Where to focus your TMUA prep
Around 87% of TMUA questions come from a handful of repeatedly-tested topics. Use the tiers below to allocate prep time without over-investing in the long tail.
What the TMUA is
The Test of Mathematics for University Admission is a 2.5-hour computer-based admissions test administered by UAT-UK and delivered by Pearson VUE. It has two 75-minute multiple-choice papers, each scored 1 to 9, no calculator allowed.
The maths content is drawn from A-Level Pure topics, but the questions test those topics in unfamiliar ways. A score of around 6.5+ on each paper is generally considered competitive at Cambridge and Oxford.
Who needs it
- Cambridge Mathematics, Computer Science, Economics
- Oxford Mathematics, Maths & Statistics, Maths & Computer Science, Maths & Philosophy, Computer Science (from 2026 entry)
- LSE Mathematics, Economics, Data Science
- Warwick Mathematics, MORSE
- Durham Mathematics, Computer Science
- Others Bath, Lancaster, Sheffield, and a growing list. Full course list →
Why this matters: the prep window is shorter than you think
Source: UAT-UK key dates page — always confirm with the official site, deadlines can shift.
Whichever sitting you target, realistic prep is a few months of part-time work — fitted around a long list of competing commitments:
There is no realistic scenario where you cover the entire 18-topic syllabus with equal depth. The tiers below are a way to spend a limited budget well:
Cover the whole syllabus, but weight your hours toward what has historically been tested most.
A quick primer: TMUA Paper 1 and Paper 2
The TMUA has two papers, each 75 minutes long with 20 multiple-choice questions and no calculator allowed. Each is scored 1 to 9 independently. Both papers draw on the same A-Level Pure Maths topic list, but they reward different skills.
Paper 1 — Mathematical Thinking
A-Level Pure Maths applied in unfamiliar contexts: algebra, calculus, trigonometry, coordinate geometry, sequences. Almost no logic or proof.
Paper 2 — Mathematical Reasoning
Same maths content as Paper 1 plus formal logic, mathematical proof, and identifying flaws in arguments — topics A-Level Maths does not formally teach. This is where most unprepared candidates lose marks.
The percentages on this page combine both papers. The dedicated "Paper 1 vs Paper 2" section further down shows how the topic mix differs between them.
Four things the data makes clear
Logic and proof are the largest topic, and they live entirely in Paper 2.
About 39% of Paper 2 is logic, mathematical proof, or identifying flaws in proofs. Paper 1 has almost none. A-Level Maths does not formally teach these, so candidates often underprep them — and lose marks in the easiest place to recover them.
Four topics carry half the test.
The Tier 1 four (The Logic of Arguments, Algebra and functions, Integration, Trigonometry) cover about 49% of all TMUA questions between them. If you only have time for serious depth on four topics, those are the four.
Integration appears more often than differentiation.
The reverse of how A-Level Pure exams usually weight calculus. The reason is structural: integration questions are easier to compress into a 90-second MCQ (often with a substitution or inspection trick), while differentiation questions tend to need multiple algebraic steps to reach a clean answer.
The GCSE-prerequisite list is mostly decorative.
The TMUA spec assumes GCSE-level material (basic geometry, ratio, statistics) as background. In practice, dedicated GCSE-style questions barely appear in past papers — these topics show up only as supporting steps inside A-Level-style questions. No need to schedule dedicated prep time at the GCSE level.
The core of the TMUA
These topics dominate every TMUA paper from 2016 onwards. Each appears multiple times per sitting, and together they account for roughly half of every test. They are also where the examiners place their trickiest, time-consuming questions — the ones that separate competitive candidates from the rest. If you only have time for one block of deep practice, make it these. Aim for fluent, automatic recall: you should be able to start a question of any of these types within seconds of reading the stem.
The Logic of Arguments
Typical style: a mathematical statement is given, and the question asks which of several other statements is a sufficient condition, a necessary condition, or logically equivalent. Quantifier scope (for all vs there exists) and the difference between converse and contrapositive carry most of the marks. Distractors are designed to catch students who quietly swap "sufficient" for "necessary".
Algebra and functions
Typical style: a polynomial inequality, a simultaneous-equation system, or a factor/remainder-theorem question where the answer is a range or set of values rather than a single number. Surds and rational indices often appear inside. Common trap: squaring an inequality and forgetting to discard extraneous roots.
Integration
Typical style: a definite integral involving a parameter, or an area-between-curves problem. The clean answer is rarely the one you reach by mechanical antiderivative — there is usually a substitution or symmetry argument that shortens the working. Common trap: evaluating at only one limit.
Trigonometry
Typical style: solve a trig equation in a restricted interval, where you first need an identity (Pythagorean, double-angle, sum-to-product) to isolate the variable. Radians are standard. Common trap: missing one of the solutions in the period, or applying an identity in the wrong direction.
Reliable supporting topics
These topics appear on most papers but in smaller doses — usually one or two questions per sitting each. The examiners often use them as the bank-the-marks questions: pitched at the easier end of the difficulty range and designed for candidates who recognise the pattern quickly. Skipping them entirely leaves easy marks on the table; over-investing here costs you Tier 1 hours that matter more. Aim for confident recognition and clean standard technique, not depth.
