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What is the TMUA?

The TMUA is a maths admissions test introduced in 2016. It consists of two multiple choice papers sat at the end of October. Although the test is not compulsory anywhere, universities such as Durham, Warwick and Lancaster will set an easier offer based on a good TMUA score.

Although the TMUA is split into two papers, there is some overlap between them. Paper 1 is very similar to advanced A-Level questions, and there are some questions of this style in Paper 2 as well. However, there are also more logic-style questions in Paper 2. In this post, I’ll explain how you should prepare for the two different styles of questions that generally appear in the TMUA. 

Hard A-Level Style Questions 

  • As lots of these questions are reliant upon a thorough understanding and knowledge of the A-Level concepts, it is obviously important to revise these fully. 
  • A good way to find extra questions that are of the TMUA-style is to approach the later questions in your textbooks. Instead of heading straight for the earlier, more elementary questions, seek out those that are more involved and applied.
  • See if your teachers can compile a selection of the hardest past A-Level questions, and try to answer these. If they can’t do this, look at the examiners’ reports, and see which questions have generally been answered badly by candidates (these are likely to be the harder ones). Attempt these questions – even if they take some time, getting to grips with them will really help you out in the long run.
  • Practice – again, this is the most important thing for the TMUA. At, you can find lots of resources to help you with this practice:
  1. Our Online Academy offers video solutions to TMUA past paper questions, as well as new questions in the style of the TMUA, written by expert tutors
  2. We also run a day course on the TMUA, which you can attend in-person. You will be exposed to new TMUA questions, and tutors will give you individual attention and advice. They will also help you develop general techniques for solving questions. 

Logic and Proof-Based Style Questions

Although it is also important to familiarise yourself with the A-Level content for these questions, they are much more difficult to prepare for – part of the point is that they are new and unseen. Having said that, there are a number ways you can acclimatise to the style:

  • Practice looking at some proofs. It is good to gain a better understanding of proofs before sitting this exam. I would recommend researching some basic proofs, to familiarise yourself with how they work and general forms. For example, look at Cantor’s proof of the infinitude of the primes, or the proof of the irrationality of square root of 2. 
  • Practice breaking proofs down into step-by-step stages. The questions in the TMUA often rely on locating specific steps in proofs. To practice viewing proofs in this way, try and break down some standard proofs into sections, and make sure you are completely confident with how every phase functions. 
  • Familiarise yourself with the classic proof techniques. Make sure you know how the main proof techniques function. These are proof by contradiction, direct proof and proof by induction:
    • Proof by contradiction is where you assume the opposite of the statement you are trying to prove, and show this cannot hold, so the opposite must hold. 
    • Direct proof is when you can prove the result directly from the hypothesis; algebraic proofs often take this form. 
    • Proof by induction is standardly in the form of: 
      • Base Case: Consider the result when your variable, often n, = 0.
      • Inductive Hypothesis: Assume the result holds for n=k. 
      • Inductive Step: Show the result holds for n=k+1.
      • You can conclude from here that the result always holds for the natural numbers. 
  • Studying foreign and complicated graphs. Look at some obscure graphs, and see what you can identify about them. Can you work out their roots? Their turning points? Are they a transformation of a more common graph? Practice these techniques with graphs such as sin(1x) or excosx.
  • Look at some basic number theory. Number theory questions, such as those about patterns in prime numbers, and the relations of different numbers to each other, are common in the TMUA, but not that common in A-Level. I would recommend reading through an elementary number theory book, such as Ogilvy and Anderson’s Excursions in Number Theory
  • Past paper questions! There are not many past paper questions for the TMUA, but we offer more at! I recommend practicing these new logic-style questions as much as you can. This is the best thing you can do to learn how they should be answered.

“Also remember, any preparation you do for the TMUA is also preparation for the MAT and vice versa. In particular, the more logic-style questions are very similar to some of the MAT multiple choice questions.”


So there you have it: a comprehensive preparation guide to this new admissions test. For more advice, visit Good luck!