Checked 18 October 2026. Links and dates read on UAT-UK’s pages on 18 October 2026. UAT-UK moves its PDF files without notice, so if a link breaks, start from its TMUA Preparation Materials page.
UAT-UK publishes 18 official TMUA papers: Paper 1 and Paper 2 for each year from 2016 to 2023, plus an early specimen pair. Every paper has its own worked answers and every year has an answer key, all free on UAT-UK’s TMUA Preparation Materials page. No paper from the computer-based sittings that began in October 2024 has been published. UAT-UK says the archive still applies: “Although the TMUA is now computer-based, both the content specification and question style are unchanged.” ProSyllabus is an independent study site, not affiliated with UAT-UK or any university. Every rule and figure here is quoted or summarised from their own pages, read on 18 October 2026.
UAT-UK TMUA page and Preparation Materials page, read 18 October 2026.
Every official TMUA paper, worked answer and answer key
| Year | Paper 1 | Paper 1 worked answers | Paper 2 | Paper 2 worked answers | Answer key |
|---|---|---|---|---|---|
| 2016 | |||||
| 2017 | |||||
| 2018 | |||||
| 2019 | |||||
| 2020 | |||||
| 2021 | |||||
| 2022 | |||||
| 2023 | |||||
| Early specimen |
Every link opens UAT-UK’s own file, as listed on its TMUA Preparation Materials page on 18 October 2026.
| Year | Paper 1 | Paper 2 | Worked answers | Answer key |
|---|---|---|---|---|
| 2016 | Yes | Yes | Both papers | Yes |
| 2017 | Yes | Yes | Both papers | Yes |
| 2018 | Yes | Yes | Both papers | Yes |
| 2019 | Yes | Yes | Both papers | Yes |
| 2020 | Yes | Yes | Both papers | Yes |
| 2021 | Yes | Yes | Both papers | Yes |
| 2022 | Yes | Yes | Both papers | Yes |
| 2023 | Yes | Yes | Both papers | Yes |
| Early specimen | Yes | Yes | Both papers | Yes |
The same list as plain text, for readers whose browser hides the links above.
Where other pages get this wrong
UAT-UK’s archive ends with the 2023 papers. There is no official “TMUA 2024” or “TMUA 2025” paper.
Commonly published instead: downloads labelled TMUA 2024 or 2025 papers, and question “dumps” said to come from recent sittings.
UAT-UK has run the TMUA on computer at Pearson VUE centres since October 2024, and its past-paper archive still ends at 2023. Check anything labelled as a 2024 or 2025 paper against UAT-UK’s own pages. UAT-UK’s free practice tests in the Pearson system are a different thing: they show the on-screen test, and one of them has explained answers.
How the old papers compare with the test you will sit
| Feature | Today (2027 entry) | Archive papers |
|---|---|---|
| Papers | Paper 1 Applications of Mathematical Knowledge, Paper 2 Mathematical Reasoning | Same two papers |
| Questions and time | 20 multiple-choice questions in 75 minutes, per paper | Check the front page of each paper |
| Calculator or dictionary | Not allowed | Check the front page of each paper |
| Wrong answers | No marks lost | Check the front page of each paper |
| How you take it | On computer at a Pearson VUE test centre | Printed papers; practise them with a timer |
| Score | One overall score, 1.0–9.0, to one decimal place | Answer keys and worked answers; UAT-UK publishes no current raw-to-scaled table |
From UAT-UK’s TMUA page, its Preparation Materials page and the TMUA Explanation of Results, read 18 October 2026. UAT-UK publishes no table that turns a raw mark out of 40 into a scaled score.
So you cannot score an old paper as “6.4”. Count your correct answers out of 20 per paper and compare yourself with yourself. For what a scaled score means for each university, see our TMUA score guide for Oxford, Cambridge, Imperial and LSE.
Where the marks are: what the specification asks
Paper 1 uses the mathematics in the specification’s Section 1, which UAT-UK says is almost all covered by AS-level pure mathematics and Higher GCSE mathematics: algebra and functions, sequences and series, coordinate geometry, trigonometry, exponentials and logarithms, differentiation, integration and graphs. Paper 2 draws on the same knowledge and adds what, in our experience, school courses teach least: the logic of arguments, mathematical proof and spotting errors in proofs. There is no formulae booklet.
| Code | What the specification asks | Why it costs marks |
|---|---|---|
| Arg1 | Use logic in simple situations: and, or (inclusive), not; “if A then B”, “A only if B”, “A if and only if B”; the converse and the contrapositive | Students treat a statement and its converse as the same |
| Arg3–Arg4 | Use “for all”, “for some” and “there exists”, and negate statements that use them | The negation of “for all” is “for some … not”, not “for none” |
| Arg2 | Understand and use the terms necessary and sufficient | The two words are swapped under time pressure |
| Prf1–Prf5 | Follow and build direct proofs, proof by cases, by contradiction, disproof by counterexample; deduce implications; put a proof’s steps in order | A proof can look right with one step out of order |
| Err1–Err2 | Find the error in a purported proof, for example claiming that ab = ac means b = c | You must find the first wrong line, not just any doubtful one |
Topic codes and wording summarised from UAT-UK’s TMUA Content Specification; the last column is our reading.
