We classified every question in the published TMUA papers from 2016 to 2023 — 320 questions across 16 papers — and split the result by paper. Paper 1 and Paper 2 are not the same exam.
Check where you stand, freeThe headline: 43.8% of TMUA Paper 2 is reasoning, not content. Logic of arguments, mathematical proof, and spotting errors in proofs together take up nearly half the paper — and that share has grown, from 38.8% across the 2016–2019 papers to 48.8% across 2020–2023. Almost none of it is assessed anywhere in a standard A-Level Maths course.
Most TMUA preparation advice treats the test as one thing. The papers say otherwise. Paper 1 is a fast, AS-level content paper where four topics carry nearly two-thirds of the questions. Paper 2 is substantially a reasoning paper wearing a maths paper's clothes — and it is the one that catches out candidates who are comfortable with the mathematics.
Below is the full breakdown for each paper: every topic, with its subtopics, as a share of that paper's questions. It is an aggregate analysis of the published papers, not a prediction of any future paper — but historical frequency is one of the more reliable signals available for deciding what to work on first.
There are two TMUA papers. You sit them one after the other in a single session, and together they produce one overall score on a scale from 1.0 to 9.0. Each paper is 75 minutes long with 20 multiple-choice questions, so the whole test is 40 questions in two and a half hours. There is no calculator and no formula booklet, every question carries equal weight, and a wrong answer costs nothing. That much is identical across the two papers. What each one asks of you is not.
| Paper 1 | Paper 2 | |
|---|---|---|
| Official title | Applications of Mathematical Knowledge | Mathematical Reasoning |
| What it tests | Applying your mathematical knowledge in new situations | Constructing and analysing mathematical arguments, plus simple ideas from elementary logic |
| Content you need | Section 1 of the specification: almost all AS-level pure maths, plus higher GCSE | Sections 1 and 2: the same maths, plus logic, proof and errors in proofs |
| Format | 20 multiple-choice questions, 75 minutes | 20 multiple-choice questions, 75 minutes |
| Biggest topics, 2016–2023 | Algebra and functions, sequences and series, integration and trigonometry: 64.4% of questions | Logic of arguments, mathematical proof and errors in proofs: 43.8% of questions |
| Largest single subtopic | Applications of integration, 11.9% | Reasoning terms (necessary and sufficient, implication), 17.5% |
The point most candidates miss: Paper 2 is not a logic paper with the maths taken out. The specification requires the whole of Section 1 for Paper 2 as well, and in the published papers more than half of Paper 2 (90 of its 160 questions) was on Section 1 topics — algebra, sequences, trigonometry and calculus, asked in a way that turns on the reasoning. So preparing for Paper 1 is also preparing for Paper 2, but not the other way round. For the reasoning half, our TMUA logic and proof notes teach each skill with a worked example.
Where the past papers are. UAT-UK publishes all sixteen papers from 2016 to 2023 free, each with worked answers and an answer key, plus two early specimen papers, in its archive of TMUA past papers. Those sixteen papers are exactly the set classified below.
The three reasoning topics at the top of this table are the ones without an A-Level equivalent. Reasoning terms alone — necessary versus sufficient conditions, implication, equivalence — is 17.5% of the paper, the single largest subtopic in either paper. If you prepare for Paper 2 the way you prepare for a maths paper, this is the part you will not have practised. We cover exactly this — necessary vs sufficient, implication vs the converse, negating quantifiers, choosing a proof method, and finding the flawed step — with full worked examples on how to answer TMUA Paper 2.
| Topic / subtopic | Questions | Share |
|---|---|---|
| Logic of Arguments | 35 | 21.9% |
| Reasoning terms | 28 | 17.5% |
| Logical connectives | 7 | 4.4% |
| Mathematical Proof | 19 | 11.9% |
| Proof methods | 14 | 8.8% |
| Constructing arguments | 5 | 3.1% |
| Algebra & Functions | 17 | 10.6% |
| Quadratics | 5 | 3.1% |
| Equations & inequalities | 5 | 3.1% |
| Functions | 3 | 1.9% |
| Indices & surds | 2 | 1.2% |
| Manipulating expressions | 2 | 1.2% |
| Errors in Proofs | 16 | 10.0% |
| Spotting flawed reasoning | 16 | 10.0% |
| Sequences & Series | 15 | 9.4% |
| Series | 6 | 3.8% |
| Sequences | 6 | 3.8% |
| Binomial expansion | 3 | 1.9% |
| Trigonometry | 10 | 6.2% |
| Trig equations | 3 | 1.9% |
| Trig functions & identities | 3 | 1.9% |
| Right-angled & triangle trig | 2 | 1.2% |
| Radians & exact values | 2 | 1.2% |
| Integration | 9 | 5.6% |
| Applications of integration | 6 | 3.8% |
| Integrating | 3 | 1.9% |
| Geometry & Measures | 8 | 5.0% |
| Angles & polygons | 4 | 2.5% |
| Congruence & similarity | 2 | 1.2% |
| Mensuration & 3D shapes | 1 | 0.6% |
| Pythagoras & coordinates | 1 | 0.6% |
| Differentiation | 8 | 5.0% |
| Derivatives | 5 | 3.1% |
| Applications of differentiation | 3 | 1.9% |
| Graphs & Transformations | 6 | 3.8% |
| Transformations | 3 | 1.9% |
| Sketching graphs | 3 | 1.9% |
| Coordinate Geometry | 6 | 3.8% |
| Straight lines | 4 | 2.5% |
| Circles | 2 | 1.2% |
| Exponentials & Logarithms | 6 | 3.8% |
| Logarithms | 4 | 2.5% |
| Exponentials | 2 | 1.2% |
| Number | 2 | 1.2% |
| Probability | 2 | 1.2% |
| Statistics | 1 | 0.6% |
Splitting Paper 2 by year shows the drift clearly. Each paper has 20 questions; the count below is how many of them sat in the three reasoning topics.
