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.
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% |
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.
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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