TMUA preparation guide

The TMUA Syllabus in Full: Section 1, Section 2 and Every Topic Boundary

Is the TMUA just GCSE-level? Do I need calculus? How deep does the Paper 2 logic go? The answers to all three sit in the official Content Specification, yet candidates routinely prepare from the wrong list. Here is the syllabus laid out in full: Section 1, shared by both papers (the AS pure maths layer MM1 to MM8 plus the GCSE layer M1 to M7); Section 2, which belongs to Paper 2 alone and covers logic and proof; and where the boundaries genuinely sit, including the misreadings that cost candidates the most marks.
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2
The TMUA Syllabus at a Glance
Structure
Two parts only: Section 1 maths knowledge (MM1 to MM8 plus M1 to M7) for both papers, Section 2 logic and proof for Paper 2 alone
Content level
Section 1 Part 1 is almost all AS pure maths; Part 2 is almost all Higher Level GCSE
Calculus
Examinable: MM6 Differentiation and MM7 Integration are listed in Section 1 Part 1
Section 2 ceiling
No symbolic notation and no formal truth tables; induction appears nowhere in the proof list
Test rules
No calculator, no dictionary, no formulae booklet, no negative marking; 20 multiple choice questions in 75 minutes per paper

All of the above follows the official TMUA Content Specification and UAT-UK's test and results pages; the current specification states on its cover that it applies to the October 2026 and January 2027 sittings.

In short
  • The syllabus has just two parts: Section 1 (maths knowledge, shared by both papers; Part 1 broadly AS pure maths, Part 2 broadly Higher Level GCSE) and Section 2 (logic and proof, Paper 2 only).
  • Calculus is examinable: MM6 Differentiation and MM7 Integration sit in Section 1 Part 1 in black and white, alongside MM5 Exponentials and logarithms and MM8 Graphs of functions, and they are the most commonly under-prepared headings.
  • Do not skip the GCSE layer: M1 to M7 carry the same Section 1 status as the MM topics, and the calculator ban makes hand arithmetic count directly towards your score.
  • Section 2 requires no symbolic notation and no truth tables, and induction appears nowhere in the official proof lists; do not import STEP or A2 habits into TMUA preparation.
  • No calculator, no formulae booklet, no negative marking: practise under real rules from day one, memorise formulae for instant recall, and answer every question.
I

Part 1 of 3

Sections 01 to 03

01

One specification, two sections, two papers

The TMUA Content Specification has a simpler shape than most candidates expect: the whole document contains only two content sections. Section 1 sets out the mathematical knowledge requirement, and the specification states that it applies to both papers of the test. Section 2 defines the scope of Paper 2 only, covering logic and proof. Mapped onto the exam itself, the two papers divide up as follows.
Paper 1Paper 2
Full titleApplications of Mathematical KnowledgeMathematical Reasoning
Time75 minutes75 minutes
Questions20 multiple choice20 multiple choice
Specification scopeSection 1Sections 1 and 2
What it testsApplying mathematical knowledge in a variety of contextsThinking mathematically: understanding and constructing arguments, drawing on Section 1 knowledge

The papers are taken one after the other in a single sitting of 2 hours 30 minutes; all questions carry equal weight and there is no penalty for wrong answers.

The TMUA syllabus

Content Specification

Section 1 Part 1MM1 to MM8 · both papers
  • The maths knowledge requirement, stated to apply to both papers
  • Content almost all within the pure maths of an AS level
  • MM6 Differentiation and MM7 Integration both sit in this layer
  • Each heading carries detailed sub-points to audit one by one
Section 1 Part 2M1 to M7 · both papers
  • Listed inside Section 1 alongside the MM topics, with equal status
  • Content almost all within a Higher Level GCSE course
  • With calculators banned, the number work here returns to pen and paper
Section 2 · logicArg1 to Arg4 · Paper 2 only
  • True and false; and, or (the inclusive or), not
  • The four conditional forms, plus converse and contrapositive
  • Necessary and sufficient; for all, for some, there exists, and their negations
  • No symbolic notation and no formal truth tables are expected
Section 2 · proof and errorsPrf1 to Prf5, Err1 and Err2 · Paper 2 only
  • Four proof types: direct deduction, cases, contradiction, counterexample
  • Deducing implications, conjecturing from small cases, ordering jumbled statements into a proof
  • Spotting errors in purported proofs and knowing the common invalid moves
  • Induction does not appear on the list

