AQA A Level Further Maths Paper 2 June 2019 question paper 7367/2

Given that z is a complex number, and that z * is the complex conjugate of z, which of the following statements is not always true?

Circle your answer.

Which of the straight lines given below is an asymptote to the curve

where a is a non-zero constant?

Circle your answer.

The positive integer k is such that

Find the value of k.

A curve has equation y cosh x

Show that the arc length of the curve from x a to x b, where 0 a b, is equal to

A circle C in the complex plane has equation z 2 5i a

The point z₁ on C has the least argument of any point on C, and arg (z₁)

The points A, B and C have coordinates A(4, 5, 2), B(3, 2, 4) and C(2, 6, 1)

The points A, B and C lie in a plane.

Find a vector equation of the plane in the form r.n k

Hence find the exact distance of the plane from the origin.

A parabola P₁ has equation y² 4ax where a 0

The line y mx is a tangent to P₂

Solutions using differentiation will be given no marks.

Find the eigenvalues and corresponding eigenvectors of the matrix

Find matrices U and D such that D is a diagonal matrix and M UDU¹

Given that Mⁿ L as n , find the matrix L.

The transformation represented by L maps all points onto a line.

Find the equation of this line.

Prove by induction that f (n) n³ 3n² 8n is divisible by 6 for all integers n 1

The line L₁ has equation

The line L₂ has equation

Find the shortest distance between the two lines, giving your answer to three significant figures.

Abel and Bonnie are trying to solve this mathematical problem:

Abel says he has solved the problem.

Bonnie says there is not enough information to solve the problem.

Abel’s solution begins as follows:

State one extra piece of information about m and p which could be added to the problem to make the beginning of Abel’s solution correct.

Prove that Bonnie is right.

Show the limiting process.

Let

where n 1

Use the method of differences to show that

where a, b and c are integers.

Two tanks, A and B, each have a capacity of 800 litres.

At time t 0 both tanks are full of pure water.

When t 0, water flows in the following ways:

Water with a salt concentration of grams per litre flows into tank A at a constant rate Water flows from tank A to tank B at a rate of 16 litres per minute Water flows from tank B to tank A at a rate of r litres per minute Water flows out of tank B through a waste pipe The amount of water in each tank remains at 800 litres.

At time t minutes (t 0) there are x grams of salt in tank A and y grams of salt in tank B.

This system is represented by the coupled differential equations

Find the value of r.

Show that 3

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