TMUA past papers: official papers from 2016 to 2023, worked answers and method

TMUA past papers (Test of Mathematics for University Admission) are published for free by UAT-UK, the organisation that runs the test: eight years of papers, from 2016 to 2023, plus a specimen paper, each with an answer key and worked answers for every paper. This page brings them together in one place, explains the order in which to use them and how to mark them, and suggests a schedule for the next sitting, from 4 to 8 January 2027.

One thing to bear in mind from the start: UAT-UK states that the TMUA is now taken on computer, but that the content specification and the style of questions are unchanged. The paper-based tests from previous years therefore remain the best practice material available. For everything about the test itself (format, universities, dates, fees), start with our complete guide to the TMUA.

 
 
 
 
  
 
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I wanted to express my immense gratitude to My Dream School for their support throughout Grégoire's university application process in England and Spain. From the very beginning, they were present, available, and remarkably professional. Thanks to their guidance, expertise, and support, Grégoire was delighted to receive offers of admission from prestigious universities such as UCL, King's College London, and IE University. We would especially like to thank Adam for his support during the release of the Baccalaureate results. Grégoire's score on the Grand Oral exam, well below our expectations, jeopardized several of his admissions, particularly that of IE, which requires a minimum Baccalaureate level. This was a particularly stressful time for our entire family. During this difficult period, My Dream School provided invaluable support. They guided us calmly and efficiently, prepared the necessary letters, advised us on the steps to take, and helped us present the situation in the best possible light. Their responsiveness, experience, and commitment made all the difference. Today, Grégoire is preparing to join IE University, after considering King's College London, where he had also been accepted. This is a fantastic opportunity for him, and we know that your support greatly contributed to this success. Thank you, Adam and Laurence, for your availability and dedication. We are very grateful. Grégoire is thrilled to begin this new adventure, and we share his joy completely.

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I am so glad I chose to work with Your Dream School. Thanks to their support, I was able to gain admission to universities like IE University and ESCP Business School, as well as other programs through UCAS. I received exceptional support from Adam and Lawrence, who played a vital role throughout the entire process. They were always available, incredibly dedicated, and encouraged me to give my best. I highly recommend Your Dream School to anyone wishing to gain admission to selective international universities. Their dedication and expertise truly make a difference.

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All official TMUA papers

Each row corresponds to one sitting. Paper 1 (Applications of Mathematical Knowledge) and Paper 2 (Mathematical Reasoning) each have 20 multiple-choice questions, to be answered in 75 minutes without a calculator. The answer keys give the correct answer to each question; the worked answers explain the reasoning expected.

Source: UAT-UK official past paper archive. The files are hosted by UAT-UK; we do not modify them.

Beyond these past papers, UAT-UK publishes the TMUA content specification, the Notes on Logic and Proof for Paper 2, Notes on Mathematics shared by the TMUA and the ESAT, and practice tests in the Pearson test player, the software used at the test centre.

In what order to use the past papers

Eighteen papers is both a lot and not very many. A lot if you work through them one after another with no method; not many if you want to keep two or three “fresh” papers for the final weeks. Here is the order we follow with our students.

1. The specimen paper, untimed. This is your diagnostic. Do both papers, taking as long as you need, then note for each question whether the mistake came from a forgotten concept, a calculation error or misreading the question. This list shapes everything that follows.
2. 2016 to 2019, topic by topic. Go back through these papers question by question, grouping the questions by section of the specification (algebra, functions, sequences, trigonometry, probability, logic). This is the time to fill the gaps identified in the diagnostic.
3. 2020 and 2021, timed, one paper at a time. One 75-minute paper at a time, timed, without a calculator. The goal is time management: spotting the long questions, knowing when to skip them and come back later.
4. 2022 and 2023, under exam conditions. Both papers on the same day, 75 minutes each, as at the test centre. Save them for the last two weeks.
5. Last, the Pearson test player practice tests. They are not for learning content but for getting to know the interface: moving between questions, flagging questions to come back to, accessibility tools. UAT-UK recommends taking them on a computer, not a phone, and under timed conditions.

How to mark a TMUA paper

The answer key isn’t enough. It tells you whether an answer is right, but not why you got it wrong, or whether you reached a correct answer by the quickest route. Useful marking takes three steps.

