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100+ Free TMUA Practice Questions

Pass your TMUA (Test of Mathematics for University Admission) exam on the first try — instant access, no signup required.

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The statement 'P if and only if Q' is true precisely when:

A
B
C
D
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Key Facts: TMUA Exam

40 questions

20 per paper across two papers

UAT-UK

2h 30m

Total time (75 min per paper)

UAT-UK

1.0-9.0

Scaled score, no pass/fail

UAT-UK

No calculator

Calculators and formula sheets banned

UAT-UK

Oct 12-16

2026 test window

UAT-UK

AS-level

Content level (Higher GCSE + AS pure maths)

UAT-UK

100

Free practice questions here

OpenExamPrep

TMUA is a 2 hour 30 minute, no-calculator admissions test of 40 multiple-choice questions across two papers — Applications of Mathematical Knowledge and Mathematical Reasoning. It is scored 1.0 to 9.0 (no pass/fail) and used by Cambridge, Imperial, Oxford, LSE, Warwick and others to assess maths, economics and computer science applicants.

Sample TMUA Practice Questions

Try these sample questions to test your TMUA exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Simplify the surd expression: √50 + √18 - √8.
A.6√2
B.8√2
C.5√2
D.4√2
Explanation: √50 = 5√2, √18 = 3√2 and √8 = 2√2. So the expression is 5√2 + 3√2 - 2√2 = 6√2.
2Rationalise the denominator of 6 / (3 - √3) and express the result in simplest form.
A.3 + √3
B.2 + √3
C.3 - √3
D.(6 + 2√3)/3
Explanation: Multiply numerator and denominator by the conjugate (3 + √3): the denominator becomes 9 - 3 = 6, and the numerator becomes 6(3 + √3). So 6(3 + √3)/6 = 3 + √3.
3Evaluate (8/27)^(-2/3).
A.9/4
B.4/9
C.27/8
D.2/3
Explanation: The negative exponent inverts the base: (27/8)^(2/3). The cube root of 27/8 is 3/2, and squaring gives 9/4.
4Solve for x: log₂(x) + log₂(x - 2) = 3.
A.4
B.8
C.2
D.6
Explanation: Combine logs: log₂(x(x-2)) = 3, so x(x-2) = 2^3 = 8. This gives x^2 - 2x - 8 = 0, factoring to (x-4)(x+2) = 0. Only x = 4 is valid since x must be positive and exceed 2.
5Given that log₃(a) = 2 and log₃(b) = 4, what is the value of log₃(a²/b)?
A.0
B.2
C.4
D.-2
Explanation: log₃(a²/b) = 2·log₃(a) - log₃(b) = 2×2 - 4 = 0.
6The quadratic x² + kx + 9 = 0 has exactly one real (repeated) root. What are the possible values of k?
A.k = 6 or k = -6
B.k = 3 or k = -3
C.k = 9 only
D.k = 6 only
Explanation: A repeated root requires discriminant zero: k² - 4(1)(9) = 0, so k² = 36 and k = ±6.
7What is the remainder when the polynomial f(x) = 2x³ - 3x² + x - 5 is divided by (x - 2)?
A.1
B.-3
C.5
D.9
Explanation: By the remainder theorem, the remainder is f(2) = 2(8) - 3(4) + 2 - 5 = 16 - 12 + 2 - 5 = 1.
8The function f is defined by f(x) = (x + 3)² - 4. What is the range of f?
A.f(x) ≥ -4
B.f(x) ≥ 0
C.f(x) ≤ -4
D.f(x) ≥ -3
Explanation: The minimum value of (x + 3)² is 0, achieved at x = -3. So the minimum of f is 0 - 4 = -4, and the range is f(x) ≥ -4.
9The line y = 2x + c is tangent to the curve y = x² + 3. What is the value of c?
A.2
B.3
C.4
D.1
Explanation: Set equal: x² + 3 = 2x + c, i.e. x² - 2x + (3 - c) = 0. Tangency means discriminant zero: 4 - 4(3 - c) = 0, so 3 - c = 1 and c = 2.
10An arithmetic sequence has first term 5 and common difference 3. What is the sum of the first 20 terms?
A.670
B.640
C.700
D.620
Explanation: Sum = (n/2)(2a + (n-1)d) = (20/2)(2×5 + 19×3) = 10(10 + 57) = 10×67 = 670.

About the TMUA Exam

The TMUA (Test of Mathematics for University Admission) is a computer-based admissions test taken by applicants to mathematics, economics, computer science and related degrees at universities including Cambridge, Imperial College London, the University of Oxford, LSE, Warwick, Durham and others. It consists of two 75-minute papers of 20 multiple-choice questions each: Paper 1 (Applications of Mathematical Knowledge) and Paper 2 (Mathematical Reasoning). No calculator is allowed, and content is drawn from Higher GCSE and AS-level pure mathematics applied in unfamiliar contexts.