Sequences and series
8.1%Typical style: a mixed problem combining arithmetic and geometric progressions, or a sum-to-infinity of a geometric series with a non-obvious common ratio. Common trap: applying the sum-to-infinity formula without checking |r| < 1.
Differentiation
7.8%Typical style: locate or classify stationary points of an awkward function (often involving rational exponents), or read a graph of f'(x) and infer something about f(x). Common trap: confusing inflection points with extrema.
Coordinate geometry in the (x,y)-plane
7.5%Typical style: a problem involving the intersection of a line and a circle, the tangent to a circle at a point, or the distance between two circles. Circle theorems (perpendicular-bisector-of-chord, tangent perpendicular to radius) often unlock the question without heavy algebra.
Graphs of functions
7.5%Typical style: a sequence of transformations is applied to a known graph (often y = x^3 or a function given graphically), and you pick which of several graphs is the result. Common trap: applying the transformations in the wrong order, especially when both x- and y-axis scalings are involved.
Exponentials and logarithms
6.6%Typical style: solve an equation that requires changing the base of a logarithm, or simplify an expression mixing a^x and log_a notation. The laws log_a(xy) = log_a x + log_a y and a^b = c ⇔ b = log_a c are the workhorses. Common trap: treating log(x+y) as log x + log y.
Low priority, but don't skip entirely
These topics appear, just rarely. A quick pass for awareness is enough; deep prep costs more time than it saves relative to Tier 1 and Tier 2.
Paper 1 vs Paper 2 are different beasts
Both papers cover the same A-Level Pure Maths topic list on paper. In practice, the topic mix is very different. About 39% of Paper 2 is logic, proof, and identifying flaws — topics that barely appear in Paper 1. Paper 1 leans heavily on calculus, algebra, and trigonometry; Paper 2 redistributes time toward the reasoning topics that A-Level Maths does not formally teach.
Treating the two papers as interchangeable — using the same prep mix for both — is the most common preparation mistake. The bar charts below show how each paper's topic distribution actually splits.
Paper 1 — Mathematical Thinking
Almost zero logic or proof. Almost entirely pure maths fluency.
- Algebra and functions 20.0%
- Integration 15.6%
- Trigonometry 13.1%
- Differentiation 11.3%
- Sequences and series 10.0%
- Coordinate geometry in the (x,y)-plane 8.8%
Paper 2 — Mathematical Reasoning
About 39% is logic, proof, or spotting errors in proofs — mostly absent from Paper 1. This is where unprepared candidates lose marks.
- The Logic of Arguments ★ 30.0%
- Identifying Errors in Proofs ★ 8.8%
- Integration 6.9%
- Trigonometry 6.9%
- Algebra and functions 6.3%
- Sequences and series 6.3%
★ logic / proof topics
Where to practice this style
Eight TMUA past papers are not a lot of practice volume. The four open resources below cover the same style and difficulty band — useful for drilling individual topics once you have worked through the official TMUA archive itself.
Oxford MAT Question 1 archive (2007 to 2025)
19 years of compulsory MAT MCQs — the closest direct ancestor of TMUA Paper 1, freely available with official solutions.
Underground Mathematics
University of Cambridge open resource. Rich problems by topic with full solutions and discussion.
STEP Question Database (1986 to 2018)
Searchable by topic. STEP is harder than TMUA but the underlying maths overlaps.
NRICH
University of Cambridge problem-solving collection. Look at Stage 5 ("Sixth Form") for TMUA-relevant material.
Method & sources
- Source papers. Every TMUA past-paper question from 2016 to 2023 (Paper 1 + Paper 2) is included. PDFs are hosted by UAT-UK and linked from this site — they are not mirrored locally. Early specimen papers were processed for completeness but are excluded from the percentages above.
- Topic taxonomy. Topics and their definitions are taken directly from the official TMUA Content Specification (April 2025) as a closed set. We did not invent or merge topics.
- Matching questions to topics. Each past-paper question was matched to its primary topic from the official syllabus list. Percentages reflect the count of primary-topic matches as a share of all questions. Accuracy caveat: we try our best to infer the topic from each question, but the labelling is not hand-graded and individual tags may be wrong. The aggregate percentages and tier rankings are robust because they average over hundreds of questions, but if you click through to a specific paper citation and feel the topic label doesn't quite fit, that's a known limitation.
- Tiers. Tier 1 = topics with share at or above 10%. Tier 2 = topics with share between 5% and 10%. Tier 3 = topics with share under 5%. These thresholds are mechanical, not editorial.
- What we publish. This page reports topic frequencies (a derivative analysis) and links to the original UAT-UK papers for the actual question content. We do not reproduce question text.
- Caveats. The TMUA spec was updated in 2018. Pre-2018 papers occasionally touch topics that have since dropped off the spec. The bulk of pre-2018 questions remain on-spec and the share numbers stay representative.