TMUA Syllabus: Every Topic on the Test, Paper by Paper

The TMUA syllabus is shorter than most people expect: the content is largely the pure mathematics you already meet at AS-level, with no statistics and no mechanics. The catch is how it is tested, which is harder, more indirect, and entirely calculator-free. This guide lays out every topic, paper by paper, so you know exactly what you are responsible for.

The test you are preparing for sits in October 2026, for 2027 entry. It is run by UAT-UK (the Cambridge and Imperial joint venture) through Pearson VUE test centres, and it consists of two papers. Each paper is 75 minutes, 20 multiple-choice questions, with no calculator and no formula booklet. Each is scored from 1.0 to 9.0, and there is no negative marking, so a wrong answer costs nothing more than a blank one.

A quick but important warning before the topic lists: the breakdown below is organised to be useful for revision, but the single authoritative source for what is examinable is the official UAT-UK specification. Read it once at the start of your preparation and again near the test, and treat it as final wherever this or any other summary is less precise. Our about page gives a condensed overview of the format and content for orientation.

How the two papers split the content

The TMUA does not have two separate syllabuses. It has one body of pure mathematics, used in two different ways.

Paper 1, Applications of Mathematical Knowledge, is pure-maths problem solving. It asks you to use the content directly to get an answer: solve, simplify, differentiate, sum, prove a value. The content sits at roughly AS-level and early A-level, assuming higher-GCSE and AS-level knowledge held at a fluent, automatic level.

Paper 2, Mathematical Reasoning, uses the very same content but tests your reasoning about mathematics rather than your ability to compute with it. Here you negate statements, judge whether a condition is necessary or sufficient, build counterexamples, and find the flawed step in an argument. The mathematics involved is drawn from the Paper 1 areas, but the skill on trial is the logic.

That structure has a strategic consequence worth stating early: knowing the syllabus is necessary but nowhere near sufficient. Almost every topic will look familiar from school. What makes the test hard is that the questions disguise routine problems, demand exact calculator-free work, and (on Paper 2) probe whether your reasoning is watertight. So use the lists below to make sure you have no content gaps, then spend most of your effort on how that content is examined.

Paper 1 topic areas

Paper 1 draws on the standard pure-maths toolkit. The table groups the content areas; the prose underneath says what each cluster involves and how it tends to show up.

Algebra, surds, indices and polynomials

This is the bedrock. You should be able to rearrange and simplify expressions, solve linear and quadratic equations (by factorising, completing the square, and the formula), and handle function notation and composition without hesitation. Surds and indices come up constantly because there is no calculator, so simplifying something like $\sqrt{50}$ to $5\sqrt{2}$ or evaluating $8^{2/3}$ has to be instant. For polynomials, expect the factor and remainder theorems, polynomial division, and using known roots to factorise a cubic. None of this is exotic, but all of it has to be fluent enough that it never becomes the bottleneck in a longer problem.

Sequences and series (including the binomial expansion)

This cluster covers arithmetic sequences and series, geometric sequences and series (including the sum to infinity where it converges), and the binomial expansion. You need the standard formulae at your fingertips, since there is no formula booklet, and you need to recognise when a wordy setup is really just an arithmetic or geometric series in disguise. The binomial expansion shows up both as a tool for expanding powers of a bracket and as a source of "find the coefficient of a particular term" questions.

Coordinate geometry: straight lines and circles

Coordinate geometry in the $(x, y)$ plane focuses on two objects: straight lines and circles. For lines, expect gradients, equations of lines, parallel and perpendicular conditions, intersections, and distances. For circles, expect the equation of a circle, finding its centre and radius (often by completing the square), tangents, and intersections of lines and circles. Diagrams are frequently implied rather than drawn, so being comfortable turning an algebraic description into a mental picture pays off.

Trigonometry

Trigonometry here means identities and equations rather than heavy modelling. You should know the exact values for common angles, the standard identities (such as $\sin^2\theta + \cos^2\theta = 1$ and the relationship $\tan\theta = \frac{\sin\theta}{\cos\theta}$), and how to solve trigonometric equations across a given interval, including remembering every solution in range rather than just the first. Because there is no calculator, exact-value work and clean use of identities matter more than numerical evaluation.

Exponentials and logarithms

This area covers the behaviour of exponential and logarithmic functions and, above all, the laws of logarithms. You should be able to combine and split logs, change a power equation into a log equation to solve it, and move fluently between exponential and logarithmic form. Expect questions that hinge on a single correct application of a log law, where a common slip (for instance, mishandling $\log_a b$ or the log of a product) is built into one of the wrong options.

Differentiation and integration

Calculus appears in its standard early-A-level form. For differentiation, expect gradients of curves, equations of tangents and normals, finding and classifying stationary points, and reasoning about where a function is increasing or decreasing. For integration, expect indefinite and definite integrals and using a definite integral to find an area under a curve. The techniques are routine; the difficulty comes from the surrounding problem and from doing the algebra and arithmetic accurately by hand.

