AQA A Level Further Maths Paper 2 November 2021 question paper 7367/2

Which of the following matrices is singular?

Circle your answer.

Find arg (4 7i) to the nearest degree.

Circle your answer.

Which of the following lines is perpendicular to the line L ?

Tick () one box.

Show that

Use the method of differences to show that

The equation

has roots , and

Find a cubic equation with roots

The ellipse E₁ has equation

Write down the equation of E₂

The ellipse E₃ has equation

Describe the transformation that maps E₂ to E₃

Explain, without doing any calculations, why L_A has an equation of the form

where c is a constant.

The diagram shows a curve known as an astroid.

The curve has parametric equations

The section of the curve from t 0 to t is rotated through 2 radians about the x-axis.

Show that the curved surface area of the shape formed is equal to where b and c are integers.

The complex number z satisfies the equations

and

where a is real.

numbers.

The line L has polar equation

Show that L is perpendicular to the initial line.

The curve C has polar equation

Find the polar coordinates of the points of intersection of L and C

Fully justify your answer.

The region R is the set of points such that

and

Find the exact area of R

In a colony of seabirds, there are y birds at time t years.

The rate of reduction in the number of birds due to birds dying or leaving the colony is proportional to the number of birds.

In one year the reduction in the number of birds due to birds dying or leaving the colony is equal to 16% of the number of birds at the start of the year.

If no birds are born or join the colony, find the constant k such that

Give your answer to three significant figures.

A wildlife protection group takes measures to support the colony.

The rate of reduction in the number of birds due to birds dying or leaving the colony is the same as in part (a), but in addition:

The rate of increase in the number of birds due to births is 20t per year.

The wildlife protection group brings 45 birds into the colony each year.

Write down a first-order differential equation for y and t

The initial number of birds is 340

Solve your differential equation from part (b) to find y in terms of t

Describe two limitations of the model you have used.

The Cartesian equation of the line L₁ is

The Cartesian equation of the line L₂ is

Calculate the values of the constants b, c, m and p

Fully justify your answers.

The integral S_n is defined by

Show that for n 2

Hence show that

Two of the solutions to the equation cos 6 0 are and

Find the other solutions to the equation cos 6 0 for 0

Use the fact that and are solutions to the equation cos 6 0 to find

a and b are integers.

Hence show that