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The University of Cambridge requires TMUA for these courses:

Computer Science

Economics

If applying for Cambridge, you must register for the October sitting of the TMUA.

Imperial College, London requires TMUA for these courses:

the Department of Computing, including joint Mathematics and Computer Science

BSc Economics, Finance and Data Science offered in Imperial College Business School

Mathematics (an offer may include STEP if TMUA has not been taken, but it is recommended that you take TMUA)

Imperial College strongly recommends registering for the October sitting of the TMUA.

Durham University strongly recommends an admissions test for its Mathematics course.

A TMUA score of 5.0 or above automatically qualifies an applicant for a reduced offer (1 grade lower than standard)

For applicants with only AS Further Maths, a TMUA score of 5.0 or above would be required to be considered for an offer.

A score is below 5.0 down to a certain threshold (determined each year but can be between 3.2 and 3.5) is viewed as positive evidence of ability, and would likely result in a standard offer.

The University of Warwick requires or strongly recommends* TMUA for these courses [see individual course pages for details]:

Computer Science

Computer Science with Business Studies

Discrete Mathematics

Economics

Economics, Politics and International Studies

Economics and Management

Mathematics

Mathematics and Statistics

Mathematics, Operational Research, Statistics and Economics

Data Science

*For some courses, STEP may be taken as an alternative to TMUA. However, TMUA is more accessible for most students.

**For students eligible for contextual offers, TMUA may not be required for some courses at Warwick - see the Warwick website for more details

London School of Economics requires TMUA for these courses:

Economics

Econometrics and Mathematical Economics

London School of Economics recommends (but does not require) TMUA for these courses:

Mathematics and Economics

Mathematics with Economics

Financial Mathematics and Statistics

Mathematics with Data Science

Mathematics, Statistics, and Business

Data Science

Actuarial Science (with and without a placement year)

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The University of Cambridge requires STEP for its Maths degree courses. See the website for more details.

Some universities will accept STEP as part of an alternative offer. For example, King's College, London, requires A Level Further Maths as part of its standard offer for Mathematics courses, but will consider students only taking AS Further Mathematics if they have also taken STEP or AEA.

Some universities may include STEP as part of their offer if the date for another exam is missed. For example, Imperial College, London will accept STEP if TMUA has not been taken (but TMUA is recommended)

See individual university websites for more details of offers including STEP.

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The University of Oxford requires students to take the MAT for these courses:

Computer Science

Computer Science and Philosophy

Mathematics

Mathematics and Statistics

Mathematics and Computer Science

Mathematics and Philosophy

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MAT is taken at a Pearson VUE test centre.

The dates for the 2025 exam are Wednesday 22nd and Thursday 23rd October 2025.

Registration is via the website and will open on 18th June.

There will be no fee to take the MAT in 2025.

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A free self-study course helping students to prepare for the MAT. Sign up here:

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Preparing for the MAT (Mathematics Admissions Test) - On Demand

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This on-demand course offers specialist tuition to support students preparing for the MAT (Mathematics Admissions Test).

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The main page with information about the MAT, including specification, test structure and past papers

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MAT (Mathematics Admissions Test) | University of Oxford

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What is the MAT?If you are applying for one of the following courses you will be required to sit the MAT: Computer ScienceComputer Science and PhilosophyMathematicsMathematics and Computer

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The 2026 MAT livestream has not yet started (in previous years it has started in June), but the 2025 recordings are here. To get the most out of the MAT livestream, first download the weekly worksheet and attempt the questions, then watch the livestream (or the recording) to see the solutions.

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Are you thinking of applying to study Mathematics at university? We're running a weekly livestream talking about maths problems and discussing problem-solving strategies, with a focus on Oxford's Mathematics Admissions Test (MAT). The sessions are free and available for everyone, and are recorded so that you can watch them later even if you can't join us live. No registration needed.

The 2023 MAT livestream is live every Thursday at 17:00-19:00 (UK time) from June to October 2023.

A recent survey of first-year Oxford Maths students found that more than half of them had engaged with the Oxford MAT Livestream while applying!

Schedule

(subject to change without notice)

*Other important dates; 19 October 2023 is the date of MAT 2023, and 29 September 2023 is the registration deadline for the MAT. The UCAS deadline for applying to Oxford is 16 October 2023.

