How to prepare for the TMUA

How the two papers differ

The TMUA takes two and a half hours. Paper 1 has 20 multiple-choice questions in 75 minutes and tests whether you can apply maths you know to situations you haven't seen before. Paper 2 has the same number of questions in the same time, but it tests mathematical reasoning and some simple ideas from logic.

You can't use a calculator or a dictionary in either paper. A wrong answer doesn't cost you a mark, so answer every question.

20 questions in 75 minutes is three minutes and 45 seconds each. That sounds generous, but one awkward question can easily take ten minutes if you let it. You need enough patience to check a claim properly, and enough discipline to pick an answer, flag it and move on when you're getting nowhere.

The official notes to read first

Start with the current TMUA specification from UAT-UK, then read the two official booklets that go with it: Notes on Mathematics and Notes on Logic and Proof. They set out the notation and wording the papers use, and the second covers the logic you need for Paper 2. Treat them as part of the syllabus.

Most of the content is maths you'll have met at higher-tier GCSE or AS level. Go through the specification, mark each point as secure, rusty or missing, then check yourself with a few questions on each. Recognising the chapter title isn't enough. The papers expect you to spot a substitution, try an extreme case, or turn a wordy condition into an inequality without being told to.

Getting faster at Paper 1

Most Paper 1 questions can be done more than one way, and the first method you think of is often the slowest. After you mark a practice question, look for a shorter one: a sketch, a substitution, symmetry, a rough bound, testing the options or trying a small number.

Take a question asking how many real solutions x³ - 3x = k has for a given k. Trying to solve the cubic gets you nowhere. A sketch of y = x³ - 3x does it in a minute: the turning points are at (-1, 2) and (1, -2), so any horizontal line y = k with k between -2 and 2 crosses the curve three times. Once you've seen this, you'll start to read other 'how many solutions' questions as sketching questions.

Look at the answer options before you start working. If they differ only in whether an endpoint is included, the question is really about that endpoint. If they're far apart, a rough estimate will pick the right one. If they're close together, you'll need to be exact. This is still proper maths, and it stops a long piece of tidy algebra from using up time you'll need later in the paper.

Learning the language of Paper 2

Paper 2 gets much easier once words like necessary, sufficient, converse, contrapositive and counterexample feel automatic. Learn them from the Notes on Logic and Proof, so that what you mean by them matches what the paper means.

Here's the sort of statement to practise on: 'if x > 2, then x² > 4'. It's true, so x > 2 is a sufficient condition for x² > 4. The converse, 'if x² > 4, then x > 2', is false, and x = -3 shows it, so x > 2 isn't a necessary condition. The contrapositive, 'if x² ≤ 4, then x ≤ 2', is true, because a statement and its contrapositive are always true or false together.

To show a claim about every number is false, one counterexample is enough, and the best ones are small. Try zero, one, a negative number, two equal values or a case right on a boundary before anything elaborate. 'If a > b, then a² > b²' falls apart with a = 1 and b = -2. If you find yourself filling a page to test a claim about all numbers, stop and try a smaller case.

For each statement, ask what's assumed, exactly what's concluded, and whether it's about all cases or only some. Watch for questions that quietly swap a statement for its converse, and remember that a few examples that work never prove a general claim.

What to do with your first full paper

Sit one full paper before you plan the rest of your revision. Use a computer if you can, put the calculator away, keep to 75 minutes and answer every question. Note which questions you flagged, where you changed your answer and which correct answers were guesses. The mark on its own can't tell you whether you know the maths but are too slow, or work quickly but make errors.

Then go through every answer you weren't sure of and write down what went wrong. It might be a topic you don't know, a problem you couldn't picture, algebra that went on far too long, a mix-up over a logic word, a condition you missed, an arithmetic error or a poor decision about time. Include correct answers that took much longer than the paper allows. Spend your next week on the problem that came up most often, whichever chapter it turned up in.

Keeping track of time over 75 minutes

Checkpoints help you notice when you're falling behind. A reasonable starting point is to check at about 25 and 50 minutes, hoping to be near question 7 and then question 14, without rushing the early questions to get there. Change these numbers once a few mocks have shown you your own pace. The aim is to spot a problem before the last ten minutes, when it's hard to catch up.