A worked solution, the way the official worked answers set it out
UAT-UK provides worked answers for every paper. Here is the same treatment for one of our own free questions (we do not reproduce official questions).
Free question · Exponentials and logarithms · original, TMUA style
What is the sum of the real solutions of 2^(2x) − 12 × 2^x + 32 = 0?
- A 3
- B log₂ 12
- C 4
- D 5
- E 6
- F 12
Show the worked solution
- Spot the hidden quadratic: 2^(2x) = (2^x)². Let y = 2^x, which is always positive.
- The equation becomes y² − 12y + 32 = 0, which factorises as (y − 4)(y − 8) = 0, so y = 4 or y = 8.
- Both are positive, so both give real x: 2^x = 4 gives x = 2, and 2^x = 8 gives x = 3.
- The question asks for the sum of the x-values: 2 + 3 = 5.
- The trap: the sum of the y-roots is 12 (option F) and their product is 32. Answer the question that was asked, about x.
Answer: D, 5.
An original ProSyllabus question, not a UAT-UK question. It is one of the five free questions at the top of our TMUA practice tests; try all five free.
A plan for using the 18 papers before January
The January 2027 sitting runs 4–8 January 2027 and booking opens on 26 October 2026, according to UAT-UK. UAT-UK says most Oxford and Cambridge applicants must sit in October; its exceptions are mature applicants to a Cambridge mature college with a January deadline and applicants to Oxford’s Astrophoria Foundation Year. UAT-UK also allows the TMUA only once in an application cycle, so if you sat it in October 2026 you cannot sit it again in January. The papers are for anyone booking January, and for next year’s applicants.
| Stage | Papers | How |
|---|---|---|
| 1. Learn the format | Early specimen Paper 1 and Paper 2 | Untimed. Read every worked answer, including the ones you got right. |
| 2. Close gaps | 2016–2019, both papers | Timed per paper (75 minutes). Log every wrong answer and every guess by topic. |
| 3. Paper 2 logic | Paper 2 from 2016–2021 | Redo every Arg, Prf and Err question you missed a week later, without notes. |
| 4. Full runs | 2020–2023, both papers back to back | Two 75-minute papers in one sitting, as on the day. No calculator. |
Our suggested order, not UAT-UK’s. Keep the most recent papers for last so they still feel new.
Track the 18 papers
Tick each paper once you have sat it timed and read every worked answer.
0 of 18 done
Ticks are saved only in this browser on this device.
When the 18 papers run out
Once you know the old papers, your score on them stops telling you much. Our TMUA practice tests add two full mocks in the same format, Paper 1 and Paper 2 each, 20 questions in 75 minutes, 80 original questions with a worked explanation for every one. The first five questions are free; the one above is question 1. The TMUA exam page has the format, dates and the universities that use the test.
Frequently asked questions
Where can I download TMUA past papers?
From UAT-UK’s TMUA Preparation Materials page at esat-tmua.ac.uk. It lists Paper 1 and Paper 2 for every year from 2016 to 2023 plus an early specimen pair, each with worked answers and an answer key.
Are there TMUA 2024 or 2025 past papers?
No official ones. UAT-UK has run the TMUA on computer since October 2024 and has not published those papers. Its archive ends with 2023.
Are the old paper-based TMUA papers still useful?
Yes. UAT-UK says that although the TMUA is now computer-based, the content specification and question style are unchanged.
How many questions are in each TMUA paper?
20 multiple-choice questions in 75 minutes. The test has two papers, 2 hours 30 minutes in total.
Can I convert my past paper mark to a TMUA score?
Not officially. UAT-UK publishes no raw-to-scaled conversion. Scores run from 1.0 to 9.0, typical candidates score around 4.5, and about 10% score above 7.0.
Is there negative marking in the TMUA?
No. You do not lose marks for wrong answers, so answer every question.
Can I use a calculator?
No calculator and no dictionary are allowed.
What is the difference between Paper 1 and Paper 2?
Paper 1, Applications of Mathematical Knowledge, applies school mathematics to new situations. Paper 2, Mathematical Reasoning, adds logic and proof: necessary and sufficient conditions, proof types and finding errors in proofs.
When is the next TMUA?
4–8 January 2027, with booking from 26 October 2026, according to UAT-UK. Most Oxford and Cambridge applicants must sit in October; UAT-UK’s exceptions are Cambridge mature colleges with a January deadline and Oxford’s Astrophoria Foundation Year.
More in this series
Sources
- official — UAT-UK: TMUA Preparation Materials (past papers, worked answers, answer keys), read 18 Oct 2026
- official — UAT-UK: TMUA (format, fees, sittings, results), read 18 Oct 2026
- official — UAT-UK: TMUA Explanation of Results (October 2025 and January 2026 sittings)
- official — UAT-UK: TMUA Content Specification
- official — UAT-UK: About UAT-UK (“The ESAT and TMUA were both offered for the first time in October 2024”)
- official — UAT-UK: Test results, read 18 Oct 2026
- official — Imperial College London: Test of Mathematics for University Admissions (TMUA), read 18 Oct 2026
- official — LSE: Test of Mathematics for University Admission (TMUA), read 18 Oct 2026
official = a document published by the conducting body. reported = a news or coaching site we could not check against an original. Where sources disagree this page says so rather than picking one.