| Year | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|---|---|---|
| Reasoning questions | 6 | 8 | 9 | 8 | 11 | 8 | 10 | 10 |
| Share of paper | 30% | 40% | 45% | 40% | 55% | 40% | 50% | 50% |
Averaged across the first four years, 38.8% of Paper 2 was reasoning. Across the last four, 48.8%. The year-to-year figure moves around, so no single paper should be read as a trend — but the direction across eight years is consistent, and it points the same way the test's own stated purpose does.
Paper 1 is far more concentrated than Paper 2. Four topics — algebra and functions, sequences and series, integration, and trigonometry — account for 64.4% of the questions. The tail is very thin: probability, statistics and ratio contribute one question each across all eight years combined.
| Topic / subtopic | Questions | Share |
|---|---|---|
| Algebra & Functions | 29 | 18.1% |
| Quadratics | 14 | 8.8% |
| Equations & inequalities | 8 | 5.0% |
| Manipulating expressions | 5 | 3.1% |
| Indices & surds | 1 | 0.6% |
| Functions | 1 | 0.6% |
| Sequences & Series | 25 | 15.6% |
| Series | 14 | 8.8% |
| Binomial expansion | 8 | 5.0% |
| Sequences | 3 | 1.9% |
| Integration | 25 | 15.6% |
| Applications of integration | 19 | 11.9% |
| Integrating | 6 | 3.8% |
| Trigonometry | 24 | 15.0% |
| Trig equations | 13 | 8.1% |
| Right-angled & triangle trig | 6 | 3.8% |
| Trig functions & identities | 5 | 3.1% |
| Differentiation | 18 | 11.2% |
| Applications of differentiation | 15 | 9.4% |
| Derivatives | 3 | 1.9% |
| Coordinate Geometry | 14 | 8.8% |
| Circles | 10 | 6.2% |
| Straight lines | 4 | 2.5% |
| Exponentials & Logarithms | 10 | 6.2% |
| Logarithms | 9 | 5.6% |
| Exponentials | 1 | 0.6% |
| Graphs & Transformations | 10 | 6.2% |
| Sketching graphs | 6 | 3.8% |
| Transformations | 4 | 2.5% |
| Geometry & Measures | 2 | 1.2% |
| Ratio & Proportion | 1 | 0.6% |
| Probability | 1 | 0.6% |
| Statistics | 1 | 0.6% |
The tables above pool eight years. Split by sitting, the same 320 questions show how much any single paper can differ from the average — useful when you sit a past paper as a mock and want to know how typical it was. Each column is one real paper of 20 questions.
| Topic | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | Total |
|---|---|---|---|---|---|---|---|---|---|
| Algebra & Functions | 1 | 3 | 3 | 3 | 10 | 3 | 3 | 3 | 29 |
| Integration | 1 | 3 | 2 | 5 | 2 | 5 | 3 | 4 | 25 |
| Sequences & Series | 4 | 4 | 3 | 3 | 2 | 2 | 3 | 4 | 25 |
| Trigonometry | 4 | 2 | 4 | 3 | 1 | 3 | 4 | 3 | 24 |
| Differentiation | 5 | 3 | 4 | 2 | 1 | 1 | 1 | 1 | 18 |
| Coordinate Geometry | 1 | 3 | 2 | 1 | 1 | 2 | 2 | 2 | 14 |
| Exponentials & Logarithms | 2 | 1 | 1 | 2 | 1 | 1 | – | 2 | 10 |
| Graphs & Transformations | – | 1 | – | 1 | 2 | 2 | 3 | 1 | 10 |
| Geometry & Measures | – | – | – | – | – | 1 | 1 | – | 2 |
| Probability | 1 | – | – | – | – | – | – | – | 1 |
| Ratio & Proportion | 1 | – | – | – | – | – | – | – | 1 |
| Statistics | – | – | 1 | – | – | – | – | – | 1 |
| Questions | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 160 |
| Topic | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | Total |
|---|---|---|---|---|---|---|---|---|---|
| Logic of Arguments | 2 | 3 | 3 | 3 | 7 | 4 | 6 | 7 | 35 |
| Mathematical Proof | 3 | 2 | 4 | 3 | 2 | 2 | 2 | 1 | 19 |
| Algebra & Functions | 2 | 1 | 2 | 5 | 2 | 2 | 1 | 2 | 17 |
| Errors in Proofs | 1 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 16 |
| Sequences & Series | 2 | 3 | 3 | 2 | 1 | – | 2 | 2 | 15 |
| Trigonometry | 1 | 1 | 2 | – | 2 | 2 | 2 | – | 10 |
| Integration | 1 | 1 | – | 1 | 1 | 2 | 1 | 2 | 9 |
| Differentiation | 1 | 1 | 2 | 1 | – | 1 | 1 | 1 | 8 |
| Geometry & Measures | 4 | – | 1 | – | 1 | – | 1 | 1 | 8 |
| Coordinate Geometry | – | 1 | – | – | – | 3 | 1 | 1 | 6 |
| Exponentials & Logarithms | – | 1 | – | 1 | – | 2 | 1 | 1 | 6 |
| Graphs & Transformations | 3 | – | 1 | 1 | 1 | – | – | – | 6 |
| Number | – | 2 | – | – | – | – | – | – | 2 |
| Probability | – | 1 | – | 1 | – | – | – | – | 2 |
| Statistics | – | – | – | – | 1 | – | – | – | 1 |
| Questions | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 160 |
