Mathematics

Interview questions

  • 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 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 applicant)
  • 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 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 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 interview, The Student Room)
  • Why do you play table tennis?  (Cambridge interview, The Student Room)
  • Do you know where the multiplication sign came from? (Oxford interview, Oxbridge Applications)
  • What is the significance of prime numbers? (Oxford interview, Oxbridge Applications)

Recommended resources for interview and university preparation


 https://nrich.maths.org/6783 Yet another University of Cambridge funded project to help people get into top universities. Look a the "prepare for university" sections to get a feel for the level of interview questions.
 
https://www.maths.ox.ac.uk/study-here/undergraduate-study/interviews
 Video of example Oxford interview in two parts.


Recommended books

 
http://www.amazon.co.uk/dp/1783741422
 Really useful book of advanced math problems which is recommended ahead of STEP, MAT, PAT etc.

"This book is intended to help candidates prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Paper). STEP is an examination used by Cambridge colleges as the basis for conditional offers. They are also used by Warwick University, and many other mathematics departments recommend that their applicants practice on the past papers even if they do not take the examination. Advanced Problems in Mathematics is recommended as preparation for any undergraduate mathematics course, even for students who do not plan to take the Sixth Term Examination Paper." 
 
http://www.amazon.co.uk/dp/0486469247
Perhaps a little old-fashioned by modern standards but great material and just the right level for university preparation.

"This volume features a complete set of problems, hints, and solutions based on Stanford University's well-known competitive examination in mathematics. It offers students at both high school and college levels an excellent mathematics workbook. Filled with rigorous problems, it assists students in developing and cultivating their logic and probability skills. 1974 edition."
 
http://www.amazon.co.uk/dp/1780747756
 Amazing book of fun (but extremely challenging!) interview-style questions (including many from our list) by Oxford Professor. Highly recommended for anyone planning to study mathematical subjects at university level. More physics problems than maths problems so less specalised than the previous two books, but still very useful for interview and university preparation.

“Entry to selective universities will often require students to demonstrate that they have engaged in super-curricular activities which develop their awareness and understanding of their subject. Thomas Povey has provided in this book enough super-curricular content to keep an aspiring mathematician, physicist, engineer or material scientist (and their teachers) happy for months. For a parent or teacher who wants to proactively support a student in preparing for entry to a competitive science or maths course this is a no-brainer purchase.” —Mike Nicholson, Director ofUndergraduate Admissions and Outreach, University of Oxford (2006-14), Director of Student Recruitment andAdmissions, University of Bath

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