# What to expect

Being interviewed at Oxford or Cambridge will undoubtedly be one of the most challenging and nerve-racking experiences for any A-level student. Unfortunately there is **no fixed format** for these interviews and different tutors in different colleges have complete freedom over the questions they decide to ask each candidate.

What remains true for each interview is that they are strongly focused on academic knowledge and an ability to think beyond the A-level syllabus. Another key feature is that these interviews are **short**; applicants must demonstrate their abilities in less than 30 minutes. Compounding this is the fact that every candidate invited to interview will have the A-level grades required to study at Maths at Oxbridge. This means that even the most academically gifted students may fail to gain a place simply by not standing out at interview, either through nerves, or through poor preparation.

This is an excerpt from Oxford’s website on what they are looking for. “We are not testing factual knowledge but ability to think. If a student expresses an interest in a particular aspect, they should be prepared to talk intelligently about it, showing knowledge of current affairs and recent developments in the field, and to have opinions about the topic. It is important, by the way, not to try to guess what the interviewer might want as an answer, but to have personal views which are logically expressed. The process is rigorous, but sympathetic, so that you can show us your best.”

Although there is no way of knowing exactly what will come up at interview, the Oxbridge Maths mentors have a good idea of the level of knowledge required, the format of questioning and the style of an interview at Oxbridge, not only through their own experiences at interview but also through the understanding of the tutorial teaching system which these interviews simulate.

In addition, these students will have spent three years with many of the tutors and will come to know their personalities, their subject of interest and their style of teaching. Having been taught by potential interviewers themselves, our mentors are ideally placed to give each applicant very specific advice on how to prepare for interview at their college of choice.

What we believe is extremely important is **practice**. Whether we see students for a single day or throughout the course of the application process our mentors will do their utmost to give each student the tools to fine-tune their interview technique in the run up to the interviews in December. Students who are prepared by our simulated interviews generally feel less intimidated on the day itself, giving them the opportunity to show their true abilities. With focused training, our candidates can even direct the course of the interview towards their stronger points and away from their weaker ones.

2. Integrate

3. What is the square root of i?

4. If I had a cube and six colours and painted each side a different colour, how many (different) ways could I paint the cube? What about if I had n colours instead of 6?

6. Integrate ln x.

7. Sketch the curve . What does it look like?

8. 3 girls and 4 boys were standing in a circle. What is the probability that two girls are together but one is not with them?

9. Prove that 1+1/2+1/3+…+1/1000<10

10. Is there such number N that 7 divided ?

12. How many squares can be made from a grid of ten by ten dots (ignore diagonal squares)?

13. Integrate tan x.

14. Pascal’s triangle (prove that every number in the triangle is the sum of the two above it)

15. Integrate 1/(1-lnx)

16. sketch

17. Prove 4n – 1 is a multiple of 3

18. How many ways there are of getting from one vertex of a cube to the opposite vertex without going over the same edge twice?

19. What shape there would be if the cube was cut in half from diagonally opposite vertices?

20. Draw x ln(x).

21. Integrate and differentiate xln(x).

22. Draw sin(1/X).

23. Differentiate

24. What do you know about triangles?

25. Find a series of consecutive integers such that the sum of the series is a power of 2.

26. Prove Ptolemy’s Theorem.

27. Find roots of the equation mx=sin x considering different values of m.

28. Integrate |sinn(x)+cosn(x)| between 0 and 2 & (pi) for cases n=1,2

29.. Prove xyz is a multiple of 60

30. Two people are playing a game which involves taking it in turns to eat chillies. There are 5 mild chillies and 1 hot chilli. Assuming the game is over when the hot chilli is eaten (and that I don’t like hot chillies), is it a disadvantage to go first? What is the probability that I will eat the chilli if I go first? How about if there are 6 mild and 2 hot?

31. kx4=x3-x Find the real roots when k=0. Sketch the graph when k is small and then when k is large, and find approximations of the real roots in both cases. When else does x have 3 real roots?

32. Sketch f(x) = (x – R(x))2, where R(x) is x rounded up or down in the usual way. then sketch g(x) = f(1/x)

33. (a+b)/2 is an integer, is the relationship transitive? (a+b)/3?

34. Differentiate 1/1+(1/1+(1/1+1/(1 + x))))

35. Sketch graph of 1/x, 1/x2, 1/(1+x2)

36. Integrate 1/(1+x2)

37. Integrate ex x2 between limits of 1 and 0. Draw that graph.

38. Integrate x -2 between limits of 1 and -1. Draw the graph. Why is it -2 and not infinity, as it appears to be on the graph?

