University Admissions Tests

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These resources for University Admissions Test preparation provide challenging questions for students anywhere studying Mathematics beyond age 16.

Update “Cambridge Assessment Admissions Testing is to withdraw from running a series of university admissions exams with effect from 2024-5. This includes BMAT (medicine), ENGAA (engineering), NSAA (natural sciences) and TMUA (mathematical skills) tests.” It is possible to sign up for BMAT updates.

See also – No Calculators Allowed which includes information on the BMAT test, and also useful resources for students of all ages to work on estimating skills.

From Nrich, Prepare for University.

To further challenge yourself, MAT, STEP and AEA questions provide an excellent source of questions. Dr Jamie Frost has created such a useful resource with his STEP, MAT and AEA questions all aligned to new A Level chapters. This document is 156 pages of categorised questions (brief answers are given). Also available is a pdf file of just the STEP questions.

For mark schemes see:

  • MAT (Maths Admissions Test), scroll down the page for past papers, solutions and additionally video solutions. For superb resources for these questions see these Underground Mathematics Review Questions where you will find not only the questions but suggestions and complete solutions.
  • STEP mark schemes can also be found on the Cambridge Assessment Preparing for STEP page. Note that STEP 1 no longer exists (from Summer 2021).
  • The AEA questions are legacy questions, extended solutions are currently available for the papers which have been used from the University of Warwick. Legacy papers and mark schemes are available on Colmanweb (I found I could access the papers in pdf format, but not Word).

Also available on Dr Frost Maths, is Mr Bowler’s Problem Solving Booklet for University Preparation; the booklet contains 151 questions to help prepare students for undergraduate admissions interviews in mathematics. Answers are also available.

The Underground Mathematics Review Questions include Oxford Mathematics Admissions Test questions and full solutions

TMUA is a newer admissions test only one question is available on the Underground Maths site, however, there is much overlap between the specifications for the TMUA and other tests such as the Oxford MAT, so these questions should provide useful resources for students taking this examination. Warwick University advises taking one of MAT, TMUA or STEP; Durham University states that:

The University uses a national Admission Test in Mathematics (TMUA), in conjunction with the Cambridge Assessment Admissions Testing (CAAT). Test results will be sent by the CAAT directly to students at the end of November, and all information concerning the Test (including whether it was taken at all) will be provided to us by the applicants on an entirely voluntarily basis: suitable performance will entitle the applicant to the reduced A*AA offer. Taking part in the TMUA can therefore only increase the chances of receiving an offer.” 

TUMA papers and mark schemes are available from Cambridge Assessment and I would highly recommend the presentation introducing the test, from Julian Gilbey. As suggested – try the questions first (pdf file) before watching the presentation.

Talking to Julian Gilbey, he recommends for the TMUA, the importance of working through the Extended specification notes on the website, to learn about the logic side. (See Test Specifications for the specification and enhanced specification with its notes on logic and proof.) He also stresses that the more Maths you can do, the more you work on stretching problems and think hard about maths the better you will get at maths. Examples he mentions for resources are any questions on the Underground Maths website, (not just the review questions already mentioned here), UKMT and olympiad problems, STEP problems.

“And essentially your ability to ‘think mathematically’ and to solve mathematical problems is all that these tests are testing”

For further resources try UKMT Senior Maths Challenge Questions. See also Dr Frost Maths Enrichment resources.

Dr Frost Maths


…and also see MadAsMaths with its many papers and solutions increasing in difficulty.

You can download a free pdf copy of  Stephen Siklos’ Advanced Problems in Mathematics. Whilst written to support students taking STEP examination papers, Advanced Problems in Mathematics is excellent preparation for any undergraduate mathematics course.

Following each question, you will find a discussion and a full solution. The clear Contents page lists all 75 problems.

STEP (Sixth Term Examination Paper) Mathematics is a well-established mathematics examination designed to test candidates on questions that are similar in style to undergraduate mathematics.

You will find free STEP solutions on MEI’s site. The STEP question papers are all available on the Cambridge Assessment website; note the STEP resources include a searchable database.  The AEA qualification from Pearson is based on the A Level specification and designed for the top 10% of students to help differentiate between the most able candidates. The Sample Assessment Materials provides a valuable source of challenging questions (with mark schemes) for A Level mathematicians.

STEP papers here. Underground Mathematics has STEP questions within their Review Questions. Each question comes with a fully worked solution.

See also, from Cambridge University, their STEP Support Programme. From the home page, access the resources, you will see STEP Support Programme Foundation modules, STEP 2 modules and STEP 3 modules.

This webinar from Nrich which In this webinar Claire explains how STEP questions can be used to help students prepare for a University Maths degree and demonstrates the resources available on the STEP Support Programme. (

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