AQA A Level Further Maths Paper 2 June 2022 question paper 7367/2

Find the imaginary part of

Circle your answer.

Prove by induction that, for all integers n 1,

Find the value of a and the value of b

Solve the inequality

Fully justify your answer.

The function f is defined as f (x) sec x

Hence find the first three non-zero terms of the Maclaurin series for f (x) sec x

Prove that

A curve passes through the point (5, 12.3) and satisfies the differential equation

Use Euler’s step by step method once, and then the midpoint formula

once, each with a step length of 0.1, to estimate the value of y when x 52

Give your answer to six significant figures.

Find the general solution of the differential equation

Given that y satisfies the differential equation in part (b)(i) and that y 123 when x 5, find the value of y when x 52

Give your answer to six significant figures.

Comment on the accuracy of your answer to part (a).

The curve C₁ has equation

The curve C₂ has equation

Find a sequence of transformations that maps the graph of C₁ onto the graph of C₂

Find the equations of the asymptotes to C₂

Give your answers in the form ax by c 0 where a, b and c are integers.

Describe how the directions of the invariant lines of the transformation represented by M are related to each other.

Fully justify your answer.

Describe fully the transformation represented by M

The shaded region shown in the diagram below is bounded by the x-axis, the curve

The shaded region is rotated through 2 radians about the x-axis to form a solid.

Show that the volume of this solid is

In the case where a 1, b 2 and

show that the volume of the solid is

where m and n are integers.

The matrix A represents a reflection in the line y mx, where m is a constant.

You may use the result in the formulae booklet.

where is the 2 2 identity matrix and k is an integer.

The diagram below shows a point P and the line y mx

Draw four lines on the diagram to demonstrate the result proved in part (b).

Label as P the image of P under the transformation represented by (BA)²

Explain how your completed diagram shows the result proved in part (b).

Find the value of m such that C BA

Fully justify your answer.

On an isolated island some rabbits have been accidently introduced.

In order to eliminate them, conservationists have introduced some birds of prey.

At time t years (t 0) there are x rabbits and y birds of prey.

At time t 0 there are 1755 rabbits and 30 birds of prey.

When t 0 it is assumed that:

the rabbits will reproduce at a rate of a% per year each bird of prey will kill, on average, b rabbits per year the death rate of the birds of prey is c birds per year the number of birds of prey will increase at a rate of d% of the rabbit population per year.

This system is represented by the coupled differential equations:

State the value of a, the value of b, the value of c and the value of d

Solve the coupled differential equations to find both x and y in terms of t

Given that x and y are both positive for 0 t 5, use your answer to part (b) to show that the conservationists’ plan will succeed.