39. Write down 3 digits, and then write the number again next to itself, eg: 145145. Why is it divisible by 13?

40. You are given a triangle with a fixed perimeter but you want to maximise the area. What shape will it be? Prove it.

41. Next you are given a quadrilateral with fixed perimeter and you want to maximise the area. What shape will it be? Prove it.

42. Integrate (1)/(x+x3), (1)/(1+x3), (1)/(1+xn)

43. How many 0′s are in 100!

44. Prove that the angle at the centre of a circle is twice that at the circumference.

45. How many ways are there in which you can colour three equal portions of a disc?

46. Integrate 1/(9 +x2)

47. Draw y=ex, then draw y=kx, then draw a graph of the numbers of solutions of x against x for ex=kx, and then find the value of k where there is only 1 solution.

48. Rubik’s cube and held it by two diagonally opposite vertices and rotated it till it reached the same position, by how many degrees did it take a turn?

49. Solve ab=ba for all real a and b.

50. There is a game with 2 players (A&B) who take turns to roll a die and have to roll a six to win. What is the probability of person A winning?

51. Sketch y=x3 and y=x5 on the same axis.

52. What the 2 sides of a rectangle (a and b) would be to maximise the area if a+b=2C (where C is a constant).

53. Can 1000003 be written as the sum of 2 square numbers?

54. Show that when you square an odd number, you always get one more than a multiple of 8.

55. Prove that 1 + 1/2 + 1/3 + 1/4 + … equals infinity

56. Prove that for n E Z ,n>2, n(n+1)>(n+1)n

57. Prove that sqrt(3) is irrational

58. What are the possible unit digits for perfect squares?

59. What are the possible remainders when a cube is divided by 9?

60. Prove that 801,279,386,104 can’t be written as the sum of 3 cubes

61. Sketch y=ln(x)/x and find the maximum.

62. What’s the probability of flipping n consecutive heads on a fair coin? What about an even number of consecutive heads?

63. Two trains starting 30km apart and travelling towards each other. They meet after 20 mins. If the faster train chases the slower train (starting 30km apart) they meet after 50mins. How fast are the trains moving?

64. A 10 digit number is made up of only 5s and 0s. It’s also divisible by 9. How many possibilities are there for the number?

65. There is a set of numbers whose sum is equal to the sum of the elements squared. What’s bigger: the sum of the cubes or the sum of the fourth powers?

66. Draw e(-x^2)

67. Draw cos(x^2)

68. What are the last two digits of the number which is formed by multiplying all the odd numbers from 1 to 1000000?

69. Prove that 1!+ 2!+ 3!+… has no square values for n>3

70. How many zeros in 365!

71. Integrate x sin2x

72. Draw ex , ln x, y=x what does show you. As x tends infinity, what does lnx/x tend to?

73. Define the term ‘prime number’

74. Find method to find if a number is prime.

75. Prove for a2 + b2 = c2 a and b can’t both be odd.

76. What are the conditions for which a cubic equation has two, one or no solutions?

77. What is the area between two circles, radius one, that go through each other’s centres?

78. If every term in a sequence S is defined by the sum of every item before it, give a formula for the nth term

79. Is 0.9 recurring = 1? Why? Prove it

80. Why are there no Pythagorean triples in which both x and y are odd?

81. draw a graph of sinx, sin2x, sin3x

82. prove infinity of primes, prove infinity of primes of form 4n+1

83. differentiate cos3(x)

84. Show (x-a)2 – (x-b)2 = 0 has no real roots if a does not equal b in as many ways as you can.

85. Hence show: i) (x-a)3 + (x-b)3 = 0 has 1 real root ii) (x-a)4 + (x-b)4 = 0 has no real roots iii) (x-a)4 + (x-b)4 = (b-a)4 has 2 real roots

86. Find the values of all the derivatives of e(-1/x^2) at x=0

87. Show that n5-n3 is divisible by 12

88. If I have a chance p of winning a point in tennis, what’s the chance of winning a game

89. Explain what integration is.

90. If n is a perfect square and its second last digit is 7, what are the possibilities for the last digit of n and can you show this will always be the case?

91. How many subsets can you form from a set of n numbers?

92. Prove that (a+b)/2 > sq.root of ab where a>0, b>0 and a does not equal b ie prove that arithmetic mean > geometric mean

93. What is 00 (i.e is it 0 or1), and solve it by drawing xx

94. If f(x+y)=f(x)f(y) show that f(0) = 1,

95. Suggest prime factors of 612612503503

96. How many faces are there on an icosahedron

97. integrate 1/(1+sin x)

98. What is the greatest value of n for which 20 factorial is divisible by 2n?

99. Prove that the product of four consecutive integers is divisible by 24.