How to prepare for the TMUA and Step 2

22nd May 2025 by Anita Naik

If you are considering studying Maths, Economics or Computer Science at University, you may need to take external exams with your A-Levels as part of your entry requirements. These exams are the TMUA - Test of Mathematics for University Admission and STEP 2 - Sixth Term Examination Paper.

The TMUA (Test of Mathematics for University Admission) and STEP 2 (Sixth Term Examination Paper) are advanced mathematics exams used at Cambridge, Warwick, Imperial, UCL and LSE. Depending on the University to which you are applying, these exams may be a compulsory part of your application or an optional one.

Ensure you know the requirements for the courses you intend to apply to, as failure to take these exams at the right time may affect your application. For example, if you're applying to Cambridge and are required to sit the TMUA, you must sit it in the October test sitting. All other candidates can choose whether to sit the test in October or January.

Preparation for both the TMUA and STEP 2 requires a strategic approach. Both tests assess advanced mathematical reasoning and problem-solving skills but differ in style and difficulty. The TMUA focuses on logical problem-solving (multiple-choice), while STEP 2 tests deep, proof-based math (long-answer).

Understanding the TMUA

The TMUA lasts 2 hours and 30 minutes and is divided into two parts:

Part 1: Applications of Mathematical Knowledge: Algebra, Calculus, Functions (20 multiple-choice questions, 75 minutes).

Part 2: Mathematical Reasoning: Proofs and problem-solving (20 multiple-choice questions, 75 minutes).

There is no pass or fail for TMUA. Your scores will be based on the number of correct answers you give, so attempt all questions.

TMUA prep plans

As the TMUA focuses on mathematical thinking and reasoning skills, it uses maths that you will have already learned in school. This means the best way to prepare is to familiarise yourself with the exam format and style of questions. Read the test specifications, use past papers and practice under timed conditions.

Official TMUA past papers (2016-2023) and online use Dr Frost Maths (TMUA section).

4 Key strategies for the TMUA

1. Master the basics: Algebra, functions, calculus, and logic (TMUA Paper 1).

2. Practice multi-step problems: TMUA Paper 2 rewards creative thinking (e.g., "spot the pattern" questions).

3. Do timed mock tests: Simulate exam conditions weekly.

4. Review mistakes: Focus on why you misapplied a concept.

When to take the TMUA?

The test is in Year 13 and can be taken in October (test sitting 1, for which you can register from 1 August). If you plan to take the test in January (test sitting 2), you can register for the TMUA from the end of October.

Understanding STEP 2

This is a 3-hour paper, with 12 questions (answer 6) on pure maths + mechanics/statistics.

It's more challenging than TMUA and more aligned with A-Level Further Maths. STEP 2 is used by Cambridge, UCL, Warwick, and Imperial (for Maths/Comp Sci).

Scoring: Grades S (top), 1, 2, 3 (aim for Grade 1/S).

STEP 2 Prep plans

To prepare for STEP 2, start early (ideally in Year 12) and follow a structured plan. Master core A-Level Further Maths topics (e.g., calculus, complex numbers, differential equations, and mechanics), then tackle past papers (2000-2024) under timed conditions to build speed and accuracy.

Past papers (2000-2024) + examiner reports and online: NRICH, STEP Database, MIT OCW for deeper theory.

4 Key Strategies for STEP 2

1. Build stamina: STEP questions take 30-45 mins each—practice under timed conditions.

2. Focus on pure maths: 75% of STEP 2 is pure (inequalities, calculus, proofs).

3. Learn "STEP tricks": Integration by substitution, substitution in proofs, and asymptotic analysis.

4. Prioritise question selection: Skip overly complex stats/mechanics if pure is your strength.

When do you take STEP 2?

The STEP 2 exam is typically taken in June of Year 13 (the final year of A-Levels). The registration deadline is early May (via your school or an approved test centre). The exam date is mid-June, and results are released the same day as A-Level results.

TMUA and STEP 2 Expert Tips

Most UK schools do not teach the TMUA or STEP 2, as these exams are specialised and only required for some university courses. Students typically rely on independent study, using resources like past papers, books (see above), and online platforms (Dr Frost Maths). Further Maths students gain indirect preparation since STEP 2 overlaps with A-Level Further Maths content.

1. TMUA: You will not be able to use a calculator

2. TMUA: Your final scores are based on the number of correct answers you give. You do not lose marks for wrong answers, so it's worth attempting all questions.

3. Review STEP 2 examiner reports for marking advice.

4. Join student forums: The Student Room (TMUA/STEP threads) or Discord study groups.

5. Seek feedback from tutors and teachers on STEP solutions.

6. If you are doing both exams, it pays to know that the TMUA and STEP 2 skillsets overlap.