Tap a branch to unfold

In other words, Section 1 is the foundation for all forty questions across the test (twenty per paper), and Section 2 is an extra layer that only Paper 2 builds on. There is no version of TMUA preparation in which you neglect Section 1 because you feel stronger on reasoning. Nor is there room to think paper by paper on scoring: the two papers are equated and scaled together into a single overall score from 1 to 9, reported to one decimal place.
02

How the test owner positions the difficulty

UAT-UK describes the content level in two statements that work at different resolutions. On the test page, the TMUA is described as based on content typically covered within Higher Level GCSE mathematics courses or AS level mathematics. The specification itself is more precise: the content of Section 1 Part 1 is almost all covered within the pure mathematics specification of an AS level in mathematics, and the content of Part 2 is almost all covered within a Higher Level GCSE mathematics course. Reading 'the content stays within AS level' as 'the test is straightforward' would be a mistake, and the right way to read the syllabus is to keep the two questions apart.
The content lists: what can be testedThe difficulty statement: how it is tested
Official wordingSection 1 Part 1 is almost all within AS pure maths, Part 2 almost all within a Higher Level GCSE courseThe tests are designed to be challenging, because they need to separate highly capable applicants clearly
What it meansThe content stays within the lists; they answer what can be testedThe difficulty comes not from deeper content but from familiar material in unfamiliar contexts, with longer chains of reasoning
What to doTick off every topic on the specification: that is the entry requirement, not the finish linePractise deploying those same topics flexibly under time pressure and without a calculator, which is where the gap opens up
On its results pages, UAT-UK is unusually blunt: the tests are designed to be challenging, because they need to separate highly capable applicants clearly, including students who have achieved top grades in school exams, and candidates are told directly not to expect to score as highly as they do at school. Content level and difficulty are two different things; keep that in mind while you work through the topic lists.
03

Section 1 Part 1: MM1 to MM8, the AS pure maths layer

Section 1 Part 1 contains eight topic headings, numbered MM1 to MM8. Since the specification states this content is almost all within the pure maths of an AS level, your AS textbook is a workable boundary marker for each heading.

MM1

Algebra and functions

MM2

Sequences and series

MM3

Coordinate geometry

MM4

Trigonometry

MM5

Exponentials and logarithms

MM6

Differentiation

MM7

Integration

MM8

Graphs of functions

Two things are worth underlining. First, these eight headings serve both papers, not just Paper 1. Section 1 is defined as the knowledge requirement for the whole test, so a Paper 2 argument question is entitled to lean on any of them: a proof built around sequences, or a true-or-false judgement about a logarithmic identity, is squarely within scope. Second, the operative wording is almost all. The specification deliberately leaves a small margin beyond the standard AS diet, which is one of the ways a discriminating admissions test stretches strong candidates; if a past-paper question uses a familiar tool in a way your course never quite practised, that is the test working as designed rather than the syllabus being breached. Under each heading the specification also lists detailed sub-points, so audit your coverage against the official document and tick sub-points, not just headings.
II

Part 2 of 3

Sections 04 to 06

04

MM5 to MM8: the half everyone underestimates (yes, calculus is examined)

The single most common misreading of the TMUA syllabus is to treat the test as a beefed-up GCSE paper and leave calculus out of the preparation plan. The specification says otherwise: MM6 Differentiation and MM7 Integration are listed in Section 1 Part 1 in black and white, alongside MM5 Exponentials and logarithms and MM8 Graphs of functions. For candidates coming straight from GCSE, or international students on curricula that sequence calculus later, these four headings are precisely where the knowledge gap sits, and they are the part that thin summaries of the test tend to skate over.
Two test-day rules raise the bar further on exactly these topics. There is no formulae booklet: students are expected to understand and recall all relevant formulae, so standard derivatives, integrals and logarithm laws must live in long-term memory rather than on a crib sheet. And calculators are not permitted, so every logarithmic manipulation and every definite integral is done by hand. Paper 1's stated purpose is to test the application of mathematical knowledge in a variety of contexts, which means the exam rewards recognising that an unfamiliar-looking problem is really a calculus problem in disguise, not merely reproducing a differentiation routine from class.
If your timeline means you have not yet met AS calculus (a common position for students preparing early), put MM5 to MM8 at the front of the study plan. The syllabus will not route around them, so your preparation cannot either.
05

Section 1 Part 2: M1 to M7, why the GCSE layer cannot be skipped

Section 1 Part 2 has seven headings, numbered M1 to M7, positioned by the specification as almost all within a Higher Level GCSE course.

