AQA A Level Further Maths Paper 1 November 2020 question paper 7367/1

Which of the integrals below is not an improper integral?

Circle your answer.

Which one of the matrices below represents a rotation of 90 about the x-axis?

Circle your answer.

The quadratic equation ax² bx c 0 (a, b, c ) has real roots and .

One of the four statements below is incorrect.

Which statement is incorrect ?

Tick () one box.

Find the value of p and the value of r.

H₁ is the locus of points such that the distance from the point (5, 0) is twice the distance from the line x 2

Show that the equation of H₁ can be written in the form

where q and r are integers.

H₂ is the hyperbola

Describe fully a sequence of two transformations which maps the graph of H₂ onto the graph of H₁

Let w be the root of the equation z⁷ 1 that has the smallest argument in the

Prove that wⁿ is also a root of the equation z⁷ 1 for any integer n.

Prove that

Three planes have equations

The planes do not meet at a unique point.

Show that k 45 is one possible value of k, and find the other possible value of k.

For each value of k found in part (a), identify the configuration of the given planes.

In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system.

The three roots of the equation

where k is a constant, form an arithmetic sequence.

Find the roots of the equation.

The function f is defined by

Find the interval (a, b) in which f (x) does not take any values.

Fully justify your answer.

Find the coordinates of the two stationary points of the graph of y f (x)

Show that the graph of y f (x) has an oblique asymptote and find its equation.

Find the general solution of the differential equation

Find the particular solution for which y 0 when x 3

Give your answer in the form y f (x)

Explain how you know that two of the lines are parallel.

Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures.

Show that the lines l₁ and l₃ meet, and find the coordinates of their point of intersection.

Use the definition of the cosh function to prove that

Ronald says that this contradicts the result given in part (a).

Explain why Ronald is wrong.

Two light elastic strings each have one end attached to a particle B of mass 3 c kg, which rests on a smooth horizontal table.

The other ends of the strings are attached to the fixed points A and C, which are 8 metres apart.

ABC is a horizontal line.

String AB has a natural length of 4 metres and a stiffness of 5 c newtons per metre.

String BC has a natural length of 1 metre and a stiffness of c newtons per metre.

The particle is pulled a distance of metre from its equilibrium position towards A, and released from rest.

Show that the particle moves with simple harmonic motion.

Find the speed of the particle when it is at a point P, a distance metre from the equilibrium position. Give your answer to two significant figures.

Hence find the sum of the series

C_n cosh x cosh 2x cosh nx

in terms of sinh x, sinh nx and sinh (n 1)x