Free Interview Resources
Free practice questions, preparation advice, and expert insights for Mathematics interviews at Imperial College London.
Real Mathematics interview questions in the style Imperial College London asks. Try answering each one aloud before you reveal the hint.
Sketch the graph of y = x^x for x > 0. Where does it attain its minimum, and what happens as x tends to 0 from above?
Problem-Solving
Hint
Write x^x as e^{x ln x} and differentiate the exponent; a strong candidate will find the turning point at x = 1/e and reason about the x ln x → 0 limit for the endpoint behaviour.
How many zeros does 100! end in? Now generalise: how many trailing zeros does n! have?
Problem-Solving
Hint
Count factors of 5 (they are scarcer than factors of 2); use Legendre's formula summing floor(n/5) + floor(n/25) + ... rather than guessing.
Evaluate the integral of 1/(1 + tan x) from 0 to pi/2 without computing an antiderivative directly.
Problem-Solving
Hint
Use the substitution x -> pi/2 - x to produce a symmetric partner integral, add the two, and let the integrand collapse to a constant.
A stick of length 1 is broken at two independent, uniformly random points. What is the probability the three pieces can form a triangle?
Problem-Solving
Hint
Set the break points as coordinates in the unit square, impose the triangle inequalities on the three lengths, and interpret the answer as an area — the answer is 1/4.
Show that among any collection of n integers there is a non-empty subset whose sum is divisible by n.
Problem-Solving
Hint
Look at the n partial sums and their residues modulo n; apply the pigeonhole principle either to hit a zero residue or to find two equal residues whose difference works.
Structured interviews that combine technical problem-solving with motivation and personal statement discussion.
Imperial interviews vary by department. Engineering and Computing tend to be technical with problem-solving elements. Medicine uses a Multiple Mini Interview (MMI) format with several short stations. Most interviews last 15-30 minutes and may include a presentation or group exercise.
Interview Invitation
Late Nov
Arrival to Interview
Early Dec
Technical Question
Mid Dec
Decision
Early Jan
Interview Invitation
Late Nov
Arrival to Interview
Early Dec
Technical Question
Mid Dec
Decision
Early Jan
What does it actually mean for a function to be continuous at a point? Can you state a definition that does not rely on the phrase 'no gaps in the graph'?
entryHint
Push the candidate towards the epsilon-delta definition and check they understand the order of the quantifiers rather than reciting it.
The derivative and the integral are said to be inverse operations. In what precise sense is that true, and where does that statement need qualification?
midHint
Distinguish the two parts of the Fundamental Theorem of Calculus and probe whether continuity or integrability assumptions matter; a strong answer notes differentiating an integral versus integrating a derivative are not symmetric statements.
Is the sum of two irrational numbers always irrational? Justify your answer, then reconsider whether the product behaves the same way.
midHint
A single counterexample settles the sum claim; the real test is whether the candidate then reasons carefully about the product rather than assuming symmetry.
Why is 0.999... recurring equal to 1, rather than merely very close to it? What is the object '0.999...' actually denoting?
midHint
Steer towards interpreting the decimal as the limit of a geometric series or a supremum; probe whether they grasp that a real number is being defined, not approximated.
You are told a function is differentiable everywhere but its derivative is not continuous. Is that possible? Try to construct or rule out an example.
hardHint
Guide them towards x^2 sin(1/x) with f(0)=0; get them to compute f'(0) from the definition and then show the derivative oscillates near 0.
Your statement mentions a piece of mathematics you read beyond the syllabus. Take one result from it and explain the idea to me as if I had not seen it before.
entryHint
Look for genuine understanding over name-dropping; a strong candidate isolates one concrete idea and builds it from something the listener already knows.
You wrote that you enjoy proof. Talk me through a proof you find elegant, and tell me precisely what makes it elegant to you rather than just correct.
midHint
Reward candidates who can articulate structure, economy, or surprise, and who can distinguish an elegant argument from a merely valid one.
Imperial Mathematics is heavily analysis- and computation-focused rather than a traditional tutorial course. What drew you to this department specifically, and how does that match how you like to work?
midHint
Probe for informed motivation: awareness of the course's applied and computational emphasis, not generic praise of the university's reputation.
Describe a mathematical problem that defeated you. What did you try, where did it break down, and what did you take away from being stuck?
midHint
This tests resilience and honest self-reflection; the best answers show a real attempt and a lesson learned, not a tidy success story.
Roughly how many piano tuners are there in London? Estimate it out loud and tell me which of your assumptions is least reliable.
entryHint
This is a Fermi-estimation test of structured reasoning under uncertainty; reward a clean decomposition and honest identification of the weakest link, not the final number.
Can you fold a standard sheet of paper in half more than seven or eight times? Explain mathematically why folding becomes impossible so quickly.
midHint
Push towards the exponential doubling of thickness against a fixed length, and get them to reason about the geometry of the rounded fold at each step.
If I pick a random real number between 0 and 1, what is the probability it is rational? Convince me your answer makes sense.
hardHint
The rationals are countable and have measure zero; the interesting part is whether the candidate can reconcile 'probability zero' with 'not impossible'.
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Book Mock Interview →Further Reading
by UAT-UK
Official historic papers and worked answers for TMUA practice.
by UAT-UK
Official information on TMUA structure, timing, scoring, and preparation.
by Imperial College London Department of Mathematics
Department guidance for admissions-process clarifications.
by UCAS
Course listing confirming code, duration, deadline, entry requirements, and TMUA requirement.
by Imperial College London
Official guidance for international applicants checking visas, English language requirements, and country information.
by Richard Courant & Herbert Robbins
A bridge from school mathematics into proof, sets, functions, and number systems.
by Kevin Houston
A practical introduction to definitions, proof techniques, and mathematical writing.
by T. W. Körner
Challenging mathematical essays that can deepen motivation and independent engagement.