Prepare for the TMUA in five stages: learn the format, check your knowledge against UAT-UK’s content specification, work through its Notes on Logic and Proof for Paper 2, sit past papers under timed conditions, and review every mistake. Everything you need is published free by UAT-UK, which does not endorse any tutoring service.
By Archie Dyer, founder of Cognivia Tutoring · Last checked 11 October 2026
Two papers, 75 minutes each20 multiple-choice questions per paper.
No calculator, no formula bookletFormulae have to be known.
A published specificationMostly AS-level pure and Higher GCSE mathematics.
Paper 2 is logic and proofUAT-UK publishes notes written for it.
Free official materialsPast papers from 2016 to 2023, with worked answers.
What is the best way to prepare for the TMUA?
The best way is the one UAT-UK, which runs the test, recommends: “read the test specification, Notes on Logic and Proof, and answer the practice test under timed conditions”. The plan below puts its free materials in the order I would use them. The facts are UAT-UK’s; the order and the advice are my own.
A five-stage TMUA preparation plan and the official free resource for each stage
Stage
What to do
Free official resource
1. Learn the format
Know the two papers, the timing and the rules before doing any questions.
Work out why each mark was lost, then go back to stage 2 or 3.
The worked answers published with each past paper.
Recommendation quoted from UAT-UK’s About the TMUA page; resources listed on its Prepare page. Checked on 11 October 2026.
Stage 1: what is the format of the TMUA?
The TMUA is two 75-minute papers of 20 multiple-choice questions each, taken one after the other on a computer at a test centre. Paper 1 is Applications of Mathematical Knowledge and Paper 2 is Mathematical Reasoning.
Calculator
Not allowed. UAT-UK: “You will not be able to use a calculator or dictionary.”
Formulae
The specification states: “There is no formulae booklet for this test; students are expected to understand and recall all relevant formulae.”
Marking
Questions across the two papers carry equal weight, and there is no penalty for a wrong answer.
Score
One overall score from 1 to 9, to one decimal place. There is no pass or fail.
Seventy-five minutes for 20 questions is 3 minutes 45 seconds a question on average, so pace has to be practised, and no question should be left blank.
Stage 2: what maths do you need to know for the TMUA?
You need the mathematics in Section 1 of the specification, which applies to both papers. UAT-UK says Part 1 “is almost all covered within the pure mathematics specification of an AS level in mathematics” and Part 2 “is almost all covered within a Higher Level GCSE mathematics course”.
Part 1: algebra and functions; sequences and series; coordinate geometry; trigonometry; exponentials and logarithms; differentiation; integration; graphs of functions.
Part 2: units; number; ratio and proportion; algebra; geometry; statistics; probability.
How I would approach it: print the specification and treat it as a checklist, marking each line as secure, shaky or not yet taught. “Almost all” is not all, so students on any curriculum, A-Level or IB, should read every line. The specification is also precise about limits: differentiation from first principles is excluded, for example.
For the shaky lines, UAT-UK publishes Notes on Mathematics for TMUA Paper 1 and ESAT Mathematics 2, set out in the same order as the specification. Its authors say: “You do not need to work through all this guide”.
Stage 3: how do you prepare for TMUA Paper 2 logic?
Prepare for Paper 2 by learning the ideas in Section 2 of the specification, using UAT-UK’s Notes on Logic and Proof. The specification says Paper 2 focuses on “the candidate’s ability to understand, and construct, mathematical arguments in a variety of contexts”, and Section 2 has three parts.
The logic of arguments: statements of the form “if A then B”, “A only if B” and “A if and only if B”; the converse and the contrapositive; necessary and sufficient; “for all”, “for some” and “there exists”; and how to negate each of them.
Mathematical proof: following direct proof, proof by cases, proof by contradiction and disproof by counterexample; deducing implications; and putting a list of statements into the right order to form a proof.
Identifying errors in proofs: spotting invalid steps, such as claiming that if ab = ac then b = c.
The specification adds that candidates “will not be expected to recognise or use symbolic notation for any of these terms, nor will they be expected to complete formal truth tables”. The notes are written, in UAT-UK’s words, “for the benefit of candidates who have not yet met them within their mathematics classes”. They warn that “skimming through everything is certainly not enough”, and advise reading “actively with a pencil and paper to hand”.
I would start this stage early. Everyday meanings get in the way: the notes point out that in mathematics “or” means one, or the other, or both.
Stages 4 and 5: where are the TMUA past papers, and how should you use them?
