AS & A Level Mathematics (9709) — May–June 2025, Paper 61

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

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Question 1 · 3 marks · The Poisson distribution (approximation to the binomial)

It is known that 1% of houses in a certain area have a wind turbine. A random sample of 400 houses in this area is chosen for a survey on domestic heating. The number of houses in…

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Question 2 · 7 marks · The Central Limit Theorem (distribution of the sample mean)

The random variable X has the distribution B(8, 3/4). A random sample of 100 values of X is chosen, and the sample mean, X-bar, is found. Find P(X-bar 6.2). You are not expected…

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Question 3 · 8 marks · Unbiased estimators of population mean and variance

The time, T minutes, for a certain daily bus journey is normally distributed. The bus company claims that the mean of T is 45. A passenger believes that the mean of T is actually…

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Question 4 · 6 marks · Linear combinations of random variables (scaling)

At an entertainment centre, the cost for using a particular video game is $0.40 per minute. The number of minutes for which people use the video game has mean 15 and variance 9.…

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Question 5 · 11 marks · The Poisson distribution (sum of independent Poisson variables)

The random variables W and X have the independent distributions Po(1.2) and Po(2.3) respectively. Find P(3 <= W + X <= 5).

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Question 6 · 7 marks · Hypothesis tests for a population proportion (binomial)

A manufacturer of cell phones claims that 25% of students own a Pumpkin phone. Jeyeraj thinks that the proportion of students at his large college who own a Pumpkin phone is less…

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Question 7 · 8 marks · Continuous random variables (probability density function)

X is a random variable with probability density function given by f(x) = 1 + cos(pi x) for 0 <= x <= 1, f(x) = 0 otherwise. Show that P(X < 1/2) = 1/2 + 1/pi.

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