University Entry Exams Explained: TMUA, ESAT, STEP, BMO and BPhO (and Where to Find Practice Questions)

If you are sitting a university admissions test this year, you have probably already discovered the problem: there is barely anything to practise with.

For GCSE and A-Level, students have decades of past papers, mark schemes, textbooks and worked solutions. For admissions tests like the TMUA, ESAT, STEP, BMO and BPhO, there are only a handful of official past papers — sometimes just two or three — and the worked solutions are thin or non-existent. Teachers rarely have spare material because these exams sit outside the normal curriculum, and most schools do not teach them at all.

That scarcity is exactly why these tests are such effective filters for top universities, and exactly why so many capable students underperform: they run out of practice weeks before the exam. The fix is not "work harder" — it is access to unlimited, exam-style questions with instant feedback, so you can keep practising until each topic clicks. That is what PeddyLoop is built to do for every exam below.

The TMUA (Test of Mathematics for University Admission) is used by leading universities for mathematics-based courses such as Mathematics, Computer Science and Economics. A strong TMUA score can unlock reduced A-Level offers at several universities, which makes it one of the highest-value tests to prepare for.

It is sat as two papers with no calculator and no formula booklet. Paper 1 (Applications of Mathematical Knowledge) tests your ability to apply core A-Level-style mathematics quickly and accurately. Paper 2 (Mathematical Reasoning) tests logic, proof and the structure of mathematical arguments — including necessary and sufficient conditions, quantifiers, and the converse, inverse and contrapositive of a statement.

What trips students up is the reasoning paper: most A-Level study never touches formal logic or proof technique. On PeddyLoop you can practise both papers across every TMUA topic — algebra, sequences, coordinate geometry, trigonometry, logarithms, calculus, graphs, mathematical logic and proof — with unlimited questions in the exam's multiple-choice style.

The ESAT (Engineering and Science Admissions Test) is used for engineering and science courses, and it is modular: you take the specific modules your course requires rather than one fixed paper. PeddyLoop covers all five modules by name.

ESAT Mathematics 1 is the core mathematics module and is compulsory for most courses — it covers units, number, ratio and proportion, algebra, geometry, statistics and probability. ESAT Mathematics 2 is the advanced mathematics module, taken for maths-heavy courses, and extends into rational indices, series, coordinate geometry, trigonometric identities, logarithms and calculus.

The three science modules — ESAT Physics, ESAT Chemistry and ESAT Biology — test the core science needed for your chosen degree. Because ESAT is relatively new, official practice material is especially limited, so generated practice that matches each module's content is particularly valuable here. PeddyLoop lets you drill exactly the modules your course needs.

STEP (the Sixth Term Examination Paper) is a long-form mathematics exam used as part of conditional offers for the most competitive mathematics courses. PeddyLoop covers STEP 2 and STEP 3.

Unlike multiple-choice admissions tests, STEP questions are extended, multi-step problems where you write full mathematical solutions and your method earns the marks. Each paper draws on pure mathematics alongside mechanics and probability and statistics, and STEP 3 reaches into Further Mathematics content.

STEP rewards depth, persistence and clean argument rather than speed. The only way to get better is to attempt many full questions and get feedback on your reasoning — which is hard when official solutions are scarce. PeddyLoop generates STEP-style problems by topic and gives step-by-step feedback on your written working, so you can build genuine problem-solving stamina.

The British Mathematical Olympiad (BMO), run by the UK Mathematics Trust, is a proof-based problem-solving competition and the next step beyond the Senior Mathematical Challenge. PeddyLoop supports both BMO Round 1 and BMO Round 2.

There is no fixed syllabus. BMO problems draw on classic olympiad areas — number theory, combinatorics, geometry, algebra and inequalities — applied in unfamiliar and creative ways. Success is about pattern recognition, rigorous proof and the willingness to sit with a hard problem.

Because there is no syllabus to "cover", the single most useful thing you can do is solve a large and varied bank of olympiad-style problems. PeddyLoop gives you that volume and variety, with feedback that helps you see the technique you were missing rather than just whether you were right.

The British Physics Olympiad (BPhO) is the physics equivalent: a demanding problem-solving competition that takes core A-Level physics and pushes it toward first-year undergraduate level. PeddyLoop covers BPhO Round 1 and Round 2.

BPhO problems reward physical intuition, confident mathematics and the ability to model unfamiliar situations from first principles. Mechanics, electricity, waves, thermal physics and more all appear, but framed in ways A-Level questions never are.

As with the maths olympiad, the bottleneck is practice volume — there are only so many official papers. PeddyLoop generates BPhO-style problems so you can keep stretching your problem-solving well past the point where the official papers run out.

The evidence on what works for hard exams is consistent: active retrieval and deliberate practice on your weak areas beat re-reading notes every time. For admissions tests, the limiting factor has always been a lack of questions to retrieve and practise with.

PeddyLoop removes that limit. For every exam above, it generates fresh, admissions-style questions for each topic on demand, marks your attempt in seconds, and explains the precise step where your method broke down — not just whether the final answer was right. Instead of rationing a handful of past papers, you practise until each topic is genuinely mastered.

A realistic plan: identify the exact exam and modules your offer requires, practise a little and often across every topic, deliberately spend extra time on the areas you find hardest, and work under timed conditions as the test approaches.

TMUA, ESAT (Mathematics 1, Mathematics 2, Physics, Chemistry and Biology), STEP, the British Mathematical Olympiad and the British Physics Olympiad are among the toughest exams in UK admissions — and the ones with the least practice material available anywhere.

PeddyLoop covers all of them by name, with unlimited exam-style questions and instant, step-by-step feedback. If finding practice questions has been the thing holding you back, this is the gap it was built to close.

Pick your exam, start a free trial, and begin practising tonight.