OCR A Level Maths A Pure Mathematics June 2023 question paper H240/01

In the triangle ABC, the length AB = 6 cm, the length AC = 15 cm and the angle BAC = 30°.

Calculate the length BC.

D is the point on AC such that the length BD = 4 cm.

Calculate the possible values of the angle ADB.

Show that + can be written in the form , where a, b and c are

Hence solve the equation + = 2.

In this question you must show detailed reasoning.

Solve the equation 2^2y - 7 × 2^y - 8 = 0.

Given that f(x) = x² + 2x, use differentiation from first principles to show that f ′(x) = 2x + 2

The gradient of a curve is given by = 2x + 2 and the curve passes through the point (- 1, 5).

Find the equation of the curve.

It is given that ABCD is a quadrilateral. The position vector of A is i + j, and the position vector of B is 3 i + 5 j.

Find the length AB.

The position vector of C is p i + p j where p is a constant greater than 1.

Given that the length AB is equal to the length BC, determine the position vector of C.

The point M is the midpoint of AC.

Given that = , determine the position vector of D.

State the name of the quadrilateral ABCD, giving a reason for your answer.

The function f (x) is defined for all values of x as f (x) = | ax - b |, where a and b are positive constants.

The graph of y = f (x) + c, where c is a constant, has a vertex at (3, 1) and crosses the y-axis at (0, 7).

Find the values of a, b and c.

Explain why f^-1(x) does not exist.

The function g(x) is defined for x ⩾ as g(x) = | px - q|, where p and q are positive constants.

Find, in terms of p and q, an expression for g^-1(x), stating the domain of g^-1(x).

State the set of values of p for which the equation g(x) = g^-1(x) has no solutions.

Determine the x-coordinates of any stationary points on the curve.

Show that the curve is convex for all values of x.

Use the result cos (A + B) = cos A cos B - sin A sin B to show that

cos (A - B) = cos A cos B + sin A sin B.

The function f (θ) is defined as cos (θ + 30 °) cos (θ - 30 °), where θ is in degrees.

Determine the following.

• The maximum value of f (θ)

• The smallest positive value of θ for which this maximum value occurs

Determine the following.

• The minimum value of f (θ)

• The smallest positive value of θ for which this minimum value occurs

Find the first three terms in the expansion of (4 + 3x in ascending powers of x.

State the range of values of x for which the expansion in part (a) is valid.

Determine the possible values of the constant a.

Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a series of weeks in the summer. A model is suggested for the number of bees, B, and the number of wildflower plants, F, at time t weeks after the start of the study.

In the model B = 20 + 2t + cos 3t and F = 50e^0.1t.

The model assumes that B and F can be treated as continuous variables.

Suggest a reason why this model may not be valid for values of t greater than 12.

The diagram shows part of the curve f(x) = + 2. The equation f(x) = 0 has a positive root α close to x = 0.3.

Explain why using the sign change method with x = 0 and x = 1 will fail to locate α.

Show that the equation f(x) = 0 can be written as x =

An alternative iterative formula is x_n+1 = F(x_n), where F(x_n) = ln(2 - 8x_n²).

By considering F ′ (0.3) explain why this iterative formula will not find α.

The owners of an online shop believe that their sales can be modelled by S = ab^t, where a and b are both positive constants, S is the number of items sold in a month and t is the number of complete months since starting their online shop.

The sales for the first six months are recorded, and the values of log₁₀S are plotted against t in the graph below. The graph is repeated in the Printed Answer Booklet.

Explain why the graph suggests that the given model is appropriate.

The owners believe that a = 120 and b = 1.15 are good estimates for the parameters in the model.

Show that the graph supports these estimates for the parameters.

Use the model S = 120 × 1.15^t to predict the number of items sold in the seventh month after

opening.

Use the model S = 120 × 1.15^t to predict the number of months after opening when the

total number of items sold after opening will first exceed 70 000.

Comment on how reliable this prediction may be.

Use the substitution u = e^x - 2 to show that