ESAT Mathematics 2, module by module

Appendix 1 of the official specification gives Mathematics 2 eight topic headings, numbered MM1 to MM8. The sentence that matters more sits in the main body: questions draw upon the topics listed for each module in Appendix 1, and all modules assume knowledge of the content of Mathematics 1. The table on page 3 spells this out module by module. Biology draws on B plus M, Chemistry on C plus M, Physics on P plus M, and Mathematics 2 on M plus MM.

So the examinable surface of your 40 minutes of Mathematics 2 is the eight MM headings together with all seven Mathematics 1 headings: Units, Number, Ratio and proportion, Algebra, Geometry, Statistics and Probability. That is not an inference, it is the stated design. Candidates who revise Mathematics 2 as "the calculus module" routinely drop marks on ratio, unit conversion and probability, which look too elementary to bother with right up until one of them costs you a question you had ninety seconds for.

There is a second consequence. If your course requires Mathematics 1 and Mathematics 2, the M content faces you twice across 54 questions. Time spent making Mathematics 1 automatic pays off in both modules. Time spent only on MM pays off in one.

The specification publishes topic headings only. The table below is a verbatim copy:

These eight are not eight parallel blocks. MM1 and MM8 are the skeleton: what a function is, how composition and inverses behave, completing the square, discriminants, factor and remainder structure, and what all of that looks like once it is drawn and then translated, stretched or reflected. MM3, MM4 and MM5 are three different languages for describing the same kinds of object: the coordinate language of lines and circles, the language of angle and periodicity, and the language of growth and decay. MM6 and MM7 are two operators that act on everything above. MM2 stands slightly apart, with arithmetic and geometric summation and convergence at its centre.

On difficulty, the specification offers one line: the test is designed to be challenging in order to differentiate effectively between able applicants, including those who might have achieved the highest possible grades in school examinations. The revision consequence, in our experience of teaching it, is that drilling one heading at a time will not get you there. Marks tend to go missing where two or three headings are bolted together, for instance differentiating a composite built from an exponential or a trigonometric function and then reading a property off the resulting graph. Topic-by-topic practice sharpens the tools; mixed practice teaches you to reach for the right one under time pressure.

One caveat worth stating plainly: the specification does not publish sub-topics under these headings, so anything advertised as an "official detailed checklist" is not official. The reading of how the headings relate to one another, and the advice about mixed practice, are our teaching judgement rather than official statements.

The official Course List for 2027 entry sets out module requirements course by course. Wherever the maths and physics density is high, Mathematics 2 is named:

Several of the Oxford engineering routes also have with Foundation Year variants, listed on their own rows in the Course List. Check against the exact UCAS code you are applying under rather than going by course name.

One rule catches people out every cycle. UAT-UK states that if you are applying to two or more courses requiring the ESAT and one of them offers a free choice of modules, your module selection must be dictated by the courses with compulsory modules listed, and that failing to register for the compulsory modules may invalidate your application to that course. A common example: Imperial Mechanical Engineering (Mathematics 2 and Physics specified) alongside Cambridge Natural Sciences (any two). Your combination is fixed at Mathematics 2 and Physics. The freedom Natural Sciences appears to offer is not freedom you actually have.

While you are checking, one thing to get straight about who uses this test at all. UAT-UK states that Imperial College London, the University of Cambridge, the University of Oxford and UCL all use the ESAT. Imperial's own ESAT page additionally lists Durham, LSE and Warwick under "ESAT for other universities", but in the official 2027 Course List those three carry TMUA requirements and no ESAT courses at all. UAT-UK separately says seven UK universities use one or more of its tests, which is exactly the number on Imperial's list, so that section reads like the membership list rather than the ESAT list. For module requirements, the Course List table is the document to trust.

In the official Course List, the row for Imperial's MEng Design Engineering (28G3) is flagged Two Modules Only, and the two are Mathematics 1 and Mathematics 2. It is the one row on the table that deserves its own paragraph.

Three things follow. Your total test time is 80 minutes rather than the 120 most candidates sit, with no break between the two modules. Mathematics 2 carries half the weight of everything the university sees, and there is no physics or chemistry score to dilute a weak one. And the overlap between your two modules is the highest of any combination available: Mathematics 1 examines the seven M topics, Mathematics 2 examines M plus MM, so the whole sitting is two samples from the same mathematical foundation.

For Design Engineering applicants that is both bad news and good news. The bad news is that if your maths wobbles, nothing else on the paper can carry you. The good news is that your preparation path is unusually clean: everything goes into maths, and you never have to ration revision time across three subjects.

In practice, treat Mathematics 1 and Mathematics 2 as one course rather than two. Get the seven M topics to the point where they require no thought, layer the eight MM topics on top, then rehearse in 40-minute blocks. The classic failure mode among candidates who revise them separately is that ratio and geometry feel solid inside a Mathematics 1 practice set, but become unrecognisable the moment they turn up wearing Mathematics 2 clothing inside a question about functions or graphs.

