TMUA past papers with worked answers
The TMUA's official past papers are free from the test's own website, each with full worked answers and an answer key. Here is every paper from 2016 to 2023, plus the early specimen papers. The test has two papers of 20 multiple-choice questions, 75 minutes each, with no calculator: Paper 1, Applications of Mathematical Knowledge, and Paper 2, Mathematical Reasoning.
TMUA past papers by year
Every file opens from the official test website. Click any file to preview it here; the preview has Open in new tab and Download. On a phone, files open in a new tab.
2023
- Paper 1Question paperWorked answers
- Paper 2Question paperWorked answers
- Both papersAnswer key
2022
- Paper 1Question paperWorked answers
- Paper 2Question paperWorked answers
- Both papersAnswer key
2021
- Paper 1Question paperWorked answers
- Paper 2Question paperWorked answers
- Both papersAnswer key
2020
- Paper 1Question paperWorked answers
- Paper 2Question paperWorked answers
- Both papersAnswer key
2019
- Paper 1Question paperWorked answers
- Paper 2Question paperWorked answers
- Both papersAnswer key
2018
- Paper 1Question paperWorked answers
- Paper 2Question paperWorked answers
- Both papersAnswer key
2017
- Paper 1Question paperWorked answers
- Paper 2Question paperWorked answers
- Both papersAnswer key
2016
- Paper 1Question paperWorked answers
- Paper 2Question paperWorked answers
- Both papersAnswer key
Early specimen papers
- Paper 1Question paperWorked answers
- Paper 2Question paperWorked answers
- Both papersAnswer key
Source: official TMUA preparation materials. All papers are © their publisher.
TMUA test format
| Paper | Questions or marks | Time |
|---|---|---|
| Paper 1: Applications of Mathematical Knowledge | 20 multiple-choice questions | 75 minutes |
| Paper 2: Mathematical Reasoning | 20 multiple-choice questions | 75 minutes |
There is no calculator or dictionary. You get one overall score on a scale from 1 (low) to 9 (high), to one decimal place, and wrong answers are not penalised, so answer every question. The test is sat at a Pearson test centre.
How to use TMUA past papers
- Answer every question: a wrong answer costs nothing, so never leave a blank.
- Time is the real challenge: 20 questions in 75 minutes is under 4 minutes each. Skip and return rather than getting stuck.
- Paper 2 tests logic and proof, such as necessary and sufficient conditions and spotting the error in an argument. Practise these as a topic.
- Use the official worked answers after each paper: they often show a faster route than the one you took.
- Save the most recent papers, 2022 and 2023, for timed mocks near your test date.
Free TMUA practice questions
Ten original questions written by our tutors in the style of the exam. Choose an answer to see if you are right and why.
Answered 0 of 10 · 0 correct
1. How many real solutions does x⁴ − 5x² + 4 = 0 have?
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D. It factorises as (x² − 1)(x² − 4) = 0, giving x = ±1 and x = ±2: four solutions.
2. What is the contrapositive of 'if n² is even, then n is even'?
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B. The contrapositive of 'if P then Q' is 'if not Q then not P'.
3. What is the sum to infinity of 6 + 3 + 1.5 + 0.75 + …?
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B. It is geometric with first term 6 and ratio ½, so the sum is 6/(1 − ½) = 12.
4. Which number is a counterexample to the claim 'every prime number is odd'?
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B. 2 is prime and even. (1, 9 and 15 are not prime.)
5. What is the minimum value of x² − 4x + 7?
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C. Completing the square: (x − 2)² + 3, which is smallest, 3, when x = 2.
6. For a real number x, the condition 'x > 3' is ___ for 'x > 1'.
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B. If x > 3 then x > 1, so it is sufficient; but x = 2 satisfies x > 1 without x > 3, so it is not necessary.
7. How many integers from 1 to 100 inclusive are divisible by 3 or by 5?
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B. 33 multiples of 3 plus 20 multiples of 5, minus the 6 multiples of 15 counted twice: 47.
8. What is ∫02 |x − 1| dx?
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C. The graph forms two triangles, each with base 1 and height 1, so the area is ½ + ½ = 1.
9. Given that log₁₀ 2 ≈ 0.301, how many digits does 220 have?
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B. log₁₀ 220 ≈ 6.02, so 220 is between 10⁶ and 10⁷ and has 7 digits.
10. If f(x) = 2x + 1, what is f(f(x))?
Show answer
C. f(f(x)) = 2(2x + 1) + 1 = 4x + 3.
Frequently asked questions
Are TMUA past papers free?
Yes. The test's owner publishes every past paper from 2016 to 2023 free, with worked answers and answer keys, and they are linked on this page.
Why are there no TMUA papers after 2023?
The TMUA moved to on-screen testing at Pearson test centres, and newer papers have not been released. The specification has not changed, so the 2016 to 2023 papers are still the best practice.
Can I use a calculator in the TMUA?
No. Neither paper allows a calculator or a dictionary.
What is a good TMUA score?
Scores run from 1 to 9. Competitive courses often look for scores well above the middle of the scale, but each university sets its own expectations, so check the course page.