6 | Differentiation | Gradients, tangents, normals, stationary points, increasing and decreasing functions |
| MM7 | Integration | Definite and indefinite integrals, area, the Fundamental Theorem of Calculus, trapezium rule |
| MM8 | Graphs of functions | Common graphs, transformations, using calculus to sketch curves, intersections |
| M1 to M7 | GCSE content | Units, number, ratio and proportion, algebra, geometry and measures, statistics, probability |
The specification lists the detail under each code, with worked illustrations, and it is worth reading in full because some items are easy to miss: the modulus function among the common graphs, the difference between a definite integral and an area, and whether the trapezium rule gives an overestimate or an underestimate.
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Enquire about this tutorSection 2: logic and proof for Paper 2
Section 2 sets out the reasoning Paper 2 tests, under three headings. UAT-UK also publishes Notes on Logic and Proof, which teach this content from first principles.
| Area | What you need to be able to do |
|---|---|
| The logic of arguments | Use true and false; and, or (inclusive), not; "if A then B", "A if B", "A only if B" and "A if and only if B"; converse and contrapositive; necessary and sufficient; "for all", "for some" and "there exists"; negating any of these |
| Mathematical proof | Follow and in simple cases construct direct proof, proof by cases, proof by contradiction and disproof by counterexample; deduce implications; make and justify conjectures from small cases; put the steps of a proof in order |
| Identifying errors in proofs | Spot errors in purported proofs, including common invalid steps such as concluding b equals c from ab equals ac, or A equals B from sin A equals sin B |
No symbolic notation or formal truth tables are required: the logic is tested in words.
Example questions for Section 1
The questions below are original, written to match the topics and style of the specification. Try each one without a calculator before reading the answer.
Example 1: the discriminant (MM1)
For which values of k does the equation x² + kx + 9 = 0 have no real solutions?
Answer: −6 < k < 6. There are no real roots when the discriminant is negative: k² − 4 × 1 × 9 < 0, so k² < 36, which gives −6 < k < 6.
Example 2: geometric series (MM2)
A geometric series has first term 6 and sum to infinity 9. What is the common ratio?
Answer: 1/3. The sum to infinity is a ÷ (1 − r), so 6 ÷ (1 − r) = 9, giving 1 − r = 2/3 and r = 1/3. The value lies between −1 and 1, as it must for the series to converge.
Example 3: the circle (MM3)
Find the centre and radius of the circle x² + y² − 6x + 4y − 12 = 0.
Answer: centre (3, −2), radius 5. Completing the square gives (x − 3)² − 9 + (y + 2)² − 4 − 12 = 0, so (x − 3)² + (y + 2)² = 25.
Example 4: counting solutions (MM4)
How many solutions does sin 2x = 1/2 have for 0° ≤ x < 360°?
Answer: 4. If 0° ≤ x < 360°, then 0° ≤ 2x < 720°. Within that range sin 2x = 1/2 when 2x = 30°, 150°, 390° or 510°, so x = 15°, 75°, 195° or 255°. Doubling the angle doubles the number of solutions, which is the step most often missed.
Example 5: exponential equations (MM5)
Solve 2 to the power (x + 1) = 3 to the power x.
Answer: x = log 2 ÷ (log 3 − log 2). Taking logarithms of both sides, (x + 1) log 2 = x log 3, so x log 2 + log 2 = x log 3, and x (log 3 − log 2) = log 2. This is the same as x = log base 1.5 of 2, which is about 1.7.
Example 6: stationary points (MM6)
Find the x-coordinates of the stationary points of y = x³ − 3x² − 9x + 5, and say which is a maximum.
Answer: x = −1 (maximum) and x = 3 (minimum). The derivative is 3x² − 6x − 9 = 3(x − 3)(x + 1), which is zero at x = −1 and x = 3. The second derivative, 6x − 6, is negative at x = −1 and positive at x = 3.
Example 7: integral or area (MM7)
Find the value of the integral of x² − 2x from 0 to 3, and the total area enclosed between y = x² − 2x and the x-axis from x = 0 to x = 3.