A handful of courses do not specify modules and hand the choice back to you. The Course List wording is Mathematics 1 plus any two modules chosen from Biology, Chemistry, Physics and Mathematics 2:

Note that Mathematics 2 is a legitimate option in every one of these. A surprising number of applicants assume Natural Sciences forces a choice between the three sciences. It does not.

Three questions decide it. First, is your A level maths clearly stronger than your second science? If so, Mathematics 2 is usually the better bet than grinding through a science you half know, because it shares its foundation with compulsory Mathematics 1 and therefore has the lowest marginal preparation cost of anything on the menu. Second, which direction are you heading in? A physical or materials route through Natural Sciences means you should be sharpening physics anyway, and Mathematics 2 pairs with it naturally; a biochemical route makes Chemistry and Biology the closer fit. Third, remember the rule from the previous section: the moment another course on your UCAS form specifies modules, this free choice evaporates and the specified combination wins.

UCL Electronic and Electrical Engineering is worth flagging in particular. The choice is formally open, and Mathematics 2 with Physics is the combination that maps most closely onto what the degree then asks of you, but that is a judgement about fit rather than something UCL states as a requirement.

Put the eight headings next to the pure maths content of A level Maths and the overlap is close to line by line: algebra and functions, sequences and series, coordinate geometry, trigonometry, exponentials and logarithms, differentiation, integration, graphs of functions. It reads like a contents page from AS and A2 pure. Equally telling is what the official list does not contain. There are no matrices, no complex numbers, no hyperbolic functions, no differential equations, no polar coordinates. None of the headings that usually mark out Further Maths appear among the eight.

For a student partway through A level Maths, then, Mathematics 2 asks for very little genuinely new content. Four other things are what actually need work.

The first is the absence of a calculator. UAT-UK states plainly that no calculator and no dictionary may be used. The difficulty for A level candidates is rarely that they cannot do the arithmetic, it is that they have become slow at it: common denominators, surd manipulation, exact trigonometric values for standard angles, changing the base of a logarithm and combining logs, squares and cubes worth knowing by sight. Two years with a calculator in reach erodes all of it faster than people expect.

The second is the seven Mathematics 1 topics. Units, number, ratio and proportion, algebra, geometry, statistics and probability are GCSE-era material for most A level students, unused for eighteen months, and Mathematics 2 assumes you have them.

The third is the multiple-choice format itself. Options can be worked backwards, orders of magnitude can be estimated, special values can eliminate candidates. None of that is trained by A level long-answer practice.

There is also one small thing the specification states outright and almost everybody skips: under Units, candidates are expected to be familiar with the SI prefixes from nano- to giga-, and with units written using negative indices, for example m s⁻¹ for velocity. Mathematics 2 assumes that too.

The fourth is speed, which gets its own section below. If you sit real timed practice and your accuracy is fine but you finish only twenty questions, the fix is arithmetic fluency and triage, not more content revision.

The arithmetic is blunt: 27 questions in 40 minutes is about 89 seconds each. Read that alongside two official rules. Each module is separately timed, and if you finish one early the time is not carried over to the next, so there is nothing to be gained by banking minutes in Mathematics 2. And there are no breaks between modules, so you go from the last question of one straight into the first question of the next.

Ninety seconds with calculus in the mix sounds brutal, and is less so than it reads, because the no-calculator rule protects you. Whoever wrote the question had to make the numbers workable by hand. That gives you a genuinely useful diagnostic: as soon as your intermediate values turn ugly, the problem is almost never that the question is hard, it is that you have taken the long road. Stop, reread the stem, and look for the shortcut, which is usually symmetry, structure, a graphical feature, or differentiating and substituting immediately rather than simplifying first.

A three-pass structure works well. Spend roughly 20 minutes on a first pass through all 27, answering anything you can see a route into and marking anything that needs sustained algebra, without ever sinking into a single question. Spend roughly 15 minutes on a second pass over the marked ones, easiest first by your own marking instinct. Keep the last 5 minutes for two jobs: filling in every remaining blank, and rechecking the handful you rushed on the first pass.

One rule underpins all of it. There is no negative marking. Your score is based on the number of correct answers and no marks are deducted for incorrect ones, so every question is worth attempting and a blank is worth exactly nothing. An unanswered question scores identically to a wrong one, which makes clearing your blanks in the final thirty seconds the highest-return action available to you all day.

The official rules are unambiguous and worth memorising. Each correct answer scores one mark and no marks are deducted for incorrect answers. Results for each module are reported on a scale from 1 (low) to 9 (high) to one decimal place. You receive a separate score for each module you take, and there is no overall ESAT total. Each module is equated and scaled separately, which is a genuine difference from the TMUA, where Papers 1 and 2 are equated and scaled together into a single score.

For Mathematics 2 that means one thing above all: the number stands on its own. A strong Physics score does nothing arithmetically for the Mathematics 2 line. An admissions tutor looking at a course that specifies Mathematics 1, Mathematics 2 and Physics sees three parallel figures, not an average. Preparation effort should therefore follow your weakest required module rather than your favourite one.

On target scores, here is the unwelcome but necessary sentence. UAT-UK states that there is no pass or fail score for the ESAT and publishes no grade boundaries; scores are used alongside the other information in your university application. The various figures circulating online for what counts as "safe" or "competitive" have no official source behind them. If someone hands you a number and calls it a threshold, ask which page of which official document it came from.

