Mathematics Oxford & Cambridge University Interview Questions

Past admissions interview questions for Mathematics

• There's a torus/ring doughnut shaped space station with 2 spacemen on a spacewalk standing diametrically oppositie each other. Can then ask a variety of questions such as if spaceman A wants to throw a spanner to spaceman B, what angle and speed should they choose (stating any assumptions made, e.g. that gravity = 0)? (submitted by Oxford university applicant) Comments

• Show that if n is an integer, n^3 - n is divisible by 6. (Submitted by Oxford Applicant) Comments

• Differentiate x^x, then sketch it. (submitted by Oxford university applicant)

• An ant starts at one vertex of a solid cube with side of unity length. caculate the distance of hte shortest route the ant can take to the furthest vertex from the starting point. (answers in this book)

• A telephone company has run a very long telephone cable all the way round the middle of the earth. Assuming the Earth to be a sphere, and without recourse to pen and paper, estimate how much additional cable would be required to raise the telephone cable to the top of the 10m tall telephone poles (answers in this book)

• a thin hoop of diameter d is thrown on to an infinitely large chessboard with squares of side L. what is the chance of the hoop enclosing two colours? (answers in this book)

• an infinitely large floor is tiles with regular hexagonal tiles of side L. Different colours of tiles are used so that no two tiles of the same colour touch. A hoop of diameter d i thrown onto the tiles. What is the chance of the hoop enclosing more than one colour? (answers in this book)

• what is the volume of the largest cube that fits entirely within a sphere of unity volume? (answers in this book)

• what is the area if an n-sided regular polygon inscribed within a circle of radius r? (answers in this book)

• for a circle inscribed in inside a regular n-sided polygon, what is the minimum n so that the ratio of the area of part outside the circle to the area of the circle is less than or equals to 1/1000? (answers in this book)

• give a vector proof that for a triangle inscribed within a semicircle, the included angle is always pi/2 (answers in this book)

• Is it possible to cover a chess-board with dominoes, when two corner squares have been removed from the chessboard and they are (a) adjacent corners, or conversely, (b) diagonally opposite. (submitted by Oxford university applicant)

• Integrate 1/(x^2) between -1 and 1. Describe any difficulties in doing this? (submitted by Oxford applicant)

• If a cannon is pointed straight at a monkey in a tree, and the monkey lets go and falls towards the ground at the same instant the cannon is fired, will the monkey be hit? Describe any assumptions you make. (submitted by applicant)

• Integrate xlog(x). (submitted by Oxford applicant)

• How many solutions to kx=e^x for different values of k?

• Prove by contradiction that when z^2 = x^2 + y^2 has whole number solutions that x and y cannot both be odd. (The Student Room)

• Sketch y=ln(x) explaining its shape. (The Student Room)

• Compare the integrals between the values e and 1: a) int[ln(x^2)]dx; b) int[(lnx)^2]dx and c) int[lnx]dx. Which is largest?. (The Student Room)

• Sketch y=(lnx)/x. (The Student Room)

• Differentiate x^x and (x^0.5)^(x^0.5). (The Student Room)

• Sketch y=cos(1/x). (The Student Room)

• What is the square root of i?

• If each face of a cube is coloured with one of 6 different colours, how many ways can it be done? (The Student Room)

• If you have n non-parallel lines in a plane, how many points of intersection are there? (The Student Room)

• Observation about (6 - (37^0.5))^20 being very small. (The Student Room)

• By considering (6 + (37^0.5))^20 + previous expression, show this second expression is very close to an integer. (The Student Room)

• Sketch Y = (x^4 - 7x^2 + 12)/(x^4 - 4x^2 +4). (The Student Room)

• Sketch y^2 = x^3 - x. (The Student Room)

• Integrate from 0 to infinity the following: Int[xe^(-x^2)]dx and Int[(x^3)e^(-x^2)]dx. (The Student Room)

• If you could have half an hour with any mathematician past or present, who would it be? (Oxbridge Applications)

• Integrate arctan x! (Cambridge applicant, The Student Room)

• Do you know where the multiplication sign came from (Oxford university applicant, mathematics and statistics, The Student Room)

• If we have 25 people, what is the likelihood that at least one of them is born each month of the year? (Oxford applicant, The Student Room)

• What makes a tennis ball spin as it's travelling through the air? (Oxford applicant, The Student Room)

• If (cos(x))^2 = 2sin(a), what are the intervals of values of a in the interval 0 ≤ a ≤ pi so that this equation has a solution? (submitted by Oxford applicant)

• If a round table has n people sitting around it, what is the probability of person A sitting exactly k seats away from person B? (submitted by Oxford applicant)

• You are given that y = t^t and x = cost. What is the value of dy/dx? (submitted by Oxford applicant)

• Differentiate y = x with respect to x^2? (submitted by Oxford applicant)

• Prove by contradiction that 2(a)^2 - b^2 is true only if a and b are both odd? (submitted by Oxford applicant)

• if your friends were here now instead of you, what would they say about you? (Cambridge interview, The Student Room)

• Whatever got you into pole dancing? (Cambridge university interview, The Student Room)

• Why do you play table tennis? (Cambridge university interview, The Student Room)

• Do you know where the multiplication sign came from? (Oxford university interview, Oxbridge Applications)

• What is the significance of prime numbers? (Oxford interview, Oxbridge Applications)

• Imagine a ladder leaning against a vertical wall with its feet on hte groun. The middle rung of the ladder has been painted a differnt colour on the side, so that we can see it when we look at the ladder from side on. What shape does that middle rung trace out as the ladder falls to the floor?

• Determine all pairs (m, n) of positive integers satisfying: (a) two of the digits of m are the same as the corresponding digits of n, while the other digit of m is 1 less than the corresponding digit of n (as in, say, 263 and 273); (b) both m and n are three-digit squares. A typical hint from the tutor would be to consider this case for 3 digit pairs (m,n) that is, Find as efficiently as possible all pairs (m,n) of positive integers satisfying the following two conditions: (a) two of the digits of m are the same as the corresponding digits of n, while the other two digits of m are both 1 less than the corresponding digits of n\ (b) both m and n are four-digit squares. (submitted by oxford university applicant)

• Show that no number in the sequence 11,111,1111,11111... is a perfect square (submitted by oxford applicant)

• If you found these sample interview questions for Maths useful, please remember to submit your questions, post interview, to oxbridgeinterview@gmail.com

Recommended reading (click images for Amazon price)

Excellent puzzle book written by Oxford professor and based upon past interview questions he has set. Hard, fun and very relevant.

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