If you want even more MAT livestream content, you can find an archive of the MAT livestream from 2020 or 2021 or 2022 at the bottom of this page.

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Are you thinking of applying to study Mathematics at university? We're running a weekly livestream talking about maths problems and discussing problem-solving strategies, with a focus on Oxford's Mathematics Admissions Test (MAT). The sessions are free and available for everyone, and are recorded so that you can watch them later even if you can't join us live.

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TMUA is taken at a Pearson VUE test centre.

There are two sittings for the exam:

1) Monday 13th or Tuesday 14th October 2025 (this sitting must be chosen if you are applying to Cambridge) - registration will be available from June

2) Thursday 8th or Friday 9th January 2026 (either sitting can be chosen if you are not applying to Cambridge - although Imperial recommends taking it in sitting 1)

There will be a fee of £75 to take the TMUA in 2025/26. There is a bursary available, which must be applied for before you book your exam slot - see the TMUA website for details: https://esat-tmua.ac.uk/bursary-scheme/

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You must live in the UK permanently at the time of application and plan to take your test at a UK Test Centre.

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A free self-study course helping students to prepare for the TMUA. Sign up here:

Preparing for the TMUA (Test of Mathematics for University Admission)

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Preparing for the TMUA (Test of Mathematics for University Admission)

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This on demand course offers specialist tuition to support students preparing for the TMUA (Test of Mathematics for University Admission).

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The main website for the TMUA, including the specification and information about the test format

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TMUA for Universities

The TMUA is currently used by the universities shown for selection to some courses in Mathematics, Economics and Computer Science.

You must refer to the relevant course pages of each institution's website to find out if the courses you are applying to require the TMUA. Failure to register for the TMUA where it is compulsory may invalidate your application.

Depending on the University to which you are applying, the TMUA may be a compulsory part of your application, or it may be optional. Make sure you know the requirements for the specific courses to which you intend to apply.
If you are 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.

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The TMUA is currently used by the universities shown for selection to some courses in Mathematics, Economics and Computer Science. You must refer to the relevant course pages of each institution's website to find out if the courses you are applying to require the TMUA.

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From the Durham University website, changes to the relative TMUA grades from 2023 to 2024:

Please note that in 2024 the scoring system changed such that:

A score of 6.5 from 2023 will equate to a score of 5.0 from 2024 onwards.

A score of 4.5 from 2023 will equate to a score of between 3.2 and 3.5 from 2024 onwards.
This is a change to the scoring scale, and not a change to the standard we're looking for.

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The content specification for TMUA

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Past papers and worked solutions

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These are historic TMUA papers. Although the TMUA is now computer-based, both the content specification and question style are unchanged, so this archive of past papers can be used when preparing for the test.

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These are historic TMUA papers. Although the TMUA is now computer-based, both the content specification and question style are unchanged, so this archive of past papers can be used when preparing for the test.

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The University of Oxford requires the PAT for these courses:

Engineering Science

Materials Science

Physics

Physics and Philosophy

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The University of Cambridge requires ESAT for these courses:

Chemical Engineering and Biotechnology

Engineering

Natural Sciences

Veterinary Medicine

Imperial College, London, requires ESAT for these courses:

Departments of Aeronautics, Chemical Engineering, Civil and Environmental Engineering, Electrical and Electronic Engineering, Design Engineering and Mechanical Engineering

Department of Physics

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Flashcards for revision for the MAT. Would also be useful for other admissions tests, such as the TMUA

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This is a set of 60 flashcards with facts from the Mathematics Admissions Test syllabus. If you print them double-sided on A4 paper then the backs will match up with the fronts (for example, the card with sin2x on the front has 1−cos2x on the back). If you would like to stick to them onto actual card, then you could print them single-sided and match the fronts to the backs by hand. The cards measure 8.5cm by 5.5cm, which should be the right size to fit onto most standard playing cards or collectible trading cards. The back of each card has spots in the corners to help you sort them out if you get them mixed up.

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STEP can be taken at an authorised test centre (this may be your school or college) The registration deadline is 4th May 2025.

STEP II is on 11th June 2025

STEP III is on 16th June 2025

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The main website for STEP, including information about the exams, past papers and worked solutions

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Sixth Term Examination Papers (STEP) | Undergraduate Admissions

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Achievement in the Sixth Term Examination Papers (STEP) normally forms part of a conditional offer to read mathematics at Cambridge. The examinations are set and administered by the OCR examination board and are usually taken in mid to late June. There is extensive free online support offered by the Faculty and NRICH to help you prepare for STEP, and many other free online resources are also available.