The one institution that has published any score data is Imperial, and three caveats come before the numbers. Its dashboard reports a raw mark labelled "Mean test score (maximum is 27)", not the 1 to 9 scaled score, and the page does not say whether that is one module or an average across the modules a candidate sat. Only the offer-holder column is solid: for the not-shortlisted column, the page does not say whether it includes applicants who registered but never sat the test. And Imperial states plainly that this is historic data, that it should not be taken as indicative of requirements or outcomes for current or future cycles, that a high score does not guarantee an offer while a low one does not rule you out, and that there are no grade boundaries or pass marks for the ESAT or TMUA. With all that attached, the 2025-entry mean marks among offer-holders were 17.5 for Aeronautics, 17.5 for Physics, 16.6 for Electrical and Electronic Engineering, 15.8 for Mechanical, 14.2 for Chemical Engineering, 12.6 for Civil and Environmental and 8.9 for Design Engineering, with the Overseas column higher in every department, 19.8 for EEE and 19.4 for Physics among them. It is a historic reference point, not a boundary.

Three practical points go with this. Results arrive approximately four weeks after your sitting, and for 2027 entry the confirmed dates are 16 November 2026 for the October window and 8 February 2027 for January. Results are matched course by course against your UCAS application and sent automatically to institutions whose courses require the test, and candidates cannot choose which institutions see them. And UAT-UK does not accept appeals against test scores.

A path that works for most candidates. First, drill the seven Mathematics 1 topics until they need no thought, since they underpin Mathematics 2 and your other module alike. Second, work through the eight MM headings to rebuild tools, focusing on fluency by hand rather than on new content. Third, move to mixed practice, so that questions welding two or three headings together become normal rather than surprising. Fourth, rehearse full 40-minute modules until the three-pass structure is automatic. Fifth, when reviewing errors, separate three causes: did not know it, knew it and miscalculated, knew it and ran out of time. The remedies are completely different, and reviewing them as one undifferentiated pile of mistakes wastes the review.

On archive papers: UAT-UK's preparation materials page provides historic ENGAA and NSAA papers and flags the questions that fall outside the current specification. Using them is sensible, provided you respect those flags rather than burning hours on material that is no longer examinable.

Three lines not to cross. Module choices cannot be changed without cancelling and rebooking, so reconcile every course on your UCAS list before you complete your registration. Sharing or discussing test content is prohibited, and it was precisely reports in October 2024 of candidates attempting to obtain and sell test content that led to the ESAT being offered on only two days in October and one in January in China, Hong Kong and Macau, a cost borne by everyone sitting in those regions since. And UAT-UK states that it is not connected to, and does not endorse or promote, commercial preparation services, and that all necessary preparatory materials are available free of charge to all candidates. FrontierVUE has no affiliation with or authorisation from UAT-UK, Pearson or any university; we build practice and explanation around the official documents, which remain your first source.

One version check before you start: the current specification is for assessment in October 2026 and January 2027. Make sure the syllabus in front of you matches the sitting you are booked for.

Two windows exist for 2027 entry: 12 to 16 October 2026, and 4 to 8 January 2027. You are permitted to sit each UAT-UK test only once per admissions cycle, and the 2027 cycle includes both windows. Attempting to register or sit a second time within the cycle, including by booking under a second UAT-UK account, is treated as misconduct. Mathematics 2 is therefore a single attempt, with no resit option.

Candidates in mainland China, Hong Kong and Macau have fixed days rather than a free choice within the window: the ESAT runs on Monday 12 or Tuesday 13 October, and on Wednesday 6 January. Book on the day registration opens. The booking windows themselves are worth diarising. For October 2026, booking opens on 20 July 2026 at 15.00 UK time and closes on 28 September 2026 at 18.00. For January 2027, booking opens on 26 October 2026 at 15.00 and closes on 21 December 2026 at 18.00. Account creation, access arrangements and bursary applications open on 1 June 2026 at 15.00 UK time.

Two further deadlines fall earlier than the booking deadline. Access arrangements close on 14 September 2026 at 18.00 for the October sitting and 7 December 2026 at 18.00 for January; bursary applications close on 21 September 2026 at 18.00 and 14 December 2026 at 18.00 respectively, all UK time. If you need either, do not leave it until booking closes.

One rule constrains the choice of window. Applicants to Cambridge or Oxford whose course requires the test will normally be required to sit in October, the exceptions being mature applicants to a Cambridge mature college with a January admissions deadline, and applicants to Oxford's Astrophoria Foundation Year, which also has a January deadline. Applicants to other institutions may decide when to sit, and UAT-UK states there is no advantage to either session. Imperial adds its own note that you take the ESAT once per cycle, in October or January as you prefer, and that only your first score is considered.

Fees depend on where your test centre is, not on your nationality or home address: £78 in the UK and Republic of Ireland, £133 elsewhere. Arrive 30 minutes early. You may be refused entry if you are more than 15 minutes late, and your fee is forfeited if that happens, which is not a risk worth running on a test you can sit only once.