Sample GCHQ Mathematics Aptitude Test
The aptitude paper will consist of 2 sections.
The first contains shorter questions, the
solutions of which typically rely on the application of one or two ideas, while the second contains
questions that will require a longer or more sophisticated train of thought. There will be plenty
more questions on the paper than you could expect to answer in the time limit – our aim is
primarily to find out those subject areas that you can do well. Therefore, you are advised to read
the whole paper before starting and focus on those questions that you can approach confidently,
as more credit is given for complete answers to a relatively small number of questions than for
a large number of partial attempts. Results from this paper will be combined with those from
the accompanying applications test.
1

Short Questions
1. Consider a 12 hour digital clock (one that takes values from 00:00 to 11:59). Look at itat a random point during a 12 hour period.What is the probability that you see at least one digit taking the value ‘1’?What is the probability that you see exactly one digit taking the value ‘1’?2. Leta, bbe positive integers and letpbe a prime factor ofab-1.Show that eithergcd(p, a-1) orgcd(b, p-1) must be greater that 1.3. Alice and Bob are given a set of five biased coins. They both estimate the probabilitythat each coin will show a head when flipped, and each coin is then flipped once. Theseare the estimates and values observed:Coin12345Alice’s estimates0.40.70.20.90.4Bob’s estimates0.20.80.30.60.3ObservedHeadsHeadsTailsTailsHeadsWhom would you say is better at estimating the bias of the coins, and why?4. Find an example of a functionffrom [0,1] to [0,1] with the following properties:i.fis continuous;ii.f(0) = 0;iii.f(1) = 1;iv.f(x) is locally constant almost everywhere.5. You have a large rectangular cake, and someone cuts out a smaller rectangular piece from

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- Spring '18
- Peter Lee
- Physics, Natural number, Prime number, Alice, Monoid