First, the answer key, for your tally. Count your correct answers for each paper, but don’t convert them into a score out of 9: UAT-UK publishes no conversion table (see our page on the TMUA score). Instead, track your number of correct answers from one past paper to the next.

Next, the worked solutions, for every question. That includes the ones you got right. On the TMUA, many questions have a long route (calculating) and a short one (reasoning about the form of the answers, eliminating options). The official worked solutions often show the short route. Learning to spot it earns you more marks on test day than getting the answer right slowly.

Finally, an error log. One line per mistake: the year, the question, the topic and the cause (concept, calculation, misreading, time). After three or four papers, the same causes keep coming back. Those are what you need to work on, not the next paper.

Paper 1 and Paper 2: what the past papers show

Paper 1 applies UK secondary-school knowledge (GCSE Higher and AS level, as set out in the specification) to unfamiliar situations. For a final-year student taking advanced mathematics, the content is rarely the problem; the challenge is working quickly without a calculator. The 2016 to 2019 papers are usually enough to rebuild your fluency.

Paper 2 tests mathematical reasoning: basic logic, necessary and sufficient conditions, negating a statement, and the structure and validity of a proof. It is the least familiar part for candidates educated outside the UK system, and the one where practice pays off most. Before tackling Paper 2, read the Notes on Logic and Proof published by UAT-UK: they define the exact terminology used in the questions.

The two papers are then scaled together to produce a single score, so a weak Paper 2 drags down your overall result just as much as a weak Paper 1.

A six-week plan before the January 2027 sitting

The October 2026 sitting, which most Oxford and Cambridge applicants must take, has passed. The second sitting runs from 4 to 8 January 2027; LSE, for example, accepts it and states that either sitting is suitable for its applicants. Booking opens on 26 October 2026 and closes on 21 December 2026 at 6 pm UK time; results are released on 8 February 2027. Check each course’s page to confirm that the January sitting is accepted for you.

Book your slot before you start this plan: places at Pearson VUE test centres fill up, and the 21 December deadline falls during the holidays.

Past-paper pitfalls

• Redoing a paper you’ve already seen. The second time round, you remember the answers. Keep 2022 and 2023 untouched for your mock exams.
• Practising with a calculator. None is allowed in the test. Mental arithmetic and estimation are part of what is assessed.
• Trusting an “estimated score”. Some websites give a score out of 9 based on your number of correct answers. UAT-UK publishes no mark scheme for this, so these figures are not official.
• Neglecting the interface. On test day, the questions appear on screen. Skipping a question, flagging it and coming back to it are all things to practise on the Pearson practice tests.
• Using unofficial past papers. Only the official papers published by UAT-UK reflect the real level and style. “Mock tests” found online may be easier or harder.

Preparing with a tutor

The past papers are free and enough for a highly self-directed student. What a tutor adds is a diagnosis of your mistakes, the short route for each type of question, and focused work on Paper 2. Our TMUA preparation is one-to-one, by video call: €1,500 incl. VAT for 10 hours, €2,800 incl. VAT for 20 hours, €3,900 incl. VAT for 30 hours. Full details are on our TMUA page and on the Oxford and Cambridge admissions tests page.

TMUA past papers: frequently asked questions

Where can you find official TMUA past papers?

UAT-UK, the organisation that runs the test, publishes the following free of charge on esat-tmua.ac.uk: the 2016 to 2023 papers (Paper 1 and Paper 2), a specimen paper, the answer keys for each year and worked solutions for each paper. These are the only official past papers; every link on this page points to these files.

Are pre-2024 past papers still relevant?

Yes, for the content. UAT-UK states that the TMUA is now taken on computer, but the content specification and question style are unchanged. What has changed is the interface, so also work through the official practice tests in the Pearson test player, the software used at the test centre.

Can you convert your number of correct answers into a TMUA score?

No, not officially. The score from 1.0 to 9.0 is calculated using a statistical model (Rasch) that scales the two papers together, and UAT-UK publishes neither a conversion table nor a mark scheme. Any score given on the basis of the number of correct answers on a past paper is only an estimate.

How many past papers should you do before the test?

There are nine sets of official papers (the specimen and 2016 to 2023), making eighteen papers in total. A realistic plan uses all of them: the oldest by topic, the most recent under timed exam conditions. What matters is not the number of papers but working through every mistake with the worked solutions.