AQA A Level Further Maths Paper 2 June 2023 question paper 7367/2

Given that y sin x sinh x, find y

Circle your answer.

It is given that f (x) cosh¹ (x 3)

Which of the sets listed below is the greatest possible domain of the function f ?

Circle your answer.

Express 5 5i in the form reⁱ, where

The point on an Argand diagram that represents 5 5i is one of the vertices of an equilateral triangle whose centre is at the origin.

Find the complex numbers represented by the other two vertices of the triangle.

Give your answers in the form reⁱ, where

Show that

where a, b and c are integers to be found.

A is a non-singular 2 2 matrix and A^T is the transpose of A

State the restriction on the possible values of k

The complex number z is such that

where k is a real number.

Find the real part of z and the imaginary part of z, giving your answers in terms of k

The region R on an Argand diagram satisfies both z 2i 3 and arg (z)

Sketch R on the Argand diagram below.

Find the maximum value of z in the region R, giving your answer in exact form.

The line l₁ passes through the points A(6, 2, 7) and B(4, 3, 7)

Find a Cartesian equation of l₁

Explain how you know that the lines l₁ and l₂ are not perpendicular.

The lines l₁ and l₂ both lie in the same plane.

Find the value of c

The quadratic equation z² 5z 8 0 has roots and

Write down the value of and the value of

Without finding the value of or the value of , show that ^{4 4} 47

Find a quadratic equation, with integer coefficients, which has roots ³

The function f is defined by

Given that z cos i sin , use de Moivre’s theorem to show that

The series S is defined as

Use part (a) to express S in the form

where each of G₁ and G₂ is a geometric series.

Hence, show that

A bungee jumper of mass m kg is attached to an elastic rope. The other end of the rope is attached to a fixed point.

The bungee jumper falls vertically from the fixed point.

At time t seconds after the rope first becomes taut, the extension of the rope is

A model for the motion while the rope remains taut assumes that the forces acting on the bungee jumper are

the weight of the bungee jumper

a tension in the rope of magnitude kx newtons

an air resistance force of magnitude Rv newtons

where k and R are constants such that 4km R²

Show that this model gives the result

You do not need to find the value of A or the value of B

A second, simpler model assumes that the air resistance is zero.

The values of k, m and g remain the same.

Find an expression for x in terms of t according to this simpler model, giving the values of all constants to two significant figures.