Original TMUA practice papers, with every answer worked.
The real numbers and satisfy and . What is the value of ?
Use the identity . Then . A common slip is to subtract only once, giving . (Indeed , works: .) So the answer is C.
Let be a real number. Which of the following is the contrapositive of the statement 'if , then '?
The contrapositive of 'if then ' is 'if not then not '. Here is and is , so the contrapositive is 'if , then '. It is logically equivalent to the original statement. 'If then ' is the converse, and 'if then ' is the inverse; both are false (try ). ' and ' is the negation. So the answer is D.
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2 full mock papers
40 questions in the real format: 20 per paper, 75 minutes each, no calculator, with a worked solution for every question.
120 more questions
A bank for weak-spot drills and for redoing the ones you got wrong, so you keep practising after the two papers.
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Fresh Paper 1 and Paper 2 style questions made on demand, leaning towards your weakest topics. Do as many sets as you like.
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Here is what we built that you will not always find in a question bank, and where we are smaller.
Every wrong answer explained
Every paid question tells you why each wrong option is wrong, and the slip that leads to it. You learn from the mistake, not just the right answer.
Checked twice, in the open
Each answer is recomputed by a script and solved again independently before it goes live. Our accuracy page says how, and where we may still be wrong. Every question has a Report a mistake link.
Practice that does not run out
When the papers are done, unlimited generated sets for both Paper 1 (applying maths) and Paper 2 (logic and proof) keep coming, aimed at your weakest topics. Every generated answer is calculated by code, so it is right.
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The weak-spot drill brings back what you got wrong and what you guessed, and the pacing panel shows where the clock beat you. It all stays on your device, with no account.
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We have 168 hand-written original questions, fewer than the biggest banks, and no teacher has marked them yet. We put the effort into explanations, checking and unlimited practice instead.
The TMUA in one minute
Paper 1: Applications of Mathematical Knowledge
20 multiple-choice questions in 75 minutes. Applying AS-level pure maths and GCSE Higher content in new situations.
Paper 2: Mathematical Reasoning
20 multiple-choice questions in 75 minutes. Logic, proof and spotting errors in arguments, using the same maths.
No calculator on either paper and no penalty for wrong answers, so attempt every question. Source: UAT-UK: TMUA.
Free, or Full Access for £9.99
| What you get | Free | £9.99 |
|---|---|---|
| Original questions | 8 | 168 |
| Full papers at the real length (20 questions, 75 minutes) | No | Yes Two papers |
| Drill bank for weak spots and mistakes | No | Yes 120 questions |
| Worked solution for every question | Yes | Yes |
| Timed on the real clock, no calculator | Yes | Yes |
| Score by topic and practice band | Yes | Yes |
| Unlimited generated practice sets | No | Yes |
| Paper 1, Paper 2, or both back to back | No | Yes |
| Progress kept on this device | Yes | Yes |
| Account or sign-up needed | None | None |
| Cost | £0 | £9.99 once |
| How long it lasts | No limit | 180 days, no renewal |
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Score calculator
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Past papers index
Every official past paper from 2016 to 2023 with worked answers, linked to UAT-UK's own downloads.
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Which courses at Cambridge, Oxford, Imperial, LSE, Warwick, UCL and Durham want the TMUA for 2027 entry.
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What do I get free?
A sample paper of 8 original questions (4 from Paper 1, 4 from Paper 2) timed at the real rate of 3 minutes 45 seconds a question, with a worked solution for every one. No sign-up. The free tools (score explainer, past papers index, universities and dates) are free too.
What does £9.99 buy?
Full Access: two full original papers of 20 questions each, timed at 75 minutes per paper, with a worked solution for every question, a score by topic and a practice band. One payment, 180 days of access, no subscription.
Are these the real TMUA questions?
No. Every question is original and written from the public content specification. We never use official papers or recalled live-test content. UAT-UK publishes its own past papers and practice tests free, and our past papers index links to them.
Do the papers match the real format?
Yes. Paper 1 and Paper 2 each have 20 multiple-choice questions in 75 minutes, with no calculator and no penalty for wrong answers, as on UAT-UK's TMUA page. The real test is taken on a computer at a Pearson VUE test centre.
Will I get a TMUA score out of 9?
No, and nobody can give you an official one from practice. UAT-UK does not publish raw-mark conversion tables because the 1.0 to 9.0 scale is set for each sitting. You get your marks, a score by topic and a rough practice band. Our score explainer shows what the official scale does and does not tell you.
How accurate are the answers?
Each question's answer is recomputed by a script and the question is solved a second time independently before it goes live. We still cannot promise it is error free, and every question has a Report a mistake link that we check.
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Payments should be made by a parent, guardian or other adult. Ask them to pay at checkout. The free sample needs no payment and no sign-up.
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