TMUA specification 2026/27: every topic you need
The full content specification for assessment in October 2026 and January 2027, as a page rather than a PDF - with how often each topic has actually come up across every official paper.
Written by the specialists behind our practice bank — over a decade writing, teaching and tracking the UK admissions tests.
Where the specification comes from
Everything on this page is taken from the official TMUA Content Specification published by UAT-UK, who run the test.
It is a 17-page PDF, free to download, and it is the only document that decides what can be asked. Read it alongside this page rather than instead of it.
This page assumes you already know the format and why you are sitting the test. If not, the complete TMUA guide covers both first.
Two companion documents go with it, and both are free too.
Here's how that breaks down:
| Document | Current edition | What it is for |
|---|---|---|
| TMUA Content Specification | For assessment October 2026 and January 2027 | The syllabus itself. Everything on this page comes from it. |
| Notes on Logic and Proof | June 2025 revision | Section 2 content, written specifically for Paper 2. |
| Notes on Mathematics | June 2026, shared with ESAT Module 2 | Section 1 content. Broader than the TMUA needs, but authoritative. |
One practical warning before you download anything, because it catches people out.
The same check is worth doing on any specification you downloaded from a revision site rather than from UAT-UK. Some are several years old.
What this page adds is the thing a specification cannot tell you: not just what is on the syllabus, but how often each topic has actually been examined across every official paper.
How the specification maps onto the two papers
The specification is split into two sections, and the split is not decorative. It determines which paper tests what.
Here's how that breaks down:
| Paper 1: Applications of Mathematical Knowledge | Paper 2: Mathematical Reasoning | |
|---|---|---|
| Time | 75 minutes | 75 minutes |
| Questions | 20 multiple choice | 20 multiple choice |
| Specification required | Section 1 only | Sections 1 and 2 |
| What it tests | Applying mathematical knowledge in a variety of contexts | Constructing and analysing mathematical arguments |
Read the third row twice. It is the single most important line in the specification.
Section 2 is additional, not alternative. Paper 2 does not swap maths for logic.
It requires everything in Section 1 and the logic and proof content on top.
The papers are taken one after the other in a single session, for 2 hours 30 minutes in total.
Three constraints apply throughout, and all three shape what "knowing" a topic has to mean.
No calculator. Every piece of arithmetic is yours to do.
No formulae booklet. The specification states that candidates "are expected to understand and recall all relevant formulae".
Nothing is given to you.
No negative marking. All 40 questions carry equal weight and there is no penalty for a wrong answer, so the specification explicitly advises attempting every question.
How each paper goes about examining this content is a separate subject, covered on the Paper 1 guide and the Paper 2 guide. This page is the syllabus.
Section 1, Part 1: the AS-level pure maths core
Part 1 is where most of the test lives. The specification describes it as content "almost all covered within the pure mathematics specification of an AS level in mathematics".
That is the level to calibrate to. Not further maths, not A2, and not university material.
Eight topic areas, and the specification numbers them MM1 to MM8. Here they are in full, with what each one means in practice.
Here's how that breaks down:
| Code | Topic | What is actually required |
|---|---|---|
| MM1 | Algebra and functions | • Laws of indices for all rational exponents • Surds, and rationalising denominators • Quadratics, the discriminant, completing the square • Simultaneous equations, including one linear and one quadratic • Linear and quadratic inequalities • Polynomial manipulation, algebraic division, the Factor and Remainder Theorems • Functions as mappings; the square root sign means the positive root; the modulus function |
| MM2 | Sequences and series | • nth-term rules and recurrence relations of the form xn+1 = f(xn) • Arithmetic series • Finite and infinite geometric series, converging when the common ratio has modulus below 1 • The binomial expansion for positive integer powers, with factorial and combination notation |
| MM3 | Coordinate geometry | • Straight lines, and the parallel and perpendicular conditions • The equation of a circle, in both standard forms • Circle properties: the perpendicular from the centre bisects a chord; tangent meets radius at a right angle; the angle at the centre; the angle in a semicircle; angles in the same segment; cyclic quadrilaterals; the alternate segment theorem |
| MM4 | Trigonometry | • Sine and cosine rules, the area formula ½ab sin C, the ambiguous case, 2D and 3D problems • Radians, with arc, sector and segment measure • Exact values at 0, 30, 45, 60 and 90 degrees • Graphs of sine, cosine and tangent, with their symmetry and periodicity • The identities tan = sin over cos, and sin² + cos² = 1 • Solving trigonometric equations within a given interval |
| MM5 | Exponentials and logarithms | • The function y = ax and its graph • The laws of logarithms • Solving ax = b, including disguised quadratics • Change of base is explicitly excluded |
| MM6 | Differentiation | • The derivative as a gradient and as a rate of change; second derivatives; the notation • Differentiating xn for rational n, including expressions you must simplify first • Tangents, normals, stationary points, increasing and decreasing functions • First principles is excluded, and so is computing points of inflexion - maxima and minima only |
| MM7 | Integration | • The difference between a definite integral and an area • Integrating xn for n not equal to −1, with pre-simplification • Both forms of the Fundamental Theorem • Combining equal or contiguous ranges • The trapezium rule, and whether it over- or under-estimates • Solving dy/dx = f(x) |
| MM8 | Graphs of functions | • Sketching standard functions, including the modulus • The transformations af(x), f(x) + a, f(x + a) and f(ax), their compositions, and composite functions f(g(x)) • The effect of the parameters in y = mx + c and y = a(x + b)² + c • Deducing shape from differentiation • Axis intersections, and the possible numbers of real roots • Intersections of graphs as simultaneous equations |
The specification notes that there is some deliberate duplication of content between Part 1 and Part 2. Do not treat the two lists as disjoint.
