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

The TMUA Content Specification has a simpler shape than most candidates expect. There are only two content sections in the whole document. 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 specification also pins down what Paper 2 is for: it tests the ability to think mathematically, with the focus on understanding and constructing mathematical arguments, and it draws on the mathematical knowledge outlined in Section 1. 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.

Two framing facts sit alongside this. The papers are taken one after the other in a single sitting of 2 hours 30 minutes. And questions across both papers carry equal weight, with no penalty for wrong answers; the two papers are then equated and scaled together into a single overall score from 1 to 9, reported to one decimal place.

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.

It would be a mistake to read 'the content stays within AS level' as 'the test is straightforward'. 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. Candidates are told directly not to expect to score as highly as they do at school.

The right way to read the syllabus, then, is to separate two questions. The content lists answer what can be tested. The difficulty statement answers how it will be tested: familiar material placed in unfamiliar contexts, with longer chains of reasoning than a standard textbook exercise. Ticking off every topic on the specification is the entry requirement for preparation, not the finish line. The gap between candidates opens up in how flexibly they can deploy those same topics under time pressure, without a calculator.

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:

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 feels like it 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 lists detailed sub-points; when you audit your own coverage, work from the official document itself and tick sub-points, not just headings.

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.

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

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.

The first half of Section 2 is the language of logic, set out in four numbered points:

Just as importantly, the specification states the ceiling: 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.

The proof half of Section 2 lists the following:

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.

Spend that 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.

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

One scoring fact completes the picture: the TMUA reports a single overall score. Papers 1 and 2 are equated and scaled together onto the 1-to-9 scale, reported to one decimal place, so there is no per-paper threshold and no strategy of sacrificing one paper to protect the other. Both papers count, so both get practised properly.

Translated into habits: 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.

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.

Step one, 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. Step two, do not walk into Paper 2 practice cold: read Notes on Logic and Proof first, because it was written precisely for candidates who have not seen logic and proof in class. Step three, 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, so 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.