Oxbridge Interview Question

Sample question:

Here is an example of a question which could come up in an Oxford interview. Normally, what happens is that the interviewer will begin with an easier question and then ask a more difficult question which builds upon the results of the previous question.`

• Starter question: Sketch the graph of f(x)=ln(x)/x, carefully labelling any turning points
• Follow-up question: Find all the integer pairs (a,b) such that a^b=b^a

Some comments:

The starter question asks you to sketch the graph of an unfamiliar function. It is very common for interviewers to ask you to sketch graphs, as graph sketching only requires knowledge of basic mathematics (e.g. differentiating to find turning points, finding the roots of equations, taking limits at infinity) but can still be quite tricky. One of the things I always stress to my students is that techniques like graph-sketching, and other problem-solving areas that repeatedly come up, need to be mastered before going into the Oxford or Cambridge maths interview.

The follow up question nearly always relies on the starter question, so you should always try to spot the link between the two questions. One obvious thing to do, in light of the starter question, is to take logs of both sides in the equation a^b=b^a and see whether this can help with answering the question.