OCR A Level Maths A Pure Mathematics and Statistics June 2025 question paper H240/02

In this question you must show detailed reasoning.

Solve the inequality x² + 3x ⩽ 10.

In this question you must show detailed reasoning.

Solve the following equations.

e^3y = 2

x⁴ - 3x² - 4 = 0

Use the trapezium rule, with intervals of 0.5, to determine an approximation to the area

of the shaded region.

Use rectangles with the same intervals as in part (i) to determine lower and upper bounds

for the exact area of the shaded region.

Write an expression for the exact area of the shaded region in terms of a limit, involving x

and intervals of width δ x.

Prove that (cos θ + sin θ)² ≡ 1 + sin 2θ.

In this question you must show detailed reasoning.

A circle has diameter PQ where P is (- 5, 1) and Q is (5, 1). The line x + 2 y = 12 meets the circle at A and B.

Find the exact length AB.

A curve has equation y = 3x⁴ - 4x³ - 6x² + 12x.

Use your answer to part (a)(i) to find the x-coordinates of the stationary points on the

curve.

Show that there is exactly one point on the curve which is a point of inflection but not a

stationary point.

An arithmetic progression has first term a and common difference d, where a and d are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio r.

Find d in terms of a.

Find the common ratio of the geometric progression.

In this question you must show detailed reasoning.

Use integration by parts to find the exact value of ln(3 - 2x)dx.

All the students in the year-group took Test 1, but some students missed Test 2.

State the number of students who missed Test 2.

Find an estimate of the number of students whose marks for Test 1 were in the range 45 to 65

inclusive.

The final parts of the graphs are horizontal.

For Test 1, explain what this means about the results.

It is required that the number of students obtaining grade A* should be the same on both tests. The minimum mark for grade A* on Test 2 was 76.

Find the minimum mark for grade A* on Test 1.

A teacher commented that the median marks on both tests were very similar. This suggests that, on average, the levels of difficulty on both tests were about the same.

Use the cumulative frequency diagram to make another comparison between Test 1 and Test

2 about the easier questions on both papers.

The marks for Test 1 are summarised in the table.

Find estimates of the mean and standard deviation of the Test 1 marks.

Use your answers to part (f)(i) to comment on whether there may be any outliers

amongst the Test 1 marks.

The random variable X has the distribution N(45, 4²).

Find P(X > 50).

The random variable Y has the distribution N(25, σ²).

Given that P(Y < 30) = 0.75, find σ.

A train company claims that a particular daily service arrives on time on 95% of days. Riley suspects that the true percentage is less than 95%. Riley tests the company’s claim by choosing a random sample of 50 days during the period November 2023 to March 2024. Riley finds that the service arrived on time on 44 of these days.

Use a binomial distribution to test at the 5% significance level whether Riley’s suspicion is

justified.

By referring to one of the necessary assumptions for using a binomial distribution, suggest a

reason why the conclusion may not be reliable.

Sam has 9 cards, each with a different non-zero digit printed on it.

Sam chooses 5 cards at random and places them in a random order in a straight line, to form a 5-digit number.

Find the probability that the 5-digit number is greater than 50 000.

Given that the 5-digit number is greater than 50 000, determine the probability that it is

greater than 58 600.

One researcher suggests that, for each of the LAs, they should compare the sum of the data for 2001 for Age 15 to 19 with the data for 2011 for Age 25 to 29.

Explain why this comparison is relevant.

The census data also includes data for the following age ranges:

Explain why none of these data ranges are helpful in this context.

The data for one of these four LAs show that some of the 0 to 4 year olds living in that LA in 2001 are definitely no longer living there in 2011.

Explain which LA this is.

One of the researchers says that there has been little movement in or out of Bournemouth between 2001 and 2011 for those who were aged 5 to 7 in 2001.

Use values from the table to show that this statement is not contradicted by the data.

Explain why the statement is not necessarily correct.

The masses of stones on a certain beach are normally distributed with mean 0.36 kg and standard deviation 0.12 kg.

Find the probability that two stones chosen at random from this beach both have masses

between 0.3 kg and 0.4 kg.

Determine the probability that the mean mass of a random sample of 40 stones from this

beach is less than 0.32 kg.

The probability distribution of the random variable X is modelled as follows.

• P(X = 1) = p where p is a constant.

• P(X = x) = 2P(X = x - 1) for x = 2, 3, 4.

• P(X = x) = 0 for all other values of x.

Three values, X₁, X₂ and X₃, of X are chosen at random.

Determine P(X₃ > X₁ + X₂).