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ESAT is taken at a Pearson VUE test centre.

There are two sittings for the exam:

1) Thursday 9th or Friday 10th October 2025 (this sitting must be chosen if you are applying to Cambridge) - registration will be available from June

2) Tuesday 6th or Wednesday 7th January 2026 (either sitting can be chosen if you are not applying to Cambridge, but Imperial recommend sitting 1)

There will be a fee of £75 to take the ESAT in 2025/26. There is a bursary available, which must be applied for before you book your exam slot - see the ESAT website for details: https://esat-tmua.ac.uk/bursary-scheme/

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You must live in the UK permanently at the time of application and plan to take your test at a UK Test Centre.

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A website with support materials for students working towards taking STEP

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STEP is an additional mathematics examination, taken at the end of Year 13, which forms part of conditional offers to applicants for mathematics and some related degrees at Cambridge and some other universities.

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PAT is taken at a Pearson VUE test centre.

The dates for the 2025 exam are Wednesday 22nd and Thursday 23rd October 2025.

Registration is via the website and will open on 18th June.

There will be no fee to take the PAT in 2025.

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STEP Support Programme - Foundation Modules

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STEP Support Programme - Foundation Modules

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These modules are designed for students to work through in preparation for STEP. The first 10-15 modules are accessible to year 12 students; it is recommended that you work through these if you are considering taking STEP.

Can be followed by the STEP 2 and STEP 3 modules (see below) in year 13

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STEP Support Programme - Foundation Modules

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STEP Support Foundation modules | STEP Support Programme

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This module involves some algebraic manipulation and some work on quadratic equations. There is also a geometry question with a surprising result.

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Main website for the ESAT, including the specification and test format

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ESAT for Universities

The ESAT is currently used by Imperial College London and the University of Cambridge for entry onto selected courses in Engineering, Physics, Natural Sciences, and Veterinary Medicine.

You must refer to the relevant course pages of each institution's website to find out if the courses you are applying to require the ESAT. Failure to register for the ESAT where it is compulsory may invalidate your application.

Different courses will require you to register for a different combination of parts. You will need to select the parts relevant to your application at the point of registration. You will not be able to change this when you get to the test centre so it is your responsibility to ensure you are correctly registered.
If you are applying to Cambridge and are required to sit the ESAT you must sit it in the October test sitting. All other candidates can choose whether to sit the test in October or January.

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The ESAT is currently used by Imperial College London and the University of Cambridge for entry onto selected courses in Engineering, Physics, Natural Sciences, and Veterinary Medicine. You must refer to the relevant course pages of each institution's website to find out if the courses you are applying to require the ESAT.

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Some universities will accept AEA as part of an alternative offer. For example, King's College, London, requires A Level Further Maths as part of its standard offer for Mathematics courses, but will consider students only taking AS Further Mathematics if they have also taken STEP or AEA.

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AEA can be taken at an authorised test centre (this may be your school or college)

The entry deadline is 21st February 2025 (late fees will apply after this date)

The 2025 exam is on 24th June.

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The main website for Edexcel's AEA qualification, including the specification

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Edexcel Advanced Extension Award Mathematics information for students and teachers, including the specification, past papers, news and support.

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Mathematics | Advanced Extension Award | Pearson qualifications

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Edexcel Advanced Extension Award Mathematics information for students and teachers, including the specification, past papers, news and support.

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A YouTube playlist of worked solutions to AEA papers by Tayyub Majeed, @tm_maths on X

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EdExcel Advanced Extension Award (AEA) in Mathematics

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Share your videos with friends, family, and the world

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TMUA includes questions on logic and proof, which is not covered in A Level Mathematics. These are some additional notes on the topic.

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The content specification for ESAT

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Specimen tests and Practice Tests for the ESAT

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Undergraduate Admissions Test UK (UAT UK) certification testing with Pearson VUE

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UAT-UK offers two computer-based undergraduate admissions tests for a range of university courses in engineering, science, and mathematics related courses: The Test of Mathematics for University Admission (TMUA) and the Engineering and Science Admissions Test (ESAT).

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The ESAT is a new exam for 2024, so there are few past paper materials. However, the past papers for the ENGAA and NSAA which were previously used for the same courses are recommended for preparation for the ESAT.

