Mintnib › ESAT
ESAT practice
The ESAT is the admissions test Cambridge, Imperial, Oxford and UCL set for engineering and the sciences. Practise it with unseen ESAT-style questions and full worked solutions.
Try Mintnib Pro free. One account covers the ESAT and the TMUA. No payment, no card.
Schools and organisations we have worked with
Shown with their permission. Not endorsements. See all
What you can do
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Practise by topic
Short sets, marked as you go, with the solution after each question.
Practise by topic -
Sit a timed paper
A full module under exam timing, marked, with every solution afterwards.
Sit a timed paper -
Start here
A dated plan for this sitting: what to do first, and what to leave.
Start here -
Formulas and facts
The things worth knowing cold, module by module, in one place.
Formulas and facts -
Sample papers
Free papers with full worked solutions, open to anyone, no account needed.
Sample papers -
Where you stand
A short diagnostic that shows which topics are costing you marks.
Where you stand
Try a question
Questions from our own method pages, with the answer one tap away. No account needed.
Sample question: Counting roots
For which values of k does x3 − 3x − k = 0 have three distinct real roots?
- A −2 ≤ k ≤ 2
- B k > 0
- C −1 < k < 1
- D −2 < k < 2
- E −3 < k < 3
The answer, and the method behind it
So the answer is −2 < k < 2, which is D.
Sample question: Binomial coefficients
Find the coefficient of x2 in (1 + 3x)4(2 − x)3.
The answer, and the method behind it
6 − 144 + 432 = 294. Three products, one line of addition, and the full expansion of degree seven never written.
Sample question: Definite integrals
Evaluate the integral from −2 to 2 of x5 − 4x3 + 3x2 + 7x + 1, with respect to x.
The answer, and the method behind it
Over an interval symmetric about zero, the odd powers x5, −4x3 and 7x integrate to zero: each one's area to the left of zero cancels its area to the right. That leaves 3x2 + 1, which is even, so the integral is twice the integral from 0 to 2. The integral of 3x2 + 1 is x3 + x, which is 8 + 2 = 10 at x = 2 and 0 at x = 0, so the answer is 2 × 10 = 20.
The methods that save time
Each method page names the question it is for, works one example in full, and shows the case where the method gives the wrong answer, because a shortcut you cannot check is a guess. Free to read, no account needed.
What students actually get wrong
Built from real answers rather than opinion: the topics where students lose the most marks, and the mistakes that cause it. Free to read, and worth reading before you plan your revision.
Your first five minutes
- Create a free accountAn email address and a password, or one tap with Google or Apple. Your free day of Mintnib Pro starts by itself. No card needed, nothing will be charged.
- Answer your first five questionsESAT-style and unseen, marked, each with its worked solution.
- Keep going where it countsEvery session after that opens on one next step, chosen from your own answers.
About the test
The ESAT is sat in October, in modules: Mathematics 1, and then two of Mathematics 2, Physics, Chemistry and Biology, chosen for the course you are applying to. Each module is multiple choice and tightly timed, which is why method matters as much as knowledge.
The specification, the dates and the rules come from the organisation that sets the test and should be read there first. Mintnib is independent of it. What the test covers, module by module.