United Kingdom · Years 12 and 13 · Sittings 12 to 16 October 2026 and 4 to 8 January 2027

ESAT maths preparation: Mathematics 1 and Mathematics 2

The Engineering and Science Admissions Test is sat in modules, and mathematics runs through all of it. Mathematics 1 is compulsory for every candidate, every other module assumes its content, and most engineering and physics applicants sit Mathematics 2 as well. Each module is 27 multiple-choice questions in 40 minutes, on screen, with no calculator. This page shows which modules each Cambridge and Imperial course asks for, maps the two mathematics modules from UAT-UK's own specification, and works through the kind of exact, quick arithmetic they reward, using examples we wrote ourselves.

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Live teaching since 2020 · 10,000+ students · ESAT facts from UAT-UK, Imperial and Cambridge pages

In short

The ESAT (Engineering and Science Admissions Test) is sat on screen at Pearson test centres and administered by UAT-UK, for entry to engineering and science degrees. It has five modules of 27 multiple-choice questions and 40 minutes each; every candidate sits Mathematics 1, and most add two more, such as Mathematics 2 and Physics. There is no calculator and no penalty for a wrong answer. For 2027 entry the October sitting runs 12 to 16 October 2026 and the January sitting 4 to 8 January 2027. Modern Age Coders teaches the two mathematics modules live online, either in a class of 15 to 20 students or with a single teacher 1-on-1.

Where to start

Three courses for an ESAT candidate

The A Level the mathematics modules are built on, a course that trains admissions-test thinking, and the Further Maths most engineering offers expect.

A-Level Maths course thumbnail

EP / 01

A Level Maths

Mathematics 2 is AS-level pure in all but name; this is where its topics are taught properly.

Open the syllabus →
TMUA and STEP Preparation (Oxford MAT Now TMUA) course thumbnail

EP / 02

Admissions-test maths

Fast multiple-choice reasoning without a calculator, the habit both mathematics modules lean on.

Open the syllabus →
A Level Further Maths course thumbnail

EP / 03

A Level Further Maths

Not examined in the ESAT, but asked for by many of the engineering courses that set it.

Open the syllabus →

How the test is built

What is the ESAT, and how many modules do you sit?

Five modules exist and most candidates sit three of them, back to back, for 120 minutes in total. Mathematics 1 is the only module every candidate sits; the other two depend on the course, and UAT-UK warns that they cannot be changed without cancelling and re-booking.

ESAT structure for 2027 entry, from the UAT-UK ESAT page and content specification (read 31 October 2026)
FeatureWhat UAT-UK says
Modules on offerMathematics 1, Biology, Chemistry, Physics, Mathematics 2
Who sits Mathematics 1All candidates
Who sits the othersMost candidates sit two of the remaining four, set by the course
Questions per module27 multiple-choice
Time per module40 minutes, each separately timed; unused minutes are not carried over
Total for most candidates120 minutes
AidsNo calculator or dictionary; an erasable booklet for working
MarkingOne mark per correct answer; none deducted for a wrong one
Score reportOne score per module, 1 (low) to 9 (high) to one decimal place
Pass markNone: "There is no pass/fail score for the ESAT."

Mathematics 1 sits under everything

UAT-UK's specification says that "All modules will assume knowledge of the content of Mathematics 1". A Physics or Chemistry question can therefore expect a student to rearrange a formula, handle standard form or work with ratio as fluently as a Mathematics 1 question does.

That makes the compulsory module the most valuable hour of revision. A biologist who skips it to concentrate on Biology is weakening both scores at once.

Scores are reported module by module

There is no single ESAT score. A Cambridge engineer receives three numbers on the 1 to 9 scale, one each for Mathematics 1, Mathematics 2 and Physics, and admissions tutors weigh them with the school record, reference and interview.

So a weak module is visible on its own; it is not averaged away. That is a reason to prepare every module the course requires, not only the one a student enjoys most.