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These are historic ENGAA Engineering Admissions Assessment (ENGAA) and Natural Sciences Admissions Assessment (NSAA) papers that contain questions of the type found in the ESAT. We have indicated in the papers where questions are not in the ESAT specification.

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The main website for the PAT, including the specification, past papers and test format

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Test preparation and practice materials
Read the candidate preparation checklist.
Taking any type of test or exam can be stressful, but you can help build your confidence by doing some preparation ahead of time. You may also do better in the real test if you've practised some past papers, and got used to the format and timings of the admissions test you have to take.

The PAT is now a hybrid test, with the questions online and a printed booklet for your answers. However, as the content hasn't changed, you will still find it helpful to practise for the test with the past papers and other practice materials on this page.

Calculators
A digital scientific calculator will be included as part of the online interface for the PAT from October 2023 onwards. Candidates must use this and may not bring their own calculators, of any description, into the exam. You will be able to try out the online digital calculator when practising the online practise test available below.

Our general advice is to follow these steps:

1. Before 29 September (the registration deadline), check that you have received email confirmation of your test registration, together with your test registration ID and log in details (username and password) for the online test platform.

2. Explore the test-specific practice materials for your test available below.

3. Watch the video demonstrating how to use the online test platform and prepare for your test. Download the transcript for preparing for your online PAT.

4. Once you have watched the videos and explored the practice materials below, we strongly recommend you have a go at your online practice test (click on the button below) using the 2022 past paper. You can download and print the sample answer booklet provided or just write your answers on available paper. You might want not to look at this past paper in advance so that you can mimic the experience of taking the online test unseen and as if for real. Having a go at the online practice test will not only allow you to practise answering some past questions and learn the structure of the test. It will also allow you to familiarise yourself with the online test platform and the tools available to support you, including the digital scientific calculator. These will also include accessibility features such as increasing font size, using coloured overlays and high contrast. This will mean that on test day you are able to focus fully on the content of your answers.

5. Please note that you will not be able to access a score or any feedback on your online practice test.

Here are our top tips for preparing specifically for the PAT:

1. Watch the our video with our Physics tutor giving insights and tips on preparing for the PAT. Download the video transcript.

2. Check the syllabus: we strongly recommend that you check the details and ensure that you have covered the relevant material. The syllabus covers A-level Maths and Physics plus knowledge of material covered at GCSE. However we cannot guarantee when the material will be covered in your school by the time of the test date. If this is the case, we recommend talking to your teacher about how best to tackle these subjects before the test. This may require you to do some independent study by reading through your textbook or looking at the online resources provided by your exam board.

3. Watch the PAT webinar on this page, which goes through some of the questions from the 2006 paper.

4. Visit the Department of Physics Youtube channel, and watch their videos about admissions and preparing for the PAT.

5. Get practice doing some problem solving/hard physics questions that are not included in your A-level syllabus. It is advisable to do questions from a range of other sources, not just A-level type questions which can be more structured in nature than the PAT. There are some links below to other websites/material which you might find helpful.

6. Sit at least one past paper in test conditions. This is really important as it will help you get used to how much time to allocate to each question and keep within the two hour limit.

Don't worry if you find the past or specimen papers very difficult - they're supposed to be! All our tests are designed to stretch you further than you have been stretched before - most candidates will find them really hard.
Practise the online PAT PAT answer booklet
Note: during the real test, for security reasons, the test platform will lock if you try and navigate away from it. If this happens accidentally, your test centre administrator will be able to unlock it again.

PAT webinar

Workbook questions

The PAT workbooks contain many questions of varying difficulty and subject matter, and the accompanying solutions manuals outline possible approaches to each question in detail.

Solutions 1 ; Solutions 2

Past or specimen papers

As you may notice when going through past papers, the PAT has undergone various changes in the past few years.

In 2015, multiple-choice questions were removed, and longer 20 mark questions were replaced by shorter 10 mark questions. In 2017, multiple-choice questions were re-introduced and the physics and maths sections were mixed up.

The PAT is now hybrid - with online questions and a paper answer booklet; however, the older papers will still be of use when preparing. Please note that from 2023, only use of the digital scientific calculators provided on the test platform will be permitted. You can explore using this via the online practice PAT.