Section 1, Part 2: the Higher tier GCSE content
Part 2 is the part almost everyone skips, on the reasonable-sounding assumption that an admissions test for mathematics degrees will not be asking GCSE material.
It is, and our analysis of the archive shows exactly where. More on that below, but the headline is that GCSE-level content accounts for a much larger share of Paper 2 than of Paper 1.
The specification describes Part 2 as content "almost all covered within a Higher Level GCSE mathematics course", numbered M1 to M7.
| Code | Topic | What is actually required |
|---|---|---|
| M1 | Units | • Standard and compound units: speed, density, pressure, rates • Unit conversion, including in algebraic contexts |
| M2 | Number | • Ordering, and the inequality symbols • The four operations, and place value • Primes, factors, multiples, HCF and LCM, unique factorisation • Inverse operations, and priority of operations • Systematic listing and counting • Squares, cubes and roots; numeric index laws; standard form • Converting between decimals, percentages and fractions, including recurring decimals • Exact calculation with fractions, surds and π • Upper and lower bounds; rounding and error intervals; estimation |
| M3 | Ratio and proportion | • Scale factors and maps • Fractions of quantities, ratio notation, dividing in a ratio • Ratio in context: conversion, mixing, concentration • Proportion, and the link between ratios, fractions and linear functions • Percentages including above 100%, percentage change, simple interest • Direct and inverse proportion, including powers • Length, area and volume ratios; similarity • Growth and decay, compound interest, iterative processes |
| M4 | Algebra at GCSE level | • Notation, index laws, substitution and vocabulary • Expanding and factorising, including quadratics; rational expressions • Changing the subject; equation versus identity • Coordinates, gradients and intercepts; roots and turning points of quadratics • Sketching and interpreting linear, quadratic, cubic, reciprocal, exponential and trigonometric graphs • Graphs in real contexts, including kinematics; gradients and areas under graphs • Setting up and solving equations, including simultaneous ones • Solving quadratics, with the formula assumed known; linear inequalities • Generating sequences; nth-term rules for linear and quadratic sequences |
| M5 | Geometry | • Terms and notation; angle facts and polygon angle sums • Triangle and quadrilateral properties • Congruence by SSS, SAS, ASA and RHS; arguments from angle facts, congruence and similarity • Transformations, invariance, vectors as translations • Pythagoras in 2D and 3D • Circle terminology, and the circle theorems with their proofs • Geometry on coordinate axes; solids, plans and elevations; maps and bearings • Area and volume formulae, with sphere, pyramid and cone formulae given if needed • Arcs and sectors; similarity in length, area and volume • Vectors, including geometric proofs |
| M6 | Statistics | • Tables and charts: two-way tables, bar charts, pie charts, pictograms, line graphs • Histograms with unequal intervals and frequency density; cumulative frequency • Mean, median, mode and range; grouped estimates; quartiles and interquartile range • Scatter graphs, correlation as distinct from causation, lines of best fit, interpolation |
| M7 | Probability | • Frequency tables and trees; expected outcomes • Relative frequency against theoretical probability • Exhaustive and mutually exclusive events summing to 1 • Systematic enumeration using tables, grids, Venn diagrams and trees; possibility spaces • The addition and multiplication rules, including conditional probability via two-way tables, trees and Venn diagrams • Formal set notation is excluded |
None of this will be unfamiliar. It is content you have met before, being asked in a way you have not - without a calculator, under time pressure, and usually as one step inside a longer problem rather than as the question itself.
Section 2: logic and proof, and why it is Paper 2 only
Section 2 is the part of the TMUA that has no equivalent anywhere in a standard A-level maths course, which is precisely why Paper 2 catches people out.
It is required for Paper 2 and not for Paper 1. That is the only asymmetry in the whole specification.