Pacing and length, worked out by us from UAT-UK's format
ItemOur figureIn the room
Time per question40 minutes / 27 = about 1 minute 29 secondsNinety seconds, then move on; a long question steals from two short ones
Two modules (Imperial Design Engineering)2 × 40 = 80 minutesMathematics 1 and Mathematics 2 only
Three modules (most courses)3 × 40 = 120 minutesMatches the 120 minutes UAT-UK gives for most candidates
Time left at the end of a moduleLost, per the specificationUse any spare minute to recheck flagged answers before the module closes

Sources: UAT-UK ESAT page and ESAT Content Specification for October 2026 and January 2027, read 31 October 2026. The pacing figures are our own division. Modern Age Coders has no connection with UAT-UK, Pearson or the universities on this page.

Which modules, for which course

Which ESAT modules does your course require?

It depends on the department, and the course page is the authority. At Cambridge, Engineering fixes all three modules while Natural Sciences lets the applicant choose two after Mathematics 1. At Imperial, every department names its modules, and one of them asks for only two.

ESAT modules by course, from the Imperial and Cambridge ESAT pages (read 31 October 2026)
University and courseModule 1Further modules
Cambridge: EngineeringMathematics 1Physics and Mathematics 2
Cambridge: Chemical Engineering and BiotechnologyMathematics 1Two chosen from Biology, Chemistry, Physics, Mathematics 2
Cambridge: Natural SciencesMathematics 1Two chosen from Biology, Chemistry, Physics, Mathematics 2
Cambridge: Veterinary MedicineMathematics 1Two chosen from Biology, Chemistry, Physics, Mathematics 2
Imperial: Aeronautics; Civil and Environmental; Electrical and Electronic; Mechanical Engineering; PhysicsMathematics 1Mathematics 2 and Physics
Imperial: Chemical EngineeringMathematics 1Mathematics 2 and Chemistry
Imperial: Dyson School of Design EngineeringMathematics 1Mathematics 2 only
Imperial: Life SciencesMathematics 1Chemistry and Biology

Applying to two ESAT courses

UAT-UK sets the rule for clashes: if one course has a free choice of modules and the other names a compulsory one, sit the compulsory module. A student applying to Cambridge Natural Sciences and Imperial Physics would therefore take Mathematics 2 and Physics, which satisfies both.

Where two courses demand modules that cannot both fit, UAT-UK asks candidates to contact it through their online account before booking, not after.

Who else uses it

UAT-UK says Imperial College London, the University of Cambridge, the University of Oxford and UCL all use the ESAT. Imperial's own page lists seven universities, adding Durham, LSE and Warwick, for certain courses in engineering, science, natural sciences and veterinary medicine.

The two lists differ, and we could not read Oxford's admissions pages on the day we checked. For anywhere beyond Cambridge and Imperial, take the course page itself as the answer.

Course and module lists from Imperial College London's ESAT page and the University of Cambridge ESAT page; the clash rule and the four-university list from the UAT-UK ESAT page; all read 31 October 2026. The Oxford admissions site refused our request that day and we did not retry it.

The two mathematics modules

What maths is in ESAT Mathematics 1 and Mathematics 2?

Mathematics 1 covers seven areas from units to probability, at roughly GCSE Higher depth. Mathematics 2 adds eight more, from algebra and functions to graphs of functions, which together look very like AS-level pure mathematics. The specification also names what is left out, which is worth knowing before revising.

Topic headings in the ESAT Content Specification for October 2026 and January 2027
Mathematics 1 (labelled M)Mathematics 2 (labelled MM)
M1 UnitsMM1 Algebra and functions
M2 NumberMM2 Sequences and series
M3 Ratio and proportionMM3 Coordinate geometry in the (x, y)-plane
M4 AlgebraMM4 Trigonometry
M5 GeometryMM5 Exponentials and logarithms
M6 StatisticsMM6 Differentiation
M7 ProbabilityMM7 Integration
MM8 Graphs of functions

Inside Mathematics 2

The detail is precise. MM1.6 includes the Factor Theorem and the Remainder Theorem; MM2.3 includes the sum to infinity of a convergent geometric series; MM4.1 expects the sine rule's ambiguous case, in two or three dimensions; MM7.5 asks whether a trapezium-rule estimate is an overestimate or an underestimate.

Mathematics 2 assumes everything labelled M as well as everything labelled MM, so a student cannot sit it on calculus alone. Weak fractions or surds will show up inside a calculus question.