Answer: integral 0, area 8/3. The integral is [x³/3 − x²] from 0 to 3, which is 9 − 9 = 0. The curve crosses the axis at x = 2, so the area is the size of the integral from 0 to 2, which is 4/3, plus the integral from 2 to 3, which is also 4/3, giving 8/3. You need to know, as the specification notes, that a definite integral and an area differ when the curve goes below the axis.
Example questions for Section 2
Example 8: necessary and sufficient (Arg2)
For real x, is "x > 2" a necessary condition, a sufficient condition, both or neither for "x² > 4"?
Answer: sufficient but not necessary. If x > 2 then x² > 4, so the condition is sufficient. It is not necessary, because x = −3 gives x² = 9 > 4 without x > 2.
Example 9: converse and contrapositive (Arg1)
Consider the true statement "if n² is even, then n is even" for integers n. Which of the following is its contrapositive, and is the converse true?
- A. If n is even, then n² is even.
- B. If n is odd, then n² is odd.
- C. If n² is odd, then n is odd.
Answer: B is the contrapositive, and the converse (A) is also true. The contrapositive of "if P then Q" is "if not Q then not P", which here is "if n is not even, then n² is not even", that is B. A is the converse, which happens to be true as well, since the square of an even number is even. C is the converse of the contrapositive, not the contrapositive itself.
Example 10: negation and counterexample (Arg4, Prf1)
Write the negation of "for every positive integer n, n² + n + 41 is prime", and decide whether the original statement is true.
Answer: "there exists a positive integer n for which n² + n + 41 is not prime"; the original is false. Taking n = 40 gives 1600 + 40 + 41 = 1681 = 41², which is not prime, so one counterexample is enough to disprove the statement.
Example 11: an error in a proof (Err1, Err2)
Here is a purported proof that 1 = 2. Which line contains the error?
- Let x = y, where x and y are non-zero.
- Then x² = xy.
- Subtracting y² from both sides, x² − y² = xy − y².
- Factorising both sides, (x + y)(x − y) = y(x − y).
- Therefore x + y = y.
- Since x = y, this gives 2y = y, and dividing by y, 2 = 1.
Answer: line 5. Moving from line 4 to line 5 divides both sides by x − y, which is zero because x = y. This is the invalid step the specification names: from ab = ac you cannot conclude b = c unless a is non-zero.
How to use the specification in preparation
The specification is a checklist as much as a syllabus. Go through it line by line before starting past papers, marking each item as secure, rusty or new, and revise the rusty and new items first. Students in Year 12 will find a few AS-level topics not yet taught at school, most often parts of integration and logarithms, and these are worth covering early.
Paper 2 is where most students have the most to learn, because necessary and sufficient conditions, contrapositives and the structure of proofs are seldom taught directly. Reading the Notes on Logic and Proof and doing the Paper 2 questions in the TMUA past papers by topic builds that quickly. The TMUA preparation plan sets out a full schedule, and the TMUA score conversion guide explains the results.
Related guides
- TMUA guide: format, scoring and which courses require it
- Applying for maths at Cambridge
- Oxford mathematics: the application and the TMUA
- Super-curricular activities for maths
- Super-curricular activities for computer science
- University admissions tests in the UK
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Frequently asked questions
- What maths do you need for the TMUA?
- The specification's Section 1 is almost all AS-level pure mathematics and Higher GCSE mathematics, covering algebra, sequences, coordinate geometry, trigonometry, logarithms, calculus and graphs, plus GCSE number, geometry, statistics and probability.
- Is there a formula sheet in the TMUA?
- No. There is no formulae booklet, so you need to recall every formula in the specification, such as the sum of a geometric series and the sine and cosine rules.
- What is tested in TMUA Paper 2?
- Paper 2 uses the mathematical knowledge from Section 1 with the logic and proof in Section 2: conditions and their converses and contrapositives, necessary and sufficient conditions, quantifiers, types of proof, and finding errors in purported proofs.
- Do you need A-level Further Maths for the TMUA?
- No. The content is within AS-level mathematics and Higher GCSE mathematics, though fluency from wider practice helps with speed.
- Can you use a calculator in the TMUA?
- No. Calculators and dictionaries are not allowed, so arithmetic, algebra and estimation all need to be quick and reliable by hand.
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