The content divides into three families: the language of logic, the methods of proof, and finding errors in proofs.
| Code | Content | Share of Paper 2 |
|---|---|---|
| Arg1 | • True and false; and, inclusive or, not • If-then; A if B; A only if B; if and only if • Converse and contrapositive, and the truth relationships between them | 4% |
| Arg2 | Necessary and sufficient conditions. | 17% |
| Arg3 | Quantifiers: for all, for some meaning at least one, and there exists. | 4% |
| Arg4 | Negating statements built from any of the above. | 2% |
| Prf1 | • Direct deductive proof • Proof by cases • Proof by contradiction • Disproof by counterexample | 15% |
| Prf2 | Deducing implications from given statements. | 11% |
| Prf3 | Forming a conjecture from small cases, then justifying it. | 3% |
| Prf4 | Rearranging given statements into a correct proof. | 1% |
| Prf5 | Problems requiring a sophisticated chain of reasoning. | 5% |
| Err1 | Identifying errors in purported proofs. | 9% |
| Err2 | Common errors in purported proofs. The specification names two: that ab = ac does not imply b = c, and that sin A = sin B does not imply A = B. | 2% |
Two things in that column are worth acting on.
Necessary and sufficient is the single most examined idea in Section 2, at roughly one in six Paper 2 questions where reasoning is the main demand. If you learn one thing from Section 2 properly, learn this one.
Proof and error-spotting together outweigh the pure logic vocabulary. Prf1, Prf2, Prf5, Err1 and Err2 account for a substantially larger share than Arg1, Arg3 and Arg4 combined.
The remaining share of Paper 2 is computational: questions where the reasoning content is light and the work is Section 1 mathematics. Paper 2 is not all logic.
The official Notes on Logic and Proof, revised June 2025, are the right place to start on this content, and they come straight from the people who set the test.
They will give you the definitions and the vocabulary. Turning that into the speed you need on the day is a question of working through enough Paper 2 questions to recognise each species on sight.
How Paper 2 actually poses these questions - and the answer formats it uses, which are unlike anything on Paper 1 - is covered on the Paper 2 guide.
What the specification explicitly excludes
Knowing what is off the table is as useful as knowing what is on it, and it saves you revising things that cannot come up.
The specification names five exclusions outright. A question requiring any of them would be off-specification.
| Excluded | Where it sits | What this means in practice |
|---|---|---|
| Change of base for logarithms | MM5.2 | You need the log laws, but not the change-of-base formula. |
| Differentiation from first principles | MM6.1 | You need to know what a derivative is, not to derive one from the limit definition. |
| Computing points of inflexion | MM6.3 | Stationary points are in scope; maxima and minima only. |
| Formal truth tables and symbolic logic notation | Arg1 | Section 2 is examined in words and mathematics, not in logical symbols. |
| Formal set-theory notation | M7.5 | Venn diagrams yes, set-builder notation no. |
Three further constraints are not exclusions from the syllabus but from the room, and they change how you have to know everything above.
No calculator, no dictionary, and no formulae booklet. Every formula in this specification has to come from memory.
That is a bigger constraint than it sounds. It means the circle theorems, the trapezium rule, the binomial coefficients and the exact trigonometric values all have to be recallable under time pressure.
What actually comes up, topic by topic
A specification tells you what could be asked. It cannot tell you what usually is, and that is the gap this section fills.
We classified every question in the official archive - 360 questions across the nine official sittings, being the 2016 to 2023 papers plus the early specimen set - against the specification codes above.
Here is where the questions actually landed.
| Code | Topic area | Paper 1 | Paper 2 |
|---|---|---|---|
| MM1 | Algebra and functions | 18% | 15% |
| MM7 | Integration | 14% | 9% |
| MM2 | Sequences and series | 13% | 8% |
| MM4 | Trigonometry | 13% | 9% |
| MM5 | Exponentials and logarithms | 11% | 7% |
| MM6 | Differentiation | 9% | 8% |
| MM8 | Graphs of functions | 9% | 7% |
| MM3 | Coordinate geometry | 8% | 5% |
| M2 | Number (GCSE) | 0% | 19% |
| M5 | Geometry (GCSE) | 1% | 6% |
| M4 | Algebra (GCSE) | 0% | 3% |
| M6 | Statistics (GCSE) | 1% | 2% |
| M7 | Probability (GCSE) | 2% | 1% |
| M3 | Ratio and proportion (GCSE) | 1% | 0% |
| M1 | Units (GCSE) | 0% | 1% |
One row in that table should change how you revise, and it is the M2 row.
The reason is not mysterious once you see it. Paper 2 tests reasoning, and number theory gives you clean, self-contained objects to reason about without needing heavy machinery.
Within that, one sub-topic dominates: primes, factors, multiples, HCF and LCM and unique factorisation accounts for roughly 11% of Paper 2 on its own.
