OCR A Level Maths A Pure Mathematics June 2025 question paper H240/01

Express each of the following in the form px^q, where p and q are constants.

The diagram shows the curve y = ax (x - b), where a and b are constants and b > 0.

Given that the curve has a stationary point at x = 3, state the value of b.

Given also that the stationary point at x = 3 is a maximum, state what can be deduced about

the value of a.

Find the y-coordinate of the stationary point, giving your answer in terms of a.

State the range of values of x for which the curve is increasing.

A sequence has terms u₁, u₂, u₃, … defined by u₁ = 3 and u_n+1 = u_n² - 5 for n ⩾ 1.

Find the values of u₂, u₃ and u₄.

Describe the behaviour of the sequence.

The position vector of the point A is 0.5 i - 0.5 j.

Determine whether 0.5 i - 0.5 j is a unit vector.

Find the direction of 0.5 i - 0.5 j, giving your answer in degrees.

The position vector of the point B is 3.5i + cj, where c is a constant.

Given that = 5, determine the possible values of c.

Scientists are comparing h, the average height of a child in cm, with t, the age of the child in years. They suggest the model h = at + b for t ⩾ 2, where a and b are constants.

They find that the average height of a 2 year old child is 87 cm and the average height of a 5 year old child is 108 cm.

Find the values of a and b that are consistent with the scientists’ findings.

Sam is 4 years old.

Use the model to predict Sam’s height.

Comment on the accuracy of this prediction.

Suggest one possible limitation of this model when predicting the average height of a 12 year

old child.

The diagram shows a triangle ABC. The arc BD is part of a circle with centre A and radius 6 cm.

The area of the sector ABD is 14.4 cm².

Show that angle BAD is 0.8 radians.

The area of the triangle ABC is three times the area of the sector ABD.

Find the length AC.

Find the perimeter of the region BCD.

In this question you must show detailed reasoning.

A curve has parametric equations x = t³ + t², y = t² + 2t for all real values of t.

The curve passes through the point P with coordinates (2, 3).

Show that the equation of the tangent to the curve at the point P can be written as 5 y = 4 x + 7

Determine the coordinates of the point where the tangent to the curve at P meets the curve

again.

Find the first three terms in the expansion of (2 + x)^-3 in ascending powers of x.

Determine the range of values of x for which the expansion in part (b) is valid, giving a

reason for your answer.

Express 3 cos 2 x + 4 sin 2 x in the form R cos (2 x - α), where R > 0 and 0 ° < α < 90 °.

In this question you must show detailed reasoning.

The graph of y = e^x can be transformed to the graph of y = e^2x-1 by a stretch parallel to the x-axis followed by a translation.

State the scale factor of the stretch.

Give full details of the translation.

Alternatively the graph of y = e^x can be transformed to the graph of y = e^2x-1 by a stretch parallel to the x-axis and a stretch parallel to the y-axis.

State the scale factor of the stretch parallel to the y-axis.

The point P lies on the curve y = e^2x-1 and has x-coordinate of

Show that the tangent to the curve y = e^2x-1 at P has equation y = 2x.

Find the exact area enclosed by the curve y = e^2x-1, the tangent to the curve at P and the

y-axis.

Assume that is rational, so it can be written as = Line 1

Hence a must be a multiple of 5, so a = 5 k, for some integer k.
Line 3

State any conditions required on Line 1.

Explain why Line 2 means that a must be a multiple of 5.

Show that the equation of the curve can be written in the form cos y = f (x), where f (x) is

fully simplified.