AS & A Level Further Mathematics (9