If you've had a proper go at a question and can't see a way in, choose the most likely answer, flag it and move on. When you come back, start with the flags where you'd already done useful work, such as narrowing to two options, drawing the graph or getting within one step of the answer. Wrong answers aren't penalised, so don't leave any blanks. A sensible guess is a normal part of pacing the paper.

Cutting down the arithmetic

Without a calculator, the best defence is to avoid arithmetic you don't need. Keep fractions exact, factorise before you expand, estimate before you calculate, and use signs and odd or even to cross out options. You should be comfortable with fractions, powers, surds and algebraic expressions without turning them into decimals.

When you make an arithmetic mistake, write down what actually happened, because 'careless' doesn't tell you how to fix it. Did you round too early, copy a condition wrongly, lose a minus sign, get a reciprocal wrong, or hurry after looking at the clock? Each has a different fix. If the same mistake keeps happening under time, even though you can do the sum easily at a desk, treat it as a problem with your method.

Using the old TMUA papers

UAT-UK keeps an archive of past TMUA papers. The test is now taken on a computer, but UAT-UK says the content and the style of question haven't changed, which makes the archive very useful. Begin with the current specification and the Pearson sample test, then use the past papers as timed practice. Mark each one and note your own mistakes before you read the official worked answers.

There's a limited number of them, so don't race through. One paper can give you a timed sitting, a slow second attempt, a search for quicker methods, entries in your list of mistakes and a second look a few weeks later. In between, work on your weak areas with original questions written to the specification, and keep some official papers unseen for later. Noetra's questions are all original. We don't reproduce real past-paper questions and we're not endorsed by UAT-UK or any university.

Eight weeks of TMUA practice

Weeks one and two: read the official documents, go through the specification and sit one Paper 1 and one Paper 2. Weeks three and four: work on your biggest Paper 1 gaps and on finding shorter methods, and practise the logic words for Paper 2 until you use them without thinking. Each week, include one timed set of mixed questions so you have to work out the topic for yourself.

Weeks five and six: do more full papers and adjust your checkpoints. Week seven: sit both 75-minute papers in one session, as you will on the day, and see how much you tire in the second. Week eight: go back over your repeated mistakes, do shorter timed sets, practise with the Pearson test software and sort out your booking and travel. If you have more than eight weeks, spend the extra time on weeks three to six, not on extra full papers.

• Weeks 1–2: official notes, the specification and one paper of each kind.
• Weeks 3–4: shorter methods for Paper 1 and logic vocabulary for Paper 2.
• Weeks 5–6: full papers and your own time checkpoints.
• Week 7: both papers in one sitting.
• Week 8: repeated mistakes, short timed sets and test-day arrangements.

What a practice score can tell you

UAT-UK gives one overall TMUA score on a scale from 1.0 to 9.0, and there's no pass or fail. Universities look at it alongside the rest of your application, and courses can use it differently. The official specimen and sample tests are there to show you the format and style. UAT-UK doesn't give them a scaled score, because scaling depends on how the candidates in a real sitting perform.

So any score from anyone else is an estimate. Noetra's 1.0–9.0 figure is our own estimate, not UAT-UK's conversion or a university's requirement. When you're deciding what to practise next, look at your accuracy over several papers, where your time went and whether the mistakes you've named are becoming less common. A single practice sitting can't tell you how an application will go.

Questions students ask

What is the TMUA format in 2026?

There are two papers of 75 minutes, each with 20 multiple-choice questions. Paper 1 tests whether you can apply maths in new situations, and Paper 2 tests mathematical reasoning and simple logic.

Can I use a calculator in the TMUA?

No. UAT-UK doesn't allow calculators or dictionaries in either paper.

Are old TMUA papers still useful?

Yes. The TMUA is now taken on a computer, but UAT-UK says the content and question style haven't changed, so the official archive is good practice.

Does the TMUA have negative marking?

No. You don't lose marks for a wrong answer, so give an answer to every question.

Official sources

Current details were checked against these primary sources on the date shown above.

• UAT-UK: TMUA format and official guidance
• UAT-UK: TMUA preparation materials and archive
• UAT-UK: preparation and score guidance
• Oxford Mathematical Institute: TMUA for 2026 applicants