How to Prepare for the TMUA

The TMUA is a challenging and time-pressured exam, but students can reliably improve their scores with efficient and focused preparation. The optimal time to begin preparing is in the summer holiday between Y12 and Y13, and in general, the earlier the better! Your strategy will depend on the amount of time you intend to spend on study, and if you are serious about the exam, we suggest that 10 hours per week over the summer holiday would be a great start. Many students start much later and still achieve acceptable results. You should decide exactly how much time you intend to spend studying each week and make a plan, detailing which resources and past papers you will be using.

We recommend the following process for exam preparation:

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Things to Memorise

Since there is no formula booklet for the TMUA, it is essential that you commit standard formulae to memory, and you should begin this straight away. The TMUA specification published by Cambridge includes a good level of detail; you should read through it thoroughly and ensure you are familiar with all of the results included. Some common pitfalls are:

• Formulae for the n’th term, or sum of first n terms of, an arithmetic or geometric sequence (including sum to infinity for geometric!)
• Circle theorems as covered at GCSE (or equivalent) level
• Sine and cosine rules
• Standard graph transformations
• The trapezium rule

You must also memorise the standard trig values. Even if you can deduce them using a triangle, there will be no time in the exam!

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1. Taught Material

Although the TMUA syllabus is mostly requires A level knowledge, there is a ‘shadow syllabus’ of exam-specific ideas that prominently feature, year after year. You can become familiar with these ideas through past paper practice, though it’s ideal to encounter them in an abstract context first, if possible. Encountering new ideas in relation to a specific problem can lock your perspective on the situation in which the idea can be applied, which is why it is best to engage with it in taught material first.

There is a great variety of taught material available that will be a valuable resource in TMUA preparation. We recommend the following resources:

• The Oxford MAT Livestream, hosted by Dr James Munro, is geared towards MAT students, but the videos may be of use to TMUA students. Q1 of the MAT is a multiple choice section so the questions are similar to the TMUA in style. The worksheet sections of these sessions are especially good, though the past paper questions can mostly be ignored by TMUA students.

• Cambridge’s ‘Notes on Logic and Proof’ provide a detailed introduction to the ideas tested on Paper 2 of the TMUA (e.g., necessary and sufficient conditions, counterexamples, negation). The notes address the ideas in a very formal way, though students should aim to cultivate an intuitive understanding of the ideas, without being overly concerned with fancy symbolism or truth tables.

• ‘Skills for success in University Admissions Tests for Mathematics’ published by Hodder Education. Of the many books on the market, this is the only one we recommend. It provides a useful overview of some key ideas, but it isn’t exam-specific, and it isn’t long enough to go into a huge amount of detail.

• The Vantage TMUA Primer Course is by far the most comprehensive taught resource available for the TMUA, recommended for the most serious students. It consists of 10 90-minute lessons, each providing a complete introduction to a key theme that appears in TMUA questions. Each lesson is accompanied by a carefully crafted worksheet, designed to ensure students have seen all the quirks and difficulties the examiners are likely to use. To learn more about the TMUA Primer Course, please read our TMUA Handbook or book a free consultation.

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2. Past Papers

TMUA past papers are (of course) the best resource available for practising questions and becoming familiar with the question style. Students should aim to complete all past papers as there are relatively few available. If you start preparing for the exam early and run our of TMUA papers, the multiple-choice section (Q1) of the MAT is also a very useful resource. MAT past papers from 1996-2022 are available to view on on our website, and we provide free answer keys to the multiple-choice section of the 1996-2006 papers, which aren't available anywhere else.

It is important to work through year by year, rather than finding questions arranged by topic. This is because a significant part of the challenge is working out what ‘topic’ the question is on! It should take approximately 4 hours to study a paper (Paper 1 + 2 together) fully, following this process:

• ‍Timed mock exam. If possible, all of the TMUA past papers should be completed as a timed mock (Paper 1 and 2 consecutively, taking 2 hours and 30 minutes). It is the most time-pressured of the maths admissions tests, so it is especially important to be strict with timings. It'll be difficult to concentrate for the full length of the exam at first, which is why it is so important to practice in timed conditions and build stamina. This also allows you to accurately track progress.‍

• Second attempt. After the timed mock, you should set the work aside and spend some time on the questions you didn’t manage to complete, in a different colour pen or on separate paper. Grappling with difficult problems and spending time trying ideas is key to developing a sharp instinct for the questions. Make sure you don’t look at the mark scheme or solutions at this point.‍

• Mark work. When you are satisfied you have completed as much as possible, mark your work using the Cambridge Assessment answer keys, and convert your raw mark to a 1.0-9.0 score using the conversion tables. These are available on our resources page.‍

• Review. You should then carefully review your second attempt to ensure a complete understanding of all the questions. Use worked solutions for this (some are provided by Cambridge assessment). Vantage TMUA preparation courses provide complete solution videos and booklets for all TMUA and MAT multiple choice questions. They teach students how to ‘come up with’ ideas in the first place, thinking as trained mathematicians. TMUA 2022 Paper 1 solution videos are available to view on YouTube for free.

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3. Individual Tuition

Many students opt for individual tuition for admissions test preparation. It provides an opportunity to smooth out any issues that were unresolvable with the available resources and ask questions. Tuition can be extremely beneficial, but students who do not receive tuition can still achieve high marks through diligent preparation. You can also ask for support from teachers, mathematically inclined friends, or online forums such as The Student Room.

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