We do not generally provide solutions to the past papers. When marking the PAT all suitable methods for solving the questions are allowed and we would not want you to feel only one specific way of solving the problem will gain you marks. Solutions to the 2009 paper can be found on the Physics website, together with a set of model solutions.

Blank answer booklet
PAT answer booklet 2022
The University does not endorse, or allow use of, its tests that are protected by copyright for commercial use

PAT Syllabus

Updated 6 June 2018 and still current

Please note that the formulae included in this syllabus do not represent an exhaustive list of formulae which may be used within the test.

Syllabus for the Mathematics content

Elementary mathematics:

Knowledge of elementary mathematics, in particular topics in arithmetic, geometry including coordinate geometry, and probability, will be assumed. Questions may require the manipulation of mathematical expressions in a physical context.

Algebra:

Knowledge of the properties of polynomials, including the solution of quadratics either using a formula or by factorising.

Graph sketching including the use of differentiation to find stationary points.

Transformations of variables.

Solutions to inequalities.

Elementary trigonometry including relationships between sine, cosine and tangent (sum and difference formulae will be stated if required).

Properties of logarithms and exponentials and how to combine logarithms, e.g. log(a) + log(b) = log(ab) .

Knowledge of the formulae for the sum of arithmetic and geometric progressions to n (or infinite) terms.

Use of the binomial expansion for expressions such as (a+bx)n, using only positive integer values of n.

Calculus:

Differentiation and integration of polynomials including fractional and negative powers.

Differentiation to find the slope of a curve, and the location of maxima and minima.

Integration as the reverse of differentiation and as finding the area under a curve.

Simplifying integrals by symmetry arguments including use of the properties of even and odd functions (where an even function has f(x)= f(-x), an odd function has f(-x)= - f(x)).

Syllabus for the Physics content

Mechanics:

Distance, velocity, speed, acceleration, and the relationships between them, eg velocity as the rate of change of distance with time, acceleration as rate of change of velocity with time. Understand the difference between vector quantities (eg velocity) and scalar quantities (eg speed). Knowledge and use of equations such as speed = distance / time, acceleration = change in velocity / time or the SUVAT equations.

Interpretation of graphs, eg force-distance, distance-time, velocity-time graphs and what the gradient of a curve or area underneath a curve represents.

Response of a system to multiple forces; Newton's laws of motion; know the difference between weight (= mg) and mass; vector addition of forces.

Circular motion including equations for centripetal force (F=mω²r or F=mv²/r) and acceleration (a=v²/r or a=ω²r).

The meaning of the terms friction, air resistance and terminal velocity and how they can be calculated.

Levers (including taking moments about a point on an object), pulleys (including calculating the tension in a rope or the overall motion in a system of ropes and pulleys) and other simple machines combining levers, springs and pulleys.

Springs, including knowledge of Hooke's law (Force = - kx) and stored potential energy ( = 1/2 kx² ).

Kinetic energy (= 1/2 mv²) and gravitational potential energy (= mgh in a constant gravitational field) and their inter-conversion; what other forms of energy exist (eg thermal, sound).

Conservation of energy and momentum (=mass x velocity); power ( = energy transfer/time) and work ( = force x distance moved in direction of force).

Waves and optics:

An understanding of the terms longitudinal and transverse waves; and that waves transfer energy without net movement of matter.

Be able to define the amplitude, frequency, period, wavelength and speed of a wave. Knowledge and use of formulae for the wave speed = wavelength x frequency and frequency = 1 / period (with units of hertz, Hz).

Basic properties of the electromagnetic spectrum, eg identify and correctly order parts of the spectrum by wavelength or frequency (radio waves, microwaves, IR, visible light, UV, X rays and gamma rays) and the nature and properties of electromagnetic waves (transverse, travel at the speed of light in a vacuum).

Description of reflection at plane mirrors, where the angle of incidence (the angle between the incident ray and the normal) = angle of reflection (angle between the reflected ray and the normal).

Refraction, including the definition of refractive index (n) as the ratio of the speed of light in a vacuum to the speed of light in a material and Snell's law n₁sinθ₁=n₂sinθ₂. Elementary properties of prisms and optical fibres including total internal reflection, where total internal reflection occurs at an angle θc when sinθc=n₂/n₁

Qualitative understanding of how interference, diffraction and standing waves can occur.

Electricity and magnetism:

Understanding of the terms current ( = charge / time), voltage (potential difference = energy / charge), charge, resistance ( = voltage / current) and links to energy and power (power = voltage x current, power = energy / time). Knowledge of transformers, including how the number of turns on the primary and secondary coils affect the voltage and current.

Understanding circuit diagrams including batteries, wires, resistors, filament lamps, diodes, capacitors, light dependent resistors and thermistors. Knowledge of current, voltage and resistance rules for series and parallel circuits.

Knowledge of the force between two point charges (Force= kQ₁Q₂/r²(where k is a constant)) and on a point charge in a constant electric field (Force = charge x electric field).

Understanding that current is a flow of electrons; the photoelectric effect, where photoelectrons are emitted if they are given sufficient energy to overcome the work function of the material, and how to find the energy of accelerated electron beams ( energy = charge x potential difference).

Natural world:

Atomic structure; that atoms consist of protons, neutrons and electrons, definition of the atomic number, Bohr model of the atom.

Basic knowledge of bodies in our Solar System, including planets, moons, comets and asteroids. (Name and relative positions of the planets should be known but detailed knowledge of their physical parameters is not required).

Know what is meant by the phrases 'phases of the moon' and 'eclipses' and how the position of the observer on the Earth affects their view of these events.

Knowledge of circular orbits under gravity including orbital speed, radius, period, centripetal acceleration, and gravitational centripetal force. This may include equating the force between two masses due to gravity (F=GM₁M₂/r²) to centripetal force of a smaller body orbiting a larger body (F=mω²r or F=mv²/r) and use of centripetal acceleration (a=v²/r or a=ω²r).

Understanding of the terms satellites; geostationary and polar orbits.

Problem solving:

Problems may be set which require problem solving based on information provided rather than knowledge about a topic.

If there are parts of the syllabus you haven't yet covered in school, we recommend talking to your teacher about how best to tackle these topics before the test date. This may require you to do some independent study by reading through your textbook or looking at the online resources provided by your exam board. You can then test yourself using the further resources outlined on this page and picking relevant questions.

Websites and resources

Visit the Physics website for a list of other helpful resources and websites to help you to prepare for the PAT.

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What is the PAT?If you are applying for one of the following courses you will be required to sit the Physics Admissions Test (PAT) - formerly known as the Physics Aptitude Test - as part of the

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The website for the Physics department at the University of Oxford, including resources for the PAT

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University of Oxford Department of Physics

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Useful websites and resources for the PAT

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Here you will find a selection of useful websites and resources that might help you to prepare for the Physics Aptitude Test (PAT)

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The Preparing for the PAT course 2025 runs online from June to October. It is for students intending on taking the PAT in October 2025.

Sign up here: Preparing for the PAT course | University of Oxford Department of Physics

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University of Oxford Department of Physics

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A course designed to ​​develop the problem solving skills necessary for the Physics Aptitude Test (PAT)

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The University of Oxford does not provide answers to PAT past papers; however, unofficial written solutions can be found on the Physics and Maths Tutor website

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Oxford Physics Aptitude Test (PAT) - PMT

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Past papers, examiner reports and model answers for the Oxford Physics Aptitude Test (PAT)

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A YouTube playlist with solutions to selected PAT problems, from the Oxford Physics department

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PAT Problems with Helena, Kathryn and Kat

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Kat, Kathryn and Helena go through a different PAT problem each week.

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An online course to prepare for the PAT

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University of Oxford Department of Physics

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A course designed to ​​develop the problem solving skills necessary for the Physics Aptitude Test (PAT)

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A talk on the PAT from an Oxford open day in June 2024, by Dr Jenny Barnes

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Department of Physics University of Oxford

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Preparing for the Physics Aptitude/Admissions Test (PAT)

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A playlist of videos from Jesus College, Oxford

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Share your videos with friends, family, and the world

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A short YouTube video on the details of the ESAT by Mr Gill's Science Stuff channel

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An unofficial guide to the PAT, updated for 2024

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PAT Oxford: The Definitive PAT Starter Guide For 2024

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Starting your PAT preparation? This guide is the perfect starting point for your PAT revision, with essential info, tips and practice questions!

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Follows on from the STEP foundation modules - it is recommended that you start these at the beginning of year 13

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Each of these STEP 2 modules consists of a selection of STEP questions, together with hints and solutions. The modules also contain "topic notes" which consists of useful formulae you should be aware of, together with some general hints and advice.

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Follows on from the STEP foundation modules and STEP 2 modules. It is recommended that you start these around January of year 13.

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The STEP 3 specifications include all of the material in the STEP 1 and STEP 2 specifications as well as some additional further mathematics topics such as hyperbolic functions and polar coordinates.

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This free online textbook by Stephen Siklos provides a commentary on how to approach and solve STEP-style problems. In can be read online or downloaded as a PDF.

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About this book

STEP

Worked Problems

Worked problem 1

Worked problem 2

Problems

P1 An integer equation

P2 Partitions of 10 and 20

P3 Mathematical deduction

P4 Divisibility

P5 The modulus function

P6 The regular Reuleaux heptagon

P7 Chain of equations

P8 Trig. equations

P9 Integration by substitution

P10 True or false

P11 Egyptian fractions

P12 Maximising with constraints

P13 Binomial expansion

P14 Sketching subsets of the plane

P15 More sketching subsets of the plane

P16 Non-linear simultaneous equations

P17 Inequalities

P18 Inequalities from cubics

P19 Logarithms

P20 Cosmological models

P21 Melting snowballs

P22 Gregory's series

P23 Intersection of ellipses

P24 Sketching x m ( 1 − x ) n

P25 Inequalities by area estimates

P26 Simultaneous integral equations

P27 Relation between coefficients of quartic for real roots

P28 Fermat numbers

P29 Telescoping series

P30 Integer solutions of cubics

P31 The harmonic series

P32 Integration by substitution

P33 More curve sketching

P34 Trig sum

P35 Roots of a cubic equation

P36 Root counting

P37 Irrationality of e

P38 Discontinuous integrands

P39 A difficult integral

P40 Estimating the value of an integral

P41 Integrating the modulus function

P42 Geometry

P43 The t substitution

P44 A differential-difference equation

P45 Lagrange's identity

P46 Bernoulli polynomials

P47 Vector geometry

P48 Solving a quartic

P49 Areas and volumes

P50 More curve sketching

P51 Spherical loaf

P52 Snowploughing

P53 Tortoise and hare

P54 How did the chicken cross the road?

P55 Hank's gold mine

P56 A chocolate orange

P57 Lorry on bend

P58 Fielding

P59 Equilibrium of rod of non-uniform density

P60 Newton's cradle

P61 Kinematics of rotating target

P62 Particle on wedge

P63 Sphere on step

P64 Elastic band on cylinder

P65 A knock-out tournament

P66 Harry the calculating horse

P67 PIN guessing

P68 Breaking plates

P69 Lottery

P70 Bodies in the fridge

P71 Choosing keys

P72 Commuting by train

P73 Collecting voles

P74 Breaking a stick

P75 Random quadratics

Syllabus

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Advanced Problems in Mathematics: Preparing for University

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This book is intended to help students prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Papers), and is recommended as preparation for any undergraduate mathematics course. The questions analysed in this book are all based on recent STEP questions, and each is followed by a comment and a full solution.

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Solutions and explanations to selected STEP questions from Underground Mathematics

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Browse stations/topics, pervasive ideas, rich resources, review questions and glossary entries.

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A searchable database where you can search for STEP questions by year, paper (STEP 1, 2 or 3) or topic

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The AMSP On-Demand Professional Development (ODPD) modules for supporting students who are preparing for admissions tests are very good, are free to teachers in UK state schools/colleges, and can be done at any time.

There is one for MAT/TMUA, and one for STEP.

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On Demand professional development offers total flexibility. Teachers can access the materials, day or night, over a period of up to one year.

It's also completely free of charge for eligible teachers.

Most on-demand courses feature a variety of content including:

short videos

practice materials

triggers for reflection and suggestions for further study

Teachers will be encouraged to interact and collaborate with others studying the same material via the online forums. An expert tutor will be monitoring the forums and will respond to queries.

We offer on-demand courses for:

This video will provide a deeper view of what's involved in this unique form of professional development:

Have you seen our PD Videos?

These videos provide ideas for how overarching themes and technology can be incorporated into lessons for GCSE Mathematics, AS/A level Mathematics and AS/A level Further Mathematics.

Skip to content

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On-Demand Professional Development - AMSP

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On Demand professional development offers total flexibility. Teachers can access the materials, day or night, over a period of up to one year.

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Information PowerPoint slides for TMUA, from a session in early spring 2026 (please check university website for the most up to date information!)

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Information PowerPoint slides for MAT, from a session in early spring 2026 (please check university website for the most up to date information!)

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Information PowerPoint slides for STEP, from a session in early spring 2025 (please check university website for the most up to date information!)

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Information PowerPoint slides for PAT, from a session in early spring 2025 (please check university website for the most up to date information!)

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Information PowerPoint slides for ESAT, from a session in early spring 2025 (please check university website for the most up to date information!)

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The University of Oxford requires TSA for these courses:

Philosophy, Politics and Economics

Economics and Management

Experimental Psychology

History and Economics

Human Sciences

Psychology, Philosophy and Linguistics

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TSA is taken at a Pearson VUE test centre.

The dates for the 2025 exam are 21st, 22nd, 23rd, 24th and 27th October 2025.

Registration is via the website and will open on 18th June.

There will be no fee to take the TSA in 2025.

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The main TSA website, including specification and past papers TSA (Thinking Skills Assessment) | University of Oxford

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Practice materials

Taking any type of test or exam can be stressful, but you can help build your confidence by doing some preparation ahead of time. The TSA is not subject-specific, so there is no curriculum to learn. It assesses the generic thinking skills you already have, and practising for the test will help you develop these skills.

Note: Candidates applying for Geography are no longer required to take the TSA but from 2023 will sit the GAT.

Here are our top tips for preparing for the TSA:

1. Watch the videos below to find out more about how to approach the TSA.

2. Review the sample papers for the TSA below. This will help you to feel familiar with the test paper and know what to expect.

3. Sit at least one past paper in test conditions. This is really important as it will help you get used to how much time to allocate to each question.

4. Check the details of the test specification and the marking criteria shown below.

5. Have a look at the TSA preparation videos on the CAAT website, to familiarise yourself with what taking the test will be like.

Don't worry if you find the past or specimen papers very difficult - they're supposed to be! All our tests are designed to stretch you further than you have been stretched before - most candidates will find them really hard.

An introduction to the Thinking Skills Assessment (TSA)

In the presentation below we explain why TSA is used by universities for admissions, and provide information about the test and how to prepare for it.

TSA Workshop video

Section 1 specimen and past papers

The first section of the TSA is made up of 50 multiple-choice questions. Below you can find a specimen paper, in addition to past papers going back to 2008. You can also download answer sheets for each paper and a score conversion chart.

Section 2 specimen and past papers

Section 2 of the TSA test is a writing task. You will have 30 minutes to write a single short essay. There will be a choice of four essay questions, on general subjects that do not require any specialised knowledge.

The writing task gives you an opportunity to show that you can communicate effectively in writing, organising your ideas and presenting them clearly and concisely. You should start by planning the essay carefully, deciding what are the main points that you want to make in the limited time available, and how to organise your answer to explain and convey them clearly. You will have plenty of space if you want to use it, but a concise and well-structured answer may be more effective than a longer essay.

It is important that your answer is relevant to the question, and addresses it directly. If the question requires you to make judgements and express your own opinions, try to provide coherent arguments to support your views, and consider the merits of possible counter-arguments. Your essay will be judged by the quality of the writing, and the way you use what you know.

The document below, written by an Oxford tutor, gives several example questions from past papers and discusses ways to approach them.

Below you can find a specimen paper and past papers going back to 2008.

Test question guide and explanation of results
Explanation of results:
Section 1 scores 1 mark per question. Scores are calculated on the TSA scale to one decimal place (running approximately 0-100). The scale is an estimate of the candidate's ability, which makes scoring comparable by factoring in the question and overall test difficulty, using the Rasch statistical technique. Marking of this section is automated.

Section 2 is reviewed by the admissions tutor(s) of the college you apply to.

Further reading

John Butterworth and Geoff Thwaites, Thinking Skills (Cambridge University Press, 2013)

Anne Thomson, Critical Reasoning: A Practical Introduction (Routledge, 2008)

Nigel Warburton, Thinking from A to Z (Routledge, 2000)

Alec Fisher, Critical Thinking: An Introduction (Cambridge University Press, 2011)

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What is the TSA? The Thinking Skills Assessment (TSA) is divided into two parts: a 90-minute, multiple-choice Thinking Skills Assessment and a 30-minute writing task. The TSA will be a paper-based