OCR A Level Maths B (MEI) Pure Mathematics and Comprehension June 2024 question paper H640/03

The function f(x) is defined by

Find an expression for f^-1(x) and state the domain of this inverse function.

Explain why g(x) = 1 + x², with domain all real numbers, has no inverse function.

In this question you must show detailed reasoning.

The diagram shows the curve with equation y = x⁵ and the square OABC where the points A, B and C have coordinates (1, 0), (1, 1) and (0, 1) respectively.

The curve cuts the square into two parts.

Show that the relationship between the areas of the two parts of the square is

In this question you must show detailed reasoning.

In this question you must show detailed reasoning.

Using the substitution u = x + 1, find the value of the positive integer c such that

In this question you must show detailed reasoning.

Solve the equation tan x - 3 cot x = 2 for values of x in the interval 0 ° ⩽ x ⩽ 360 °.

Prove that sin 8 θ tan 4 θ + cos 8 θ = 1.

In this question you must show detailed reasoning.

Express cos x + sin x in the form R sin(x + α), where R > 0 and 0 < α < . Give the

This question is about the equation f(x) = 0, where f(x) = x⁴ - x - Fig. 9.1 shows the curve y = f (x).

Show, by calculation, that the equation f (x) = 0 has a root between x = 1 and x = 2.

Fig. 9.2 shows part of a spreadsheet being used to find a root of the equation.

Write down a suitable number to use as the next value of x in the spreadsheet.

Determine a root of the equation f (x) = 0. Give your answer correct to 1 decimal place.

Fig. 9.3 shows a similar spreadsheet being used to search for another root of f (x) = 0.

Explain why it looks from rows 2 and 3 of the spreadsheet as if there is a root between

0 and 1.

Explain why this process will not find a root between 0 and 1.

The diagram below shows the curve y = f (x).

Sketch the graph of the gradient function, y = f ′(x), on the copy of the diagram in the Printed

Answer Booklet.

Fig. 11.1 shows the curve with equation y = g (x) where g (x) = x sin x + cos x and the curve of the gradient function y = g ′ (x) for - 2 π ⩽ x ⩽ 2 π.

Fig. 11.1

Show that the x‑coordinates of the points on the curve y = g(x) where the gradient is 1 satisfy the equation - cos x = 0.

Fig. 11.2 shows part of the curve with equation y = - cos x.

Fig. 11.2

Use the Newton‑Raphson method with a suitable starting value to find the smallest positive x ‑coordinate of a point on the curve y = x sin x + cos x where the gradient is 1.

You should write down at least the following.

• The iteration you use

• The starting value

• The solution correct to 4 decimal places

Explain why x₁ = 3 is not a suitable starting value for the Newton‑Raphson method in

part (b).

The diagram shows the curve with parametric equations

x = sin 2 θ + 2, y = 2 cos θ + cos 2 θ, for 0 ⩽ θ < 2 π.

In this question you must show detailed reasoning.

Determine the exact coordinates of all the stationary points on the curve.

Write down the equation of the line of symmetry of the curve.

Substitute appropriate values of t₁ and t₂ to verify that t₁t₂ gives the correct value for the

y ‑coordinate of the point of intersection of the tangents at the points A and B in Fig. C1.

in Fig. C2.

Show that, for the curve y = ax² + bx + c, the equation of the tangent at the point with

Hence show that for the curve with equation y = ax² + bx + c, the tangents at two points, P and Q, on the curve cross at a point which has x ‑coordinate equal to the mean of the

x ‑coordinates of points P and Q, as given in lines 11 to 14.

A student is investigating the intersection points of tangents to the curve y = 6x² - 7x + 1. She uses software to draw tangents at pairs of points with x ‑coordinates differing by 5.

Find the equation of the curve that all the intersection points lie on.