M1

Units

M2

Number

M3

Ratio and proportion

M4

Algebra

M5

Geometry

M6

Statistics

M7

Probability

A level students are the group most tempted to wave this layer through on the grounds that GCSE is finished business. Three reasons to resist that. First, status: these headings sit inside Section 1 alongside the MM topics, and Section 1 is the knowledge requirement for both papers, with no small print discounting the GCSE layer. Second, fit: Paper 1 tests the application of knowledge in new situations, and ratio, probability, statistics and geometry are natural raw material for building those situations; marks lost here are rarely lost to ignorance, they are lost to rustiness.
Third, arithmetic: with calculators banned, the number work in M2 and M3 comes back to pen and paper. A year of AS maths with a calculator to hand quietly erodes mental and written arithmetic, and the speed you recover here is time you can spend on the genuinely hard questions. A short, honest audit (timed drills on fractions, surds, percentages and ratio without a calculator) usually reveals more decay than candidates expect, and fixing it is cheap.
06

Section 2, part one: elementary logic (Arg1 to Arg4)

The first half of Section 2 is the language of logic, set out in four numbered Arg points. Here they are one by one.
Arg1 · true and false, connectives, conditionals
The terms true and false; and, or (explicitly the inclusive or), not; the four conditional forms if A then B, A if B, A only if B, and A if and only if B; the converse and the contrapositive of a statement, and how their truth relates to the truth of the original statement.
Arg2 · necessary and sufficient
Understanding and using the terms necessary and sufficient.
Arg3 · quantifiers
Understanding and using the terms for all, for some (which the specification glosses as meaning for at least one), and there exists.
Arg4 · negation
Negating statements that use any of the terms above.
The ceiling is stated just as clearly: candidates are not expected to recognise or use symbolic notation for any of these terms, nor to complete formal truth tables. You do not need a formal logic course. What you do need is precision in ordinary language: the direction of only if, the exact force of for some, and the negation of quantified statements are the places where otherwise fluent mathematicians slip, and they repay deliberate practice rather than a single read-through.
UAT-UK also publishes a companion document, Notes on Logic and Proof, which explains that the formal side of mathematics (theorems and proofs) is the main focus of Paper 2, and that the notes exist as a brief introduction for candidates who have not yet met these ideas in class or in wider reading. It is the natural first read before touching any Paper 2 practice.
III

Part 3 of 3

Sections 07 to 09

07

Section 2, part two: proof, conjecture and error-spotting (Prf1 to Prf5, Err1 and Err2)

The proof half of Section 2 comes in two groups, Prf and Err. The official points are these.

Prf1

Four proof types

Direct deductive proof, proof by cases, proof by contradiction, disproof by counterexample

Prf2

Deducing implications

Deducing implications from given statements

Prf3

Conjecture and justification

Making conjectures based on small cases, then justifying them

Prf4

Ordering a proof

Rearranging a jumbled sequence of statements into the correct order to form a proof

Prf5

Sophisticated reasoning

Problems requiring a sophisticated chain of reasoning to solve

Err1 and Err2

Errors and invalid moves

Identifying errors in purported proofs, and knowing the common invalid moves

The specification's own examples of invalid moves are deducing b = c from ab = ac, and deducing A = B from sin A = sin B, neither of which is a valid step. Spend your preparation time on the list itself, especially Prf4-style proof-ordering and Err-style error-spotting. Both are formats that school mathematics rarely trains in any systematic way, both feel awkward on first contact, and both reward deliberate, repeated practice with the official materials more than any other part of the test.
A word on mathematical induction, because it is the most persistent myth about this syllabus: induction does not appear in the Prf1 list, and the word does not occur anywhere in the Content Specification or in the official Notes on Logic and Proof. Importing it from STEP or A2 course habits wastes preparation time that Section 2 actually needs.
08

Three rules outside the topic lists that shape how you practise

Alongside the topic lists, the specification and the official test pages fix a handful of rules that change day-to-day preparation more than any single topic does.

