AQA A Level Further Maths Paper 2 June 2024 question paper 7367/2

It is given that

where λ is a constant.

Find the value of λ

Circle your answer.

–28 –8 8 28

The function f is a quartic function with real coefficients.

The complex number 5i is a root of the equation f(x) = 0

Which one of the following must be a factor of f(x)?

Circle your answer.

(x² – 25)
(x² – 5)
(x² + 5)
(x² + 25)

The first four terms of the series S can be written as

S = (1 × 2) + (2 × 3) + (3 × 4) + (4 × 5) +...

Show that the sum of the first n terms of S is equal to

The cubic equation x³ + 5x² – 4x + 2 = 0

has roots α, β and γ

Find a cubic equation, with integer coefficients, whose roots are 3α, 3β and 3γ

The matrices A and B are defined as follows.

Find the values of p such that A and B are commutative under matrix multiplication.

Fully justify your answer.

The matrix C is defined by

Prove that the transformation represented by C has no invariant lines of the form y = kx

Latifa and Sam are studying polynomial equations of degree greater than 2, with real coefficients and no repeated roots.

Latifa says that if such an equation has exactly one real root, it must be of degree 3

Sam says that this is not correct.

State, giving reasons, whether Latifa or Sam is right.

Use the method of differences to show that

Find the smallest integer n such that

The matrix M is defined as

where k is a constant.

State any restrictions on the value of k

The diagram shows the line y = 5 – x

On the diagram above, sketch the graph of y = │x² – 4x│, including all parts of the graph where it intersects the line y = 5 – x

(You do not need to show the coordinates of the points of intersection.)

Find the solution of the inequality

│x² – 4x│> 5 – x

Give your answer in an exact form.

The function f is defined by

where a and b are constants.

The graph of y = f(x) has asymptotes x = –2 and y = 3

Write down the value of a and the value of b

The diagram shows the graph of y = f(x) and its asymptotes.

The shaded region R is enclosed by the graph of y = f(x), the x-axis and the y-axis.

The shaded region R is rotated through 360° about the x-axis to form a solid.

Find the volume of this solid.

Give your answer to three significant figures.

The shaded region R is rotated through 360° about the y-axis to form a solid.

Find the volume of this solid.

Give your answer to three significant figures.

The Argand diagram below shows a circle C

Write down the equation of the locus of C in the form

│ z – w │ = a

where w is a complex number whose real and imaginary parts are integers, and a is an integer.

It is given that z₁ is a complex number representing a point on C. Of all the complex numbers which represent points on C, z₁ has the least argument.

Find │z₁│

Give your answer in an exact form.

Solve the differential equation

given that y = and = 0 when x = 0

The integral I_n is defined by

Use the result from part (a) to show that

where a and b are integers to be found.