Graphs, transformations and inequalities

You should be able to sketch and interpret graphs of the standard functions (polynomials, reciprocals, exponentials, logarithms, trigonometric functions) and apply transformations: translations, stretches and reflections, and how each changes the equation. Inequalities are tested both as something to solve (linear, quadratic, and inequalities involving simple algebraic fractions) and as something to reason about, which makes them a natural bridge towards the logical thinking of Paper 2.

Number and divisibility

Finally, Paper 1 includes basic work with numbers and divisibility: factors, multiples, and simple divisibility arguments. This is not number theory in any deep sense, but a question may turn on recognising, say, that a quantity is always even, or always a multiple of a particular number. It also overlaps neatly with Paper 2, where divisibility is a favourite setting for proofs and counterexamples.

A clarifying note on level: every area above is examined at roughly AS-level depth, assumed at a fluent standard. If you want a topic-by-topic feel for how that level translates into actual questions, the fastest route is to work through real ones, filtered by topic and difficulty, in CrackTMUA's free question bank.

Here is a real Paper 1 question built on these core topics. The content is familiar from A-level; notice how much more indirect the question is than a textbook exercise:

Paper 2 reasoning skills

Paper 2 reuses the content above, but the questions are about reasoning. The table lists the core skills; the prose explains each one.

The logic of arguments and methods of proof

Paper 2 expects you to reason carefully about whether a conclusion genuinely follows from the steps that precede it. That includes recognising the main methods of proof and when each applies: direct proof (deriving the result step by step), disproof by counterexample (a single case that breaks a universal claim), and proof by contradiction (assuming the opposite and deriving an impossibility). You do not need to write proofs out in full, because the format is multiple choice, but you do need to read, evaluate and complete arguments built from these methods.

Necessary and sufficient conditions

This distinction is central to Paper 2. A condition is sufficient for a result if its truth guarantees the result, and necessary if the result cannot hold without it. A condition can be one, the other, both, or neither, and many questions offer an option that quietly swaps the two directions. The reliable habit is to ask, for every "if-then" you meet, which way the implication actually runs, and to test both directions before committing.

Negation and quantifiers

Negation is where intuition most often misleads people, so it is worth training by rule rather than by feel. The negation of "for all $x$, $P(x)$" is "there exists an $x$ for which $P(x)$ is false"; the negation of "there exists" is a statement about "all". The negation of "all are" is "at least one is not" (not "none are"), and the negation of "if $A$ then $B$" is "$A$ is true and $B$ is false". Handling quantifiers ("for all", "there exists") cleanly, and flipping statements correctly, is a recurring source of marks.

Spotting the flawed step in a proof

A signature Paper 2 format hands you a proof of something false and asks for the single step where the reasoning first breaks. Every line looks plausible, which is the trap. Common culprits are dividing by an expression that could be zero, taking a square root and silently dropping the negative case, or treating an implication as if it ran both ways. The skill is locating the first invalid step, not merely any suspicious-looking line.

A fuller treatment of how to train these reasoning habits, and why Paper 2 is the most improvable part of the test, lives in our guide to Paper 1 vs Paper 2.

What is NOT on the TMUA

Knowing what to ignore saves real time. The TMUA does not include statistics and does not include mechanics, so probability distributions, hypothesis testing, kinematics, forces and the rest of applied A-level maths are all off the syllabus. It also does not reach beyond the defined pure content into university-level material: no formal calculus of limits, no linear algebra, no abstract structures. And, to repeat the constraint that shapes everything, there is no calculator and no formula booklet, so anything that would only be tractable with a calculator is not what is being asked.

This is genuinely good news for planning. The content footprint is small and familiar, which means you can stop worrying about coverage relatively quickly and pour your remaining time into the harder, calculator-free, more indirect way the test uses that content.

Turning the syllabus into a study plan

Because the content is mostly familiar, the syllabus is best treated as a checklist for finding gaps, not as the main event. Run through the Paper 1 areas above and honestly mark any that are rusty (logs and the binomial expansion are common weak spots), then close those gaps quickly. After that, shift your energy to two things: exact, fast, calculator-free fluency for Paper 1, and the explicit reasoning toolkit for Paper 2. For a structured way to sequence all of this over the weeks before the test, read how to prepare for the TMUA.

The most efficient way to pressure-test your syllabus knowledge is volume on real questions with worked solutions that explain the trap and the quickest route. CrackTMUA gives you a free, interactive bank of every official TMUA question, filterable by paper, topic and difficulty, with spaced repetition built in so the patterns stick. Premium unlocks everything for £37 one-time (12 months access). Start practising now, work through every topic above, and confirm anything examinable against the official UAT-UK specification.

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