AS & A Level Mathematics (9709) — May–June 2025, Paper 63

7 questions · every question answerable online with instant point-by-point AI marking against the official mark scheme.

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Question 1 · 8 marks · The Poisson distribution

At a certain shop, customers arrive independently and randomly at a constant average rate of 23.4 per hour. Find the probability that, in a randomly chosen 1-minute period, at…

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Question 2 · 7 marks · Hypothesis tests (mean of a normal population)

The lengths of pencils made at a factory are normally distributed. The standard deviation of the lengths is sigma cm, and the mean is supposed to be 10 cm. An inspector thinks…

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Question 3 · 8 marks · Estimation (unbiased estimates of population mean and variance)

A machine dispenses coffee into cups. The volume, V cm^3, of coffee in a cup was measured for a random sample of 150 cups. The results were summarised as follows. sum v = 46350…

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Question 4 · 4 marks · Random selection and sampling

Emma needs to choose one person at random from three people, P, Q and R. She plans to throw two fair coins and note the number, n, of heads. If n is 0, she will choose P. If n is…

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Question 5 · 4 marks · Linear combinations of normal random variables

In Urberia, the masses, in kilograms, of men have the distribution N(70.3, 5.9^2). A certain footbridge in Urberia can take a maximum safe load of 1500 kg. When n men stand on the…

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Question 6 · 9 marks · Continuous random variables (probability density functions)

A random variable X has probability density function given by f(x) = ax for 0 <= x <= b, f(x) = 0 otherwise, where a and b are constants. Show that a = 2/b^2.

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Question 7 · 10 marks · Hypothesis tests (binomial; critical region)

In the past, one quarter of job applicants at a certain firm had first-class degrees. A change is made in the job description and a director of the firm believes that, on average,…

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