TMUA past papers are free on UAT-UK’s website. Its archive holds Paper 1 and Paper 2 for each year from 2016 to 2023, plus an early specimen paper, each with worked answers and an answer key. UAT-UK says: “Although the TMUA is now computer-based, both the content specification and question style are unchanged, so this archive of past papers can be used when preparing for the test.”
There are also a specimen test and a practice test that open in the Pearson test player, the software used at the test centre. UAT-UK recommends completing the practice test under timed conditions and viewing it on a desktop computer, not a phone.
Practice tests do not give a score. UAT-UK says this is intentional, because the score scale in any year is based on that year’s candidates, “with the median fixed at 4.5 and the 90th percentile fixed at 7.0”. It adds that “performance on a practice test will not necessarily be indicative of performance under exam conditions”. Be wary of anyone converting a practice mark into a predicted score.
How I would use them:
Sit papers whole and timed: 75 minutes, no calculator, no formula sheet.
Keep some papers unseen for the last few weeks, so there is still fresh material when it matters most.
Review before moving on. For each question that was wrong or guessed, decide whether the cause was a gap in knowledge, a misread question, a step of reasoning or time. Then read the worked answer and return to the specification or the logic notes.
UAT-UK is frank that the test is designed to be challenging: “You should not expect to score as highly on these tests as you would in your school exams.”
There is no official figure, and it depends on your starting point. UAT-UK says the TMUA “will use maths that you are likely to have already learned in your school studies, so you shouldn’t need to do too much additional preparation”.
My own judgement is that the sensible starting point is early enough to finish the specification check and the logic notes before the timed papers begin, with a little done each week instead of everything in the final fortnight. I would not trust a fixed number of hours from anyone, including me.
Quotation from UAT-UK’s Prepare page. Checked on 11 October 2026.
Do you need a tutor for the TMUA?
Not necessarily. UAT-UK states: “All necessary preparatory materials are available, free of charge, to all candidates”, and many candidates prepare with those alone. A tutor does not replace that work. What one-to-one help adds is someone to explain the worked answer that still does not make sense, to talk a Paper 2 argument through aloud, and to spot the habit that keeps costing time.
How Cognivia can help
Cognivia Tutoring is independent, and is not affiliated with or endorsed by UAT-UK, Pearson VUE or any university. A one-to-one lesson is a way to work through the official materials with someone: a worked answer that still does not make sense, or a Paper 2 argument talked through aloud. Archie Dyer, a second-year Theoretical Physics student at UCL and the only tutor at Cognivia, teaches the TMUA in one-hour lessons at £45 online on Google Meet or £55 in person around Marylebone, London. A 5-pack of five lessons saves £15, at £210 online or £260 in person, and a 30-minute online trial is free with no card needed. None of this is a promise about a score.
There is no official figure. UAT-UK says the TMUA uses maths you are likely to have already learned at school, so you should not need too much additional preparation. How long it takes depends on how secure the AS-level pure content is and on whether the logic in Paper 2 is new to you.
Yes. UAT-UK publishes an archive of TMUA Paper 1 and Paper 2 for each year from 2016 to 2023, plus an early specimen paper, with worked answers and answer keys. It also links to a specimen test and a practice test in the Pearson test player. All of it is free of charge.
Paper 2 is Mathematical Reasoning: 20 multiple-choice questions in 75 minutes. It uses the same mathematics as Paper 1, plus Section 2 of the specification: the logic of arguments, mathematical proof, and identifying errors in proofs. UAT-UK publishes Notes on Logic and Proof to prepare for it.
No to both. UAT-UK’s specification states that there is no formulae booklet and that students are expected to understand and recall all relevant formulae. Calculators may not be used.
No. UAT-UK says its specimen and practice tests show which answers were right and wrong but do not award a score, because the score scale in any year is based on that year’s candidates. It also says performance on a practice test will not necessarily indicate performance in the real test.
Not necessarily. UAT-UK makes all its preparation materials free, and many candidates prepare with those alone. A tutor does not replace that work; one-to-one help adds someone to explain the answers that still do not make sense, talk through Paper 2 arguments and spot habits that cost time.
Yes. Archie Dyer teaches the TMUA one to one in one-hour lessons, at £45 online or £55 in person around Marylebone, London, with a free 30-minute online trial. Cognivia Tutoring is independent of UAT-UK and makes no promise about scores.
Where to go next
The pages below cover the test itself, the dates and the tutoring.