Single papers are lumpy. The largest concentration in any one paper was algebra & functions in the 2020 Paper 1: 10 of its 20 questions. The clearest drift in Paper 1 is differentiation, which fell from 14 questions across the 2016–2019 papers to 4 across 2020–2023. Neither is a forecast — but both are a reason to sit several years of papers rather than one, and to read any single mock score against the paper it came from.
Do not split your time evenly between the papers. They reward different work. Paper 1 rewards fluency and speed on a narrow, predictable set of AS-level techniques — and because applications of integration and applications of differentiation are the two biggest subtopics, that means calculus applied to areas, volumes, rates and stationary points, not the differentiation and integration rules in isolation.
Paper 2 rewards something your A-Level does not teach. Nearly half of it is formal reasoning: deciding whether a condition is necessary, sufficient, both or neither; recognising which proof method a statement calls for; and reading a flawed proof closely enough to name the line where it breaks. Those are trainable skills, but only if you know to train them. A candidate who spends their preparation revising AS content is preparing thoroughly for a little over half of Paper 2.
The thin tail is a genuine saving. Probability and statistics have contributed two questions in total across all sixteen published papers. That is a real permission to deprioritise, and it is the kind of decision this analysis exists to support.
Want to know which of these you are actually weak on? Our free TMUA readiness check is a short adaptive set of TMUA-style questions. It gives you a readiness band and a topic-by-topic map across both papers, so you can point this breakdown at your own gaps rather than guessing.
Two. Paper 1 (Applications of Mathematical Knowledge) and Paper 2 (Mathematical Reasoning) are sat one after the other in a single session and together produce one overall score on a 1.0 to 9.0 scale. Each paper is 75 minutes long with 20 multiple-choice questions.
40 in total: 20 multiple-choice questions in each paper, all carrying equal weight. There is no penalty for a wrong answer, no calculator and no formula booklet.
Paper 1 tests applying mathematical knowledge — almost all AS-level pure maths, plus higher GCSE — in unfamiliar situations. Paper 2 tests mathematical reasoning: it assumes the same maths and adds logic, proof and spotting errors in proofs. In the published papers, 43.8% of Paper 2 was reasoning and the rest (90 of 160 questions) was Section 1 maths content.
Four topics account for 64.4% of Paper 1: algebra and functions (18.1%), sequences and series (15.6%), integration (15.6%) and trigonometry (15.0%). The largest single subtopic is applications of integration, at 11.9%.
Across the published papers from 2016 to 2023, 43.8% of Paper 2 questions sit in the three reasoning topics: logic of arguments (21.9%), mathematical proof (11.9%) and identifying errors in proofs (10.0%). The single largest subtopic is reasoning terms — necessary and sufficient conditions, implication and equivalence — at 17.5% of the paper.
It has been getting more reasoning-heavy. In the 2016 to 2019 papers, 38.8% of Paper 2 questions were reasoning questions. In the 2020 to 2023 papers that rose to 48.8%. Pure content knowledge covers a shrinking share of the paper.
The mathematical content stays within AS-level material, but Paper 2 tests formal reasoning that is not assessed anywhere in a standard A-Level Maths course — necessary versus sufficient conditions, the logical structure of a proof, and locating the exact step where a flawed proof fails. That is why strong A-Level candidates are often surprised by Paper 2.
UAT-UK publishes every paper from 2016 to 2023 free, each with worked answers and an answer key, plus two early specimen papers, in its archive of TMUA past papers. Specimen and timed practice tests in the on-screen test player are on its preparation page.
The breakdown tells you what the test asks. The free check tells you where you are against it — and a tutor turns that into a route with an order and a deadline.
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