What the specification leaves out

UAT-UK says plainly that "Questions requiring knowledge of the change of base formula will not be set", that differentiation from first principles is excluded, and that points of inflexion "will not be examined", though a qualitative sense of them is expected.

There is nothing from Further Maths, no mechanics as such, and no statistics beyond Mathematics 1. Revising those for the ESAT is time spent somewhere else.

Units are mathematics here. The specification expects the SI prefixes from nano to giga and negative indices in units such as m s−1. Converting 72 km/h to 20 m/s, or 1 m3 to 106 cm3, is a Mathematics 1 skill that Physics and Chemistry questions quietly depend on.

Topic headings, inclusions and exclusions from the UAT-UK ESAT Content Specification, read 31 October 2026. The comparisons with GCSE Higher and AS-level are our own reading of it, not UAT-UK's wording. The two unit conversions are our examples.

Worked examples, ours

How do you answer ESAT maths questions without a calculator?

By keeping numbers exact for as long as possible and reaching for structure before arithmetic. Every example below is ours, written for this page in the spirit of the specification; none is an ESAT, ENGAA or NSAA question.

Surds stay surds

Find √48 + √27. Pull out squares first: √48 = 4√3 and √27 = 3√3, so the sum is 7√3. Squaring it gives 147. No decimal ever appears, which is how a no-calculator question wants to be answered.

Disguised quadratics

Solve 4x − 6 × 2x + 8 = 0. Write y = 2x, so 4x = y2, and the equation is y2 − 6y + 8 = 0, giving y = 2 or y = 4. Then x = 1 or x = 2. Spotting the substitution is the whole question.

Over or under?

Estimate the area under y = x2 from 0 to 2 with two trapezium strips: (1/2)(0 + 2 × 1 + 4) = 3. The exact value is 8/3. The curve bends upwards, so every straight top sits above it and the estimate is too large.

Our ambiguous-case example. In triangle ABC, a = 5, b = 7 and angle A = 30°. How many triangles fit? The sine rule gives sin B = 7 × sin 30° / 5 = 0.7. Two angles between 0° and 180° have a sine of 0.7, one acute and one obtuse, and they add to 180°.

Finishing the triangle

The acute angle is a little under 45°, so the obtuse one is a little over 135°. Add angle A: 30° plus just over 135° is still under 180°, so the obtuse case leaves room for angle C. Both triangles exist, and the answer is two.

Nobody needs the angle to a decimal place. Knowing that sin 45° is about 0.71 is enough to place the acute angle, and that is the kind of number sense the module rewards.

The ninety-second habit

Each example above can be done in well under 90 seconds by a student who looks for structure first: squares inside surds, a hidden quadratic, the shape of a curve, the sum of angles. The same examples take three or four minutes by brute force.

Our lessons train that first look. Before writing anything, the student says out loud what kind of question it is; only then do they start working.

Seven questions we wrote, answers alongside
#Our questionAnswer
1Simplify (√12 + √3)2.27, since √12 + √3 = 3√3
2Find the remainder when x3 − 2x + 5 is divided by (x − 2).9, by the Remainder Theorem: 8 − 4 + 5
3Find the sum to infinity of 6 + 3 + 1.5 + ...12, since 6 / (1 − 1/2) = 12
4How many solutions does sin 2x = 1/2 have for 0° ≤ x < 360°?4: x = 15°, 75°, 195°, 255°
5Evaluate log2 8 + log2 (1/4).1, since 3 + (−2) = 1
6Find the coefficient of x2 in (1 + 2x)5.40, since 10 × 22 = 40
7Find the gradient of the normal to y = x3 at x = 1.−1/3, since the tangent gradient is 3

We wrote every example and practice question here on 31 October 2026; none comes from an ESAT, ENGAA or NSAA paper. UAT-UK publishes an ESAT Guide for each module, a specimen test, and historic ENGAA and NSAA papers from 2016 to 2023, marked where questions fall outside the ESAT specification, all free on its preparation page.

Dates for 2027 entry

When is the ESAT in 2026 and 2027, and when do you book?