So the candidate who revises AS pure thoroughly and skips Part 2 has prepared well for one paper out of two.
On Paper 1, the concentration sits elsewhere. Four sub-topics come up far more often than the rest.
| Code | Sub-topic | Share of Paper 1 |
|---|---|---|
| MM1.3 | Quadratics, the discriminant, completing the square | 9% |
| MM6.3 | Tangents, normals and stationary points | 8% |
| MM4.6 | Solving trigonometric equations in an interval | 8% |
| MM5.3 | Solving ax = b, including disguised quadratics | 6% |
Those four together account for roughly 31% of Paper 1, from a specification listing more than forty sub-topics.
None of this means a topic outside the table will not appear. The specification is the contract, and everything in it is examinable.
What the data tells you is where to spend the marginal hour, which is a different and more useful question.
These figures come from the archive, which runs to 2023.
UAT-UK state that the content specification and question style are unchanged since the handover, which is the basis for treating the archive as representative - but no paper from 2024 onward has been released, so this is an inference from their statement rather than something we can verify directly.
Turning the specification into a revision plan
Everything above is specification and evidence. This section is advice, and worth labelling as such.
The instinct with a syllabus is to work through it from the top. That is the wrong order for this test, for two reasons.
The first is that the specification is organised by topic, not by difficulty or by frequency. MM1 is first because it is called MM1.
The second is that you already know most of Section 1 Part 1 from your A-level course. Time spent re-reading it is time not spent on the parts that are genuinely new.
A more useful order, based on what the archive shows:
Start with Section 2. It is the only content with no A-level equivalent and it carries roughly half of Paper 2, so it is where unfamiliarity costs you most.
The official Notes on Logic and Proof are a good foundation to build from.
Then audit Part 2, the GCSE content - specifically number. At almost a third of Paper 2, it earns more attention than its label suggests.
The gap is rarely knowledge. You use factors and bounds and standard form constantly at A-level, just as machinery inside bigger problems.
What Paper 2 does is make them the point of the question. Being asked to reason directly about why a number must have an odd number of factors is a different skill from using factorisation to simplify an expression, and it is worth practicing as its own thing.
Then work the four heavy Paper 1 sub-topics in the table above, under exam conditions rather than as textbook exercises.
Then fill gaps from the specification proper, using it as a checklist to find what you have not covered rather than as a syllabus to teach yourself from scratch.
One constraint should shape all of it: no calculator and no formulae booklet. Practicing with either is practicing a different test.
To work through the actual questions behind every figure on this page, the whole archive is free with official worked answers: every official TMUA past paper, 2016 to 2023.
Frequently asked questions
What is on the TMUA specification?
Two sections. Section 1 is mathematical knowledge, split into Part 1 at roughly AS-level pure standard and Part 2 at Higher tier GCSE. Section 2 is logic and proof, and is required for Paper 2 only.
Which TMUA specification is current?
The edition whose cover page reads "For assessment in October 2026 and January 2027". The URL is misleading because UAT-UK overwrite the file in place, so check the cover page rather than the link.
Is the TMUA A-level or GCSE standard?
Both. Section 1 Part 1 is described as almost all within AS-level pure mathematics, and Part 2 as almost all within a Higher tier GCSE course. In the archive, GCSE-level content is about 4% of Paper 1 but 31% of Paper 2.
Do I need Further Maths for the TMUA?
No. Nothing in the specification requires Further Mathematics. The difficulty comes from how the content is combined and the time pressure, not from the level of the content itself.
What is the difference between Paper 1 and Paper 2 content?
Paper 1 requires Section 1 only. Paper 2 requires Sections 1 and 2, so it needs everything Paper 1 needs plus logic and proof on top.
Is calculus on the TMUA?
Yes. Differentiation and integration are both in Section 1 Part 1, and together account for roughly 23% of Paper 1. Differentiation from first principles and computing points of inflexion are explicitly excluded.
Can I use a calculator or a formulae booklet?
Neither, and no dictionary either. The specification states that candidates are expected to understand and recall all relevant formulae.
What is explicitly not on the TMUA?
Five things are excluded by name: change of base for logarithms, differentiation from first principles, computing points of inflexion, formal truth tables and symbolic logic notation, and formal set-theory notation.
How many questions are there, and how long is the test?
Two papers of 20 multiple-choice questions in 75 minutes each, taken one after the other, for 2 hours 30 minutes in total. There is no negative marking, so every question is worth attempting.
Has the specification changed since the test moved to computer?
No. UAT-UK state that both the content specification and the question style are unchanged, which is why the 2016 to 2023 archive is still the right thing to practice on.
Practice the TMUA specification, topic by topic
A syllabus tells you what to learn. The hard part is finding enough questions on the topics you are weakest at, in the format the real test uses.
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