Calculator and dictionary

Both banned

All numerical work is done by hand, and dictionaries are not allowed into the test either

Formulae booklet

None provided

Students must understand and recall all relevant formulae, from trigonometric identities to standard derivatives and integrals

Wrong answers

No penalty

Questions across the two papers carry equal weight, and UAT-UK explicitly advises attempting every question

Per-paper threshold

None

The papers are equated and scaled together onto the 1-to-9 scale, reported to one decimal place

With no per-paper threshold there is no strategy of sacrificing one paper to protect the other: both papers count, so both get practised properly. Translated into habits, that means three things. Work without a calculator from day one rather than planning to adjust later. Embed formulae through repeated retrieval during practice rather than a cramming session before test day. And in every timed set, force yourself to commit to an answer on every question, because that is exactly what the marking rules reward.
09

Using the syllabus to drive preparation: the official route, used well

UAT-UK's own preparation advice is short: read the test specification, read Notes on Logic and Proof, and answer practice papers under timed conditions. Plain as it sounds, the order and the method matter.
01Step 1

Use the specification as a checklist

Go through MM1 to MM8 and M1 to M7 heading by heading and sub-point by sub-point, sorting each into secure, met-but-rusty, and not-yet-met. For most candidates the not-yet-met pile is some mix of MM6, MM7 and the whole of Section 2; that pile gets scheduled first.
02Step 2

Read Notes on Logic and Proof first

Do not walk into Paper 2 practice cold. The notes were written precisely for candidates who have not seen logic and proof in class, so read them before you start on Paper 2 questions.
03Step 3

Practise under real conditions from the start

Timed, calculator-free, and answering everything. Past papers from 2016 to 2023 remain valid preparation material: the test has since moved on screen, but the syllabus and question style are unchanged.
Finally, check you are reading the current document: the cover of the present Content Specification states it applies to assessment in October 2026 and January 2027. Download the file for your own cycle from the official site rather than relying on an old PDF of unknown provenance.
FrontierVue's TMUA practice is organised around the same specification headings, with topic-filtered drills and timed mock papers, so you can work directly against the checklist above and target the headings that need it.
FAQ

Frequently asked questions

Does the TMUA include calculus?
Yes. MM6 Differentiation and MM7 Integration are listed in Section 1 Part 1 of the official specification, alongside MM5 Exponentials and logarithms and MM8 Graphs of functions. There is no formulae booklet and no calculator, so the standard results must be memorised and applied by hand.
Is the TMUA GCSE level or A level?
Officially it is based on content from Higher Level GCSE or AS maths: Section 1 Part 1 is almost all AS pure maths and Part 2 almost all Higher GCSE. But UAT-UK also states the tests are deliberately designed to be challenging in order to separate highly capable applicants, so content level is not the same thing as difficulty.
What is the difference between the Paper 1 and Paper 2 syllabus?
Both papers share the whole of Section 1. Paper 2 additionally examines Section 2 (the Arg, Prf and Err points on logic and proof), and the specification states that Paper 2 draws on Section 1 knowledge. So Section 1 is prepared for both papers, Section 2 for Paper 2 alone.
Do I need proof by induction for the TMUA?
No. The specification's Prf1 lists exactly four proof types: direct deductive proof, proof by cases, contradiction, and disproof by counterexample. The word induction appears nowhere in the Content Specification or in the official Notes on Logic and Proof. It belongs to STEP and A2 preparation, not the TMUA.
How deep does the Section 2 logic go? Do I need truth tables?
The specification is explicit: no symbolic notation and no formal truth tables are expected. What is required is command of Arg1 to Arg4 in ordinary language: the four conditional forms, converse and contrapositive, necessary and sufficient, and quantifiers with their negations. Start with the official Notes on Logic and Proof.
Is there a formulae booklet for the TMUA? Can I use a calculator?
No to both. There is no formulae booklet, and students are expected to understand and recall all relevant formulae; calculators and dictionaries are not allowed at any point. Practise under exactly those conditions: by hand, from memory, against the clock.
Which sittings does the current specification cover, and are old past papers still useful?
The cover of the current Content Specification states it applies to assessment in October 2026 and January 2027. Past papers from 2016 to 2023 remain useful preparation: the test has moved on screen, but the syllabus and question style are unchanged.
SOURCES

Official sources

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