October is the main sitting, from 12 to 16 October 2026, and almost every Cambridge and Oxford applicant must use it. January, from 4 to 8 January 2027, is the second chance for other courses. October bookings shut on 28 September 2026; the January list has been open since 28 October 2026 and shuts at six in the evening, UK time, on 21 December 2026.

ESAT sittings and deadlines for 2027 entry, from the Imperial, Cambridge and UAT-UK ESAT pages
Sitting 1Sitting 2
Days on which to sitMonday 12 October to Friday 16 October 2026Monday 4 January to Friday 8 January 2027
Candidates in China, Hong Kong or MacauOnly 12 to 13 October 2026Only 6 January 2027
Bookings accepted from3pm UK time on 20 July 20263pm UK time on 28 October 2026
Last moment to book6pm UK time on 28 September 20266pm UK time on 21 December 2026
Cambridge bursary voucher deadline21 September 2026See UAT-UK
Cambridge and Oxford applicantsRequired, apart from the exceptions belowOnly Cambridge mature-college January applicants and Oxford's Astrophoria Foundation Year
ResultsAbout four weeks after the sitting, sent to the universities automaticallyThe same
  1. Check every course page. Write down the modules each course requires before opening the booking system, because UAT-UK will not change them at the test centre.
  2. Open a UAT-UK account. Accounts could be created from 1 June, and a returning candidate keeps the same one.
  3. Ask for a bursary or access arrangements first. Both must be approved before booking; UAT-UK says it cannot refund a test already paid for under a later bursary.
  4. Book the sitting, the modules and a Pearson centre. UAT-UK advises that earlier bookings are more likely to find a slot at the preferred centre.
  5. Bring compliant photo ID on the day. UAT-UK points candidates to its ID policy and a Pearson video on what to expect.
  6. Wait about four weeks. Results arrive in the UAT-UK account and go automatically to every university that asked for the ESAT.

One sitting per cycle

Imperial is explicit that the ESAT counts once in each application cycle: whichever sitting comes first is the score universities use. Booking January as a back-up for a disappointing October does not work.

Cambridge and Oxford applicants are tied to October in any case, which means their October modules travel to Imperial, UCL and anywhere else on the UCAS form.

What registration mistakes cost

UAT-UK is blunt: "Not registering for the ESAT when it's required might make your application invalid." Missing a deadline is not an inconvenience; it can end the application to that course.

The same applies to the wrong modules. A Cambridge engineer who books Chemistry instead of Mathematics 2 has sat a test the course cannot use.

Dates, booking times and the one-score rule from Imperial's ESAT page; the Cambridge bursary, access arrangement and booking deadlines from the Cambridge ESAT page; the January exceptions, results timing and registration warnings from the UAT-UK ESAT page; all read 31 October 2026. Test fees are set by UAT-UK by test centre location and are not printed here.

How much preparation

Do you need an ESAT tutor, and what does a good score look like?

UAT-UK and Cambridge both say good preparation means knowing the format and revising any specification topics you have not met. Neither publishes a target score. Cambridge says almost no one gets full marks and that some strong applicants do not finish in the time.

Preparation and scores in the words of UAT-UK and Cambridge (read 31 October 2026)
WhoTheir words
UAT-UKPreparation means knowing the format and style of questions, reading the specification for unfamiliar topics, then doing the specimen test "under timed conditions"
UAT-UKThe test "is designed to be challenging in order to differentiate effectively between able applicants", including those with top school grades
Cambridge"we recommend that you revise relevant topics and familiarise yourself with the format of the test"
Cambridge"Almost no one gets full marks." and "Some strong applicants may not even complete the paper in the time given."
CambridgeThere is no pass mark, but "a strong performance is usually needed to get through to interview"
Cambridge"Sharing admissions assessment or interview questions is strictly prohibited."

When a class earns its place

A Year 12 student who has not yet reached AS-level calculus, logarithms or the circle theorems in MM3.3 will meet Mathematics 2 topics for the first time while preparing. Some school courses teach them late in Year 12; some international curricula order them differently.

A second case is the student who knows the content but works slowly without a calculator. Ninety seconds a question is a skill, and it improves fastest with feedback on each attempt.