Questions

100 scored questions

Time Limit

2 hours 30 minutes total (75 minutes per paper)

Passing Score

No pass/fail — reported on a 1.0 to 9.0 scale; competitive scores are typically 6.5+ for shortlisting at top universities

Exam Fee

Approximately £75 in the UK/EU and £130 outside the UK/EU; bursary/fee reimbursement available for eligible candidates (UAT-UK (delivered by Pearson VUE))

TMUA Exam Content Outline

50%

Paper 1: Applications of Mathematical Knowledge

Algebra and surds, laws of indices, logarithms and exponentials, quadratics and the discriminant, polynomials and the remainder theorem, sequences and series (arithmetic, geometric, binomial), functions and transformations, coordinate geometry and circles, trigonometry and identities, differentiation, integration and probability

50%

Paper 2: Mathematical Reasoning

Elementary logic and truth values, the connectives and/or/not and De Morgan's laws, implication, converse, inverse and contrapositive, necessary and sufficient conditions, quantifiers (for all, there exists) and their negation, validity of arguments (modus ponens and modus tollens), and methods of proof including direct, contradiction, contrapositive, induction, exhaustion and counterexample

How to Pass the TMUA Exam

What You Need to Know

  • Passing score: No pass/fail — reported on a 1.0 to 9.0 scale; competitive scores are typically 6.5+ for shortlisting at top universities
  • Exam length: 100 questions
  • Time limit: 2 hours 30 minutes total (75 minutes per paper)
  • Exam fee: Approximately £75 in the UK/EU and £130 outside the UK/EU; bursary/fee reimbursement available for eligible candidates

Keys to Passing

  • Complete 500+ practice questions
  • Score 80%+ consistently before scheduling
  • Focus on highest-weighted sections
  • Use our AI tutor for tough concepts

TMUA Study Tips from Top Performers

1Master AS-level pure maths fluency first — surds, indices, logarithms, quadratics, sequences and calculus must be automatic since no calculator is allowed
2Practise Paper 2 logic vocabulary precisely: converse, inverse, contrapositive, necessary, sufficient, and the negation of 'for all' and 'there exists'
3Learn the standard proof methods (direct, contradiction, contrapositive, induction, exhaustion) and when each is the cleanest approach
4Work official UAT-UK specimen and past papers under timed conditions — about 3.75 minutes per question — to build speed and stamina
5Since there is no negative marking, never leave a question blank; eliminate wrong options and make an educated guess if time runs short
6Review every mistake to identify whether it was a knowledge gap, a reasoning slip or a careless arithmetic error, and drill that weakness specifically

Frequently Asked Questions

What is the TMUA and who takes it?

The TMUA (Test of Mathematics for University Admission) is a UK admissions test for applicants to mathematics, economics, computer science and related degrees. It is used by universities including Cambridge, Imperial College London, the University of Oxford, LSE, Warwick and Durham to help select and shortlist candidates.

How is the TMUA structured?

The test lasts 2 hours 30 minutes and has two papers of 75 minutes each. Paper 1 (Applications of Mathematical Knowledge) and Paper 2 (Mathematical Reasoning) each contain 20 multiple-choice questions, giving 40 questions and a maximum raw score of 40. No calculator or formula sheet is permitted.

How is the TMUA scored?

There is no pass or fail. Raw marks are converted to a scaled score from 1.0 (low) to 9.0 (high), reported to one decimal place. There is no penalty for incorrect answers, so candidates should attempt every question. A score of around 6.5 or above is widely considered competitive.

What is the difference between Paper 1 and Paper 2?

Paper 1 tests your ability to apply pure mathematics — algebra, calculus, trigonometry, sequences and probability — to unfamiliar problems. Paper 2 tests mathematical reasoning and elementary logic: implication, converse and contrapositive, necessary and sufficient conditions, quantifiers, and recognising valid proofs and counterexamples.

What level of mathematics does the TMUA cover?

The content is drawn from Higher GCSE and the first year of A-level (AS-level) pure mathematics. No knowledge beyond AS-level is required, but questions apply this material in unfamiliar and reasoning-heavy ways rather than testing rote recall.

When is the TMUA taken and can I use a calculator?

The TMUA is sat in October each year (12-16 October in 2026) at Pearson VUE test centres on computer. Calculators, dictionaries and formula sheets are not permitted, so arithmetic and manipulation must be done by hand.