Oxbridge Maths - Interview Questions

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Reply 20

76. Should be straightforward, but the key thing is to organise how you count and map it to a simple series
77. If unsure, try a few simple values of k and spot which elementary function gives the result.
78. Could do it a few ways, mod (divisibilty), induction, algebra (factors), …
79. Again, organize the count and one way is to think about which lengths are possible then count all combinations of a particular length.
80. Think about the polygons with the smallest and largest number of possible sides, then how can you generate ones inbetween.
81. Sketch x and log(x) and think about their product. Axis cross point, where it crosses y=x, … What happens when x<1 and maybe differentiate to check behaviour. Integral is fairly standard by parts.
82. Google if necessary, but there are a few famous ones and the key thing is to think about their properties.
83. The Gauss trick for summing arithmetic series should help think about the argument (or using the usual formula). Exploring a few simple series doesnt really give a lot here.
84. Dont really understand question, apart from to note its something to do with apparent weight being small.
85. Can prove using geometry or using complex (polar) numbers.

Reply 21

mqb2766

86. A sketch helps you think about the crucial values of m which correspond to tangents to the sin curve passing through 0. Problem cant be solved exactly using elementary functions to get the roots (could bracket them or solve numerically), though it could be noted that its equivalent to m = sinc(x) = sin(x)/x, for x!=0, which is a famous function.
87. For n=1, you should note it can be transformed into harmonic form, and a quick sketch should convince you that the integral is simple to evaluate (without having to evaluate the phase offset or alpha). Probably even simpler for n=2. For n is odd, same integration limits spotting and consider the sin^3 and cos^3 terms separately (pythagorean sub). For n is even, by parts/reduction.
88. Its obviously related to pythagorean triples. 60=3.4.5 (basic triple), so consider squares mod 3, 4 and 5 and reason about different combinations.
89. Simply writing down the probabilities (for one person) for the two different games gives the result. Obv related to a geometric distribution but without replacement. Note, a bit of spotting might say you only need to observe the first two picks and a bit of reasoning, rather than the whole game.
90. Should be reasonably straightforward. The very local (taylor/maclaurin series) of this polynomial at the origin is x and -x^3+x and as the polynomial must tend to +ive for large |x|, this (sketch) pretty much gives the behaviour in terms of the roots. It can also help to give their approximate values for large/small k.
91. Should be reasonably straightforward to sketch.
92. Similar to a cauchy distribution and should be easy to sketch. Much slower tails than a bell/normal distribution.
93. Obv difference for odd/even exponents and related to the inverse of Q21).
94. Could “rationalise” the denominator / use pythagoras or a weierstrass (tan half angle) substitution.
95. Straightforward factorial question

(edited 2 years ago)

Reply 22