What we do not do

We do not teach the Biology or Chemistry modules, and our Physics help is limited to the mathematics inside physics questions. Students sitting those modules should use UAT-UK's free guides for them.

We do not circulate questions from past sittings, which Cambridge forbids sharing, and we do not promise a score on the 1 to 9 scale. The ESAT has no pass mark and we will not invent one.

Quotations from the UAT-UK ESAT page and specification, the Cambridge ESAT page, and the Cambridge Admission tests and assessments page, read 31 October 2026. Score scale details for the ESAT are also explained on Imperial's page about understanding ESAT and TMUA scores.

From Year 12 to the October sitting

Four stages before the ESAT

The order we suggest for an October sitting at the start of Year 13. A mature applicant using January 2027 runs the same four stages in a few weeks. Every stage is finished by a skill, never by a date.

Our ESAT mathematics plan
WhenStageHow you know it is done
Year 12, spring term1. Mathematics 1 made automaticFractions, surds, standard form, ratio and unit conversions done exactly, without a calculator
Year 12, summer term2. Mathematics 2 topics all metEvery MM heading has been taught at least once, including logs, the trapezium rule and circle theorems
Year 13, September3. The UAT-UK guides and specimen test, untimedMost answers right with no clock, and each mistake explained
Final three weeks4. Timed modules, back to backThree 40-minute modules in a row with no blanks left

If the sitting is January 2027

Only a mature Cambridge applicant in the January round or an Oxford Astrophoria applicant can use it for those universities. Others should check that their courses accept January.

With under ten weeks, skip stage 2 for topics already taught at school and go straight to the guides and timed modules.

If the sitting is next October

Spend Year 12 on secure AS-level pure in ordinary lessons; test papers can wait until September.

Students also sitting the TMUA for a joint course will find the mathematics overlaps, but the ESAT gives about half the time per question, so speed work is not interchangeable.

The full list

What we teach an ESAT candidate

Grouped by what the student needs: the mathematics modules, the A Levels behind them, and wider skills for engineering applicants.

I

The mathematics modules

Mathematics 1 and 2

EP / T / 01

Admissions-test maths

No calculator, multiple choice, under time.

Open the syllabus

EP / T / 02

GCSE Higher mathematics

The ground Mathematics 1 stands on.

Open the syllabus
II

The A Levels behind it

A Level and IB

EP / A / 02

A Level Further Maths

Asked for by many engineering offers.

Open the syllabus

EP / A / 03

IB Mathematics

Analysis and Approaches covers both maths modules.

Open the syllabus
III

For future engineers

Beyond the test

EP / E / 01

Olympiad mathematics

Unfamiliar problems, the ESAT's favourite kind.

Open the syllabus

EP / E / 02

Python for teens

The language most engineering courses teach first.

Open the syllabus

EP / E / 03

High school mathematics

A full refresh for a gap-year applicant.

Open the syllabus

How lessons run

Live ESAT maths classes from summer to January

Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays.

Weekday evenings

One specification heading per lesson.

Weekend mornings

A timed 40-minute module, then the review.

1-on-1

A single applicant with a near sitting.

Exact first

Answers stay as fractions, surds and powers until the very last line.

Name it, then solve it

Students say what kind of question it is before they write a symbol.

Ninety seconds

The clock comes in once accuracy is steady, never in the first weeks.

Groups of 15 to 20

Comparing two routes to one answer shows which route is quicker.

Our own questions

We write the practice; students work through UAT-UK's free guides on their own.

No score promised

There is no ESAT pass mark, and we name no target on the 1 to 9 scale.

Student work

Work our students have published

Some of the maths, code and machine-learning projects our students have built and shared. More are in the student labs.

NutriLife AI nutrition coach project screenshot

AI and ML

NutriLife

An AI nutrition coach that reads what you eat and works you toward a target.

by Bhavya · Open it

Misti AI chatbot for maths and coding screenshot

AI and ML

Misti

A chatbot that answers mathematics and programming questions, built and deployed by a student.

by Harshit · Open it

GuardianX AI internet safety assistant screenshot

AI and ML

GuardianX

An assistant that helps a young person recognise unsafe situations online.

by Vivaan · Open it

SkyCast weather forecast application screenshot

Web app

SkyCast

A weather forecasting site with live conditions for any location.

by Krish

Fees

Fees

The ESAT fee is paid to UAT-UK, varies with the test centre, and is covered in full by a bursary for eligible UK candidates. Our classes are a separate monthly fee with no joining charge, and most ESAT students stop once their sitting is over.

