Two papers · Multiple choice · Pure mathematics
TMUA Preparation
The TMUA, the Test of Mathematics for University Admission, is two papers of twenty multiple-choice questions. The first tests whether you can apply mathematics you already know; the second tests whether you can reason about it: logic, proof, and necessary and sufficient conditions. The content is AS-level pure mathematics and Higher GCSE, with no mechanics, and there are no method marks. They are different skills, and most candidates prepare for only the first.
Taught by Lucas Rey · Oxford DPhil candidate, Rolls-Royce sponsored · Cambridge MRes
Not sure where to start? 5 places to go
Start with your question
Why visitors arrive: You have a TMUA requirement in an offer and want to know what the test is and how to prepare.
Your question: What does the TMUA test, and how should I prepare for it?
You may also be asking
- What is the difference between Paper 1 and Paper 2?
- How should I split my preparation between the two papers?
- Do I need Further Maths?
- Are there free practice papers?
- How is it different from the ESAT?
Where to go next
- The TMUA diagnostic: Fifteen questions, scored instantly, to find the weaker paper
- TMUA reasoning mock: The paper most candidates neglect
- TMUA applications mock: Twenty questions with worked solutions
- ESAT library: 150 hand-checked questions on overlapping content
- Enquire about one-to-one: Direct TMUA preparation
- What it is: the Test of Mathematics for University Admission, used by a number of UK universities for mathematics-heavy courses, including Cambridge Computer Science and Economics, and mathematics and economics courses at Imperial, Warwick, LSE and Durham among others.
- Published here: no questions yet: the questions written for it are held for review before they ship.
- Also here: the ESAT library, 150 questions written and checked by hand, which is the deeper resource on this site.
- Dates, deadlines and which course needs what: these change between cycles, so check the official specification rather than any tutor's summary of it.
- One-to-one preparation: by application, limited places.
What does the TMUA test?
[01]
Two papers, not one
Paper 1 is applications of mathematical knowledge. Paper 2 is mathematical reasoning: logic, proof, negation, necessary and sufficient conditions. The second is the one candidates walk into cold.
[02]
Multiple choice throughout
No method marks. An answer is right or it is not, which makes the arithmetic slip that would cost you one mark in an A-level paper cost you the whole question here.
[03]
Narrower than A-level Maths
The published content is AS-level pure mathematics plus Higher GCSE material, which brings in some probability and data handling and no mechanics at all. The narrowness is the point: it is what lets the questions be harder than the content is.
How do you prepare for the TMUA?
- Paper 2 first, not second. Paper 1 rewards fluency you are already building at school. Paper 2 rewards a way of thinking that school does not teach: what makes a counterexample valid, why a converse does not follow, when a condition is necessary rather than sufficient. Start there, because it is the part with the most room to improve.
- Learn to read the options. In a multiple-choice paper the options are information. A pair of options differing only by a sign tells you the examiner expects a sign error. Reading the five before starting the algebra often eliminates three.
- Never leave a blank. Eliminate what you can, then commit. An unanswered question cannot score, and narrowing five options to two before guessing is a skill in its own right. For the current marking rules, read the official specification rather than any tutor's summary of it.
Practice material on this site
The TMUA papers are being rebuilt on the same machine that produces the ESAT material, so they will be measured against the official material, mapped to the specification question by question, gated before they ship and read by a person, exactly as those are. Nothing is published here until they are. The ESAT papers and worked solutions are free and complete today, and are the same writing these will be.
Alongside the papers
- The TMUA method library Every method behind these solutions, with the questions that drill each one
- The TMUA diagnostic Fifteen questions across both papers, scored instantly
- The preparation plan Dated, working back from the sitting window
Found an error in one of them? Tell me and it will be fixed or withdrawn.
What the practice material covers
Counted from the questions themselves, not transcribed from a specification. For what is actually examinable in your cycle, check the official specification: it is revised between cycles and no tutor's summary of it should be trusted over the source.
| Topic | Questions |
|---|---|
| Conditional statements | 4 |
| Counterexamples | 4 |
| Negation | 4 |
| Algebra and functions | 3 |
| Necessary and sufficient conditions | 3 |
| Proof by contradiction | 3 |
| Counting and probability | 2 |
| Differentiation | 2 |
| Exponentials and logarithms | 2 |
| Flawed arguments | 2 |
| Integration | 2 |
| Quadratic equations and inequalities | 2 |
| Sequences and series | 2 |
| Trigonometry | 2 |
| Binomial expansion | 1 |
| Coordinate geometry | 1 |
| Graphs and transformations | 1 |
40 questions across 17 topics. Thin coverage of a topic here does not mean it is unimportant in the real test; it means this practice set is small.
Common questions
- What is the difference between TMUA Paper 1 and Paper 2? Paper 1 asks you to apply mathematics you already know. Paper 2 asks you to reason about mathematics: negation, counterexamples, necessary and sufficient conditions, proof by contradiction, and spotting why an argument fails. Paper 2 is the one most candidates walk into cold, because school does not usually teach it.
- Do I need Further Maths for the TMUA? No. The content is drawn from AS-level pure mathematics and is narrower than a full A-level Maths course. The difficulty is in how the questions combine that content, not in how advanced it is.
- How should I split my preparation between the two papers? Not evenly, and not in the order most candidates use. Paper 1 rewards fluency you are already building at school, so it improves slowly with practice. Paper 2 rewards a way of thinking school rarely teaches: what makes a counterexample valid, why a converse does not follow, when a condition is necessary rather than sufficient. That is where the room to improve is, so start there and come back to Paper 1.
- How is the TMUA different from the ESAT? The ESAT covers mathematics alongside physics, chemistry and biology, and is used mainly for engineering and science courses. The TMUA is pure mathematics only and is used for mathematics-heavy courses. The pure content overlaps, so ESAT practice is useful preparation for TMUA Paper 1.
- Are there practice papers on this site for the TMUA? Not at the moment. They are being rebuilt on the machine that produces the ESAT papers, and nothing is published here until that is done. The official past papers remain the final word on the real paper either way.
Sitting more than one of these?
Candidates often sit two admissions tests in the same cycle, and the overlap is larger than it looks: the pure mathematics is shared, and the habits that earn marks in one paper usually earn them in another. Each page below covers what its test rewards and what free practice exists for it.
- MAT: Mathematics Admissions Test Retired after 2025 and Oxford now uses the TMUA
- PAT: Physics Aptitude Test Retired after October 2025 and Oxford now uses the ESAT
- STEP: Sixth Term Examination Paper Cambridge and Warwick and STEP 2 and STEP 3 and Long-form proof
- ESAT: Engineering and Science Admissions Test The deepest library on this site, written and checked by hand
Part of
This page sits within the broader knowledge structure on lucasrey.com:
- Teaching - Engineering-informed teaching, mentoring, and quantitative reasoning.
Connected work
Related content from the same research and engineering work:
Where to go from here
Every paper, practice set and guide on this site is free to read. These are the things worth doing next.
- Find out where you actually stand - Fifteen questions, twenty minutes, scored instantly, with a worked solution for every one you miss
- Work a practice set at full paper depth - Ten sets of five questions, two in every module, each one worked in full
- See what can actually be checked about this tutor - What is published, counted from the pages themselves, and the credentials behind it
- Enquire about one-to-one preparation - For candidates who want the gap closed rather than mapped. Places are limited and taken by application
- Books for admissions test preparation - The ESAT Move Register, a book of shortcuts for the ESAT, with a preview of its pages
The diagnostic and the practice sets linked above are free and stay free. One-to-one places are limited and taken by application, not by the hour.