AQA A Level Further Maths Paper 1 June 2022 question paper 7367/1

The displacement of a particle from its equilibrium position is x metres at time t seconds.

The motion of the particle obeys the differential equation

Calculate the period of its motion in seconds.

Circle your answer.

Simplify

Tick () one box.

The vector v is an eigenvector of the matrix N with corresponding eigenvalue 4

The vector v is also an eigenvector of the matrix M with corresponding eigenvalue 3

Given that

Circle your answer.

10 24 36 144

Find the other three roots.

Given that x 1, prove that

Solve the equation

Give your answer in logarithmic form.

The matrix M is defined as

where k is a constant.

State any restrictions on the value of k

Using your answer to part (a)(i), solve

The complex number w is such that

It is given that w x iy, where x and y are real and x 0

Find an equation for y in terms of x

The complex number z satisfies both

The region R is the locus of z

Sketch the region R on the Argand diagram below.

z₁ is the point in R at which z is minimum.

Calculate the exact value of z₁

Express z₁ in the form a ib, where a and b are real.

Roberto is solving this mathematics problem:

Roberto’s solution is as follows:

Sketch the curve C₁

Explain what Roberto has done wrong.

Find the area enclosed by C₁

P and Q are distinct points on C₁ for which r is a maximum. P is above the initial line.

Find the polar coordinates of P and Q

T maps C₁ onto a curve C₂

T maps P onto the point P

Find the polar coordinates of P

Find the area enclosed by C₂

Fully justify your answer.

In this question all measurements are in centimetres.

A small, thin laser pen is set up with one end at A(7, 2, 3) and the other end at

A laser beam travels from A to B and continues in a straight line towards a large thin sheet of glass.

The sheet of glass lies within a plane ₁ which is modelled by the equation

where p is an integer.

A second large sheet of glass lies on the other side of ₁

This second sheet lies within a plane ₂ which is modelled by the equation

Calculate the distance between the sheets of glass.

The point A(7, 2, 3) is reflected in ₁

Find the coordinates of the image of A after reflection in ₁

In this question use as 10 m s²

A smooth plane is inclined at 30 to the horizontal. The fixed points A and B are 3.6 metres apart on the line of greatest slope of the plane, with A higher than B

A particle P of mass 0.32 kg is attached to one end of each of two light elastic strings. The other ends of these strings are attached to the points A and B respectively.

The particle P moves on a straight line that passes through A and B

The natural length of the string AP is 1.4 metres. When the extension of the string AP is e_A metres, the tension in the string AP is

The natural length of the string BP is 1 metre. When the extension of the string BP is e_B metres, the tension in the string BP is

The particle P is held at the point between A and B which is 0.2 metres from its equilibrium position and lower than its equilibrium position. The particle P is then released from rest.

At time t seconds after P is released, its displacement towards B from its equilibrium position is x metres.

Show that during the subsequent motion the object satisfies the equation

Fully justify your answer.

The experiment is repeated in a large tank of oil. During the motion the oil causes a resistive force of kv newtons to act on the particle,

The oil causes critical damping to occur.

Find x in terms of t, giving your answer in exact form.

Calculate the maximum speed of the particle.

The Argand diagram shows the solutions to the equation z⁵ 1

Solve the equation

giving your answers in the form z cos i sin , where 0 2

Explain why the points on an Argand diagram which represent the solutions found in part (a) are the vertices of a regular pentagon.

Show that if c cos , where z cos i sin is a solution to the equation z⁵ 1, then c satisfies the equation

The Argand diagram on page 22 is repeated below.

Explain, with reference to the Argand diagram, why the expression

has a repeated quadratic factor.