Free first class

USD 0

no card required

  • A free first class
  • Five of our Mathematics 2 questions, worked together
  • No card details needed
Book it

Group batch

USD 100

a month, billed in US dollars

  • Groups of 15 to 20
  • One teacher all term
  • Timed modules marked
  • Monthly, stop after the sitting
Start here

One to one

USD 150

a month, billed in US dollars

  • One applicant, one teacher
  • Planned from your weakest headings
  • For a sitting weeks away
Enquire

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ESAT questions

What ESAT applicants and parents ask

What is the ESAT?

The Engineering and Science Admissions Test, a computer-based test run by UAT-UK at Pearson test centres. Cambridge and Imperial, among others, use it for engineering, physical and natural sciences and veterinary medicine courses.

What replaced the ENGAA and NSAA?

The ESAT. UAT-UK still publishes ENGAA and NSAA papers from 2016 to 2023 as practice, because they contain questions of the type found in the ESAT, and marks the questions that fall outside the ESAT specification.

How long is the ESAT?

Each module lasts 40 minutes and has 27 multiple-choice questions. Most candidates sit three modules, so the test is 120 minutes long; a course that asks for only Mathematics 1 and Mathematics 2, such as Imperial Design Engineering, means 80 minutes.

Is Mathematics 1 compulsory in the ESAT?

Yes. UAT-UK says all candidates need to sit Mathematics 1, and its specification says every other module assumes Mathematics 1 content. Whether you also sit Mathematics 2 depends on the course.

Can you use a calculator in the ESAT?

No. The specification states that calculators may not be used, and the only aid is an erasable booklet for rough work. That is why exact arithmetic with fractions and surds matters so much.

Is there negative marking in the ESAT?

No. The specification says each correct answer scores one mark and nothing is deducted for an incorrect answer, so a candidate should answer every question before a module closes.

When is the ESAT for 2027 entry?

UAT-UK gives a week in each window: 12 to 16 October 2026, which Cambridge and Oxford applicants use, and 4 to 8 January 2027. Imperial says January bookings close at 6pm UK time on 21 December 2026.

What is a good ESAT score?

Each module gets its own score between 1 and 9, given to one decimal place, and UAT-UK sets no pass mark. Cambridge says almost no one gets full marks and that a strong performance is usually needed for interview, but publishes no target figure.

Do I need an ESAT tutor?

UAT-UK recommends familiarity with the format, a check of the specification, and the specimen test under timed conditions. A class helps most when Mathematics 2 topics have not been taught yet or when working without a calculator is slow.

What days are your classes on?

Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays.

Are you connected with UAT-UK?

No. We are an independent online school with no link to UAT-UK, Pearson or any university on this page. Every candidate registers, picks modules and pays UAT-UK directly.

Related pages

Other pages for engineering and science applicants

The school mathematics the modules draw on, and stretch beyond it.

A Level Maths tuition online

The pure mathematics that Mathematics 2 tests.

Further Maths tuition online

Outside the ESAT, inside most engineering offers.

IB Maths tuition online

IB candidates aiming at Cambridge or Imperial engineering.

UKMT maths challenge tutoring

Fast multiple-choice problem solving, a year or two earlier.

British Mathematical Olympiad

For applicants who want harder problems than any test sets.

Maths tuition in London

Live lessons for London sixth formers, wherever they apply.

Related pages in this guide

Part of the guide: UK university admissions: maths tests and computing applications

Admissions tests and interviews

Computer science applications

More guides: UK maths exam boards and specification codes · UK maths year by year, Reception to Year 11 · UK selective school maths tests: 7+, 11+, 13+ and 16+ · UK maths enrichment, competitions and able pupils

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Tell us the courses you are applying for and the sitting you have booked. In the free class we work through five of our Mathematics 2 questions with you and point out which specification headings need the most time.

If you would like to look around first, see the course list or how a lesson runs on how we teach.

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