AQA A Level Further Maths Paper 1 November 2021 question paper 7367/1

Find

Circle your answer.

2450 2660 5320 43680

Given that z 1 3i is one root of the equation z² pz r 0, where p and r are real, find the value of r.

Circle your answer.

The curve C has polar equation

Find a Cartesian equation for C.

Circle your answer.

Show that the solutions to the equation

can be expressed in the form

where a and b are integers to be found.

Show that the equation

has exactly four solutions and state these solutions.

Plot the four solutions to the equation in part (a) on the Argand diagram below and join them together to form a quadrilateral with one line of symmetry.

The diagram below shows the graph of y f (x) ( 4 x 4)

The graph meets the x-axis at x 1 and x 3

The graph meets the y-axis at y 2

Show any axis intercepts and asymptotes.

Sketch the graph of y f (x) on the axes below.

Show any axis intercepts.

Use l’Hoˆ pital’s rule to show that

Fully justify your answer.

Evaluate the improper integral

showing the limiting process.

Find the acute angle between the lines L₁ and L₂, giving your answer to the

Find the equation of ₁, giving your answer in the form r.n d

Hence find the shortest distance of the plane ₁ from the origin.

State any restrictions on the value of p.

The equations below represent three planes.

Find, in terms of p, the coordinates of the point of intersection of the three planes.

In the case where p 2, show that the planes are mutually perpendicular.

Kamla transforms the graphs of various functions by applying first S, then T.

Leo says that, for some graphs, Kamla would get a different result if she applied first T, then S.

Kamla disagrees.

State who is correct.

Fully justify your answer.

The hyperbola H has equation y² x² 16

The circle C has equation x² y² 32

The diagram below shows part of the graph of H and part of the graph of C.

Show that the shaded region in the first quadrant enclosed by H, C, the x-axis and the y-axis has area

In this question use = 9.8 m s²

A particle P of mass m is attached to two light elastic strings, AP and BP.

The other ends of the strings, A and B, are attached to fixed points which are 4 metres apart on a rough horizontal surface at the bottom of a container.

The coefficient of friction between P and the surface is 0.68

When the extension of string AP is e_A metres, the tension in AP is 24me_A

When the extension of string BP is e_B metres, the tension in BP is 10me_B

The natural length of string AP is 1 metre

The natural length of string BP is 1.3 metres

Show that when AP 1.5 metres, the tension in AP is equal to the tension in BP.

P is held at the point between A and B where AP 1.9 metres, and then released from rest.

At time t seconds after P is released, AP (1.5 x) metres.

Show that when P is moving towards A,