ESAT Worked Solutions

Who writes the questions, and how they are checked

Our researchers write them. Doctoral researchers and recent graduates in engineering, physics and mathematics at Oxford and Cambridge, who teach the subjects. Every question here was written by one of them.

Then every question is checked by software we developed ourselves, in house, for exactly this work: fixed rules where an answer has to be exact, and a review pass where the question needs judgement. This is the part we would point at first. Someone reading several hundred questions is sharp on the first and tired by the hundredth; the checks are not. On every question, every time a paper is built, they:

• work the question out again from scratch, without being told the intended answer, and have to arrive at the same one
• confirm no second option is also correct
• confirm every wrong option is one a student could really reach, and name the mistake that gets them there

Nothing goes into a paper until it has passed all of it. That is why the worked solutions can tell a student what reasoning led them to C, rather than only that the answer was B.

The Engineering and Science Admissions Test (ESAT) is a computer-based admissions test set by UAT-UK, a collaboration between Imperial College London and the University of Cambridge, and delivered by Pearson VUE at test centres worldwide. It is used by Imperial College London, the University of Cambridge, the University of Oxford and UCL for a range of engineering and science courses.

It replaced the older ENGAA and NSAA papers, and at Oxford it replaces the Physics Aptitude Test (PAT) for Engineering Science, Physics, Physics and Philosophy, and Biomedical Sciences. If you sat mock papers labelled ENGAA or NSAA, they remain useful practice: the ESAT draws on the same material.

How the test is built

How the score works

The anchor points are worth pausing on, because almost nothing else about the scale is published. A 4.5 is the middle of the field. A 9.0 is the top tenth. Everything between those two is where admissions decisions are actually made, and it is a narrow band.

Securing an offer requires rapid pattern recognition and execution under pressure. The worked solutions on this site focus on the fastest correct route to each answer, because at 89 seconds a question the method you choose decides the score more often than the mathematics you know.

There are two sittings in the cycle. Cambridge and Oxford applicants sit in October; the January window is for Imperial and UCL applicants, and at Cambridge and Oxford only for a small number of mature college and foundation year routes, so check your own course page before assuming you have a choice. Failing to register for a test your course requires can invalidate the application, and no university will make an exception for it.

Every deadline in the 2026 to 2027 cycle

Two of these are earlier than people expect. Access arrangements close on 14 September, a fortnight before the 28 September booking deadline, and the bursary closes on 21 September, a week before it. Both have to be granted before you can book, so a candidate who needs either one is working to a mid September deadline, not a late September one.

Cost, bursaries and what you cannot change

Mathematics 1 is fixed. The two remaining choices are where candidates lose marks before the test even begins, and you cannot change your modules on test day, so the decision is made at booking.

Many courses specify which modules you must take. Where they do, there is no choice to make: take what the course names. Check your own course page before booking, because these requirements are revised between cycles.

Who requires the ESAT, and which modules

Four universities require the ESAT for 2027 entry. The tables below are transcribed from UAT-UK's own Course List for 2027 entry, last updated April 2026, which is the only document that lists every ESAT course at every institution in one place. Durham, LSE and Warwick appear on that list too, but for the TMUA and the TARA, not the ESAT, which is worth knowing because some university pages name all seven institutions without separating the tests.

Two rules from that document decide bookings and are easy to miss. If you apply to two ESAT courses and one of them names compulsory modules, those compulsory modules govern your booking, even if the other course leaves the choice open. And module selection cannot be changed after booking: it means cancelling and rebooking, subject to whatever test centre slots are left.

Imperial College London

University of Cambridge

Natural Sciences applicants should choose the two optional modules that match the subjects they intend to study. Cambridge applicants sit in October.

University of Oxford

At Oxford the ESAT has replaced the Physics Aptitude Test. Oxford's own Physics department now calls the PAT its previous admissions test, and the PAT archive remains worthwhile preparation for interview even though the test itself is retired.

UCL

If you genuinely have a free choice

• Take Mathematics 2 if you are able to. It is pure mathematics with no statistics and no probability at all, it overlaps heavily with Mathematics 1, and preparation for one improves the other. It is also the module where a strong candidate can most reliably separate themselves.
• Pick the science you are furthest ahead in, not the one you enjoy most. The syllabus is fixed and shallow; what varies is how much of it you have already met.
• Check the overlap with your A-level timetable. A module whose content you are covering in class in September is far cheaper to prepare than one you finished last year.
• Do not choose a module because it looks easier. Scores are reported per module and read in context; there is no advantage in an easy module you have not prepared.

Module requirements and test dates are set by the universities and change between admissions cycles. Treat everything above as a starting point and confirm it against the official pages linked at the foot of this page before you book.

Most preparation material is written from an impression of the test. This library is written against a measurement of it. UAT-UK publishes no ESAT past papers, but it does point candidates at the ENGAA and NSAA Section 1 papers from 2016 to 2023, which is the closest thing to a real corpus that exists. UAT-UK describes them as papers that “contain questions of the type found in the ESAT”, and notes that it has marked the questions in them that fall outside the ESAT specification. Those 648 questions across 16 papers were parsed and measured, every question counted, and the tables below are that measurement.

A note on the instrument, because it decides whether any of this means anything. A stem's length is counted with every run of mathematics collapsing to one word, so $x^2 + 3x - 4$ counts as one and not five. Counting it the other way makes an algebra question look twice the length of a prose one. The same rule is used on both sides of every comparison here.

What a real question looks like

Two things in that table surprise almost everyone. A real Biology question runs to a median of 80 words and 45% of them carry a figure, against 22 words and 4% for Mathematics 2. They are the same 27 questions in the same 40 minutes, but they are not the same kind of reading. And the 90th percentile column is the one that decides your pacing: in Biology one question in ten runs past 165 words, which at 89 seconds a question is most of your time gone before you have done any biology.

How many options a real question offers

Only 18% of real questions offer five options. The commonest is eight. This matters more than it looks: on a five-option paper a blind guess pays 20%, but across the real distribution it pays about 15%. A raw mark on any five-option mock, including the ones on this page, is therefore a mild overestimate of what the same performance scores on the day. It is worth knowing rather than discovering.

This library, measured the same way

Where this library differs from the real thing

Every module here reads shorter than the real thing, and Chemistry and Biology carry far fewer figures. That is a real limitation and it points one way: use these to drill method and speed, and use the official ENGAA and NSAA papers to practise reading a long stem under time. The gap is largest in Biology and Chemistry, which is exactly where a candidate who has only ever practised on short questions gets hurt.

Figure share is detected from the stem, because a paper has to tell a candidate that a diagram is there. That method understates Chemistry, whose figures are often structural formulae printed as the options rather than referred to in the stem, so treat the Chemistry figure numbers as a floor. Every other number in these tables is counted from the questions themselves.

Or go straight to a single module. Advanced Mathematics below is the official Mathematics 2 module:

Everything on this page is written from the documents below. Where a preparation site and an official page disagree, the official page wins, and requirements are revised between admissions cycles, so check these before you book anything.

The test itself: UAT-UK

• About the ESAT: structure, scoring, test dates, registration deadlines and the list of universities that use it.
• ESAT Content Specification (PDF): the definitive syllabus. Appendix 1 lists every assessable sub-topic, and is the document the tables above are counted from.
• Preparation materials: official subject guides, plus the full archive of ENGAA and NSAA past papers from 2016 to 2023 with answer keys. These are the closest thing to real past papers that exists.
• ESAT Guide: Mathematics 1 (PDF) and Notes on Mathematics for Mathematics 2 (PDF): the two documents that matter most for the questions on this page.
• Course List for 2027 entry (PDF): every course at every UAT-UK institution that requires a test, with the UCAS code and the exact modules. This is the document the requirement tables above are transcribed from, and it settles arguments that university pages do not.
• Deadlines: booking, access arrangements, bursary and results dates for both sittings, with times. The source for the dates table above.
• Scoring and results: how the 1.0 to 9.0 scale is anchored, when results appear, and the fact that you cannot choose which institutions see them.
• What the test costs, and the bursary scheme for UK candidates in financial hardship, which covers the full fee.

The universities

• Imperial College London, the ESAT. The clearest of the three on module requirements: it names the modules department by department, and it is the only one of Cambridge, Oxford and Imperial that lets you sit the January window.
• University of Cambridge, admissions tests. Lists which courses require the ESAT; the individual course pages carry the module detail.
• Oxford, Engineering Science and the ESAT and Oxford Physics, the ESAT. The Physics department page also keeps the PAT past papers online, which remain useful preparation even though the PAT itself has been replaced.
• UCL Faculty of Engineering: the ESAT. UCL requires it for two courses only, BEng and MEng Electronic and Electrical Engineering, and uses the score alongside the rest of the UCAS application rather than as a threshold.

External links open in a new tab and are maintained by the universities and UAT-UK, not by this site. If one of them contradicts anything written here, follow the official page and please let me know so it can be corrected.

Both are built against the official material rather than against an impression of it. The stem lengths, the number of worked steps and the share of questions carrying a figure are measured from the ENGAA and NSAA papers UAT-UK points candidates at, and each paper is commissioned to match that distribution question by question. Every question is then independently re-derived: the key confirmed as uniquely correct, and every wrong option traced to a specific named mistake that produces that exact value.

One limit, stated rather than buried: these are printed papers and the real ESAT is sat on screen at a Pearson VUE centre, so a printed mock trains the mathematics and the pacing but not the interface. Use UAT-UK's free on-screen practice at least once for that. Each pack also ships with a page listing its measured differences from the real test.