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?
- Is there negative marking?
- Do I need Further Maths?
- Are there free practice papers?
- How is it different from the ESAT?
Where to go next
- TMUA reasoning mock — The paper most candidates neglect
- TMUA applications mock — Twenty questions with worked solutions
- ESAT library — 1,053 hand-checked questions on overlapping content
- Enquire about one-to-one — Direct TMUA preparation
Two papers · Multiple choice · Pure mathematics
TMUA Preparation
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. They are different skills, and most candidates prepare for only the first.
Taught by Lucas Rey · Oxford DPhil candidate, Rolls-Royce sponsored · Cambridge MRes
- 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: 40 practice questions with worked solutions, free, no signup. AI-drafted, 30 of 40 answer keys machine-checked. Read the notice on the paper before using them.
- Also here: the ESAT library, 1,053 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 the TMUA actually tests
[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]
Pure mathematics only
No statistics, no mechanics. The content is narrower than A-level Maths, which is precisely why the questions can afford to be harder.
How to prepare for it
- 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.
- No negative marking means never leave a blank. Eliminate what you can and commit. A blank scores zero; a guess between two options scores half a mark on average.
Practice material on this site
These papers were drafted with AI assistance and have not been through the review process the ESAT material goes through. The answer keys were checked by computer algebra where the notice on each paper says so; the difficulty and the explanations were not. They are extra practice, not a predictor, and not a substitute for real past papers.
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.
- Is there negative marking? No. That makes leaving a question blank strictly worse than guessing: a blank scores zero, and a guess between two remaining options scores half a mark on average. Eliminating options and committing is a skill worth practising.
- 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 the practice papers on this site official? No. They are mock questions drafted with AI assistance, with the answer keys checked by computer algebra. Every page says so at the top. They are extra practice, not a predictor, and not a substitute for official past papers.
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 Oxford and Imperial and Multiple choice and long-form
- PAT: Physics Aptitude Test Oxford and Physics and mathematics
- STEP: Sixth Term Examination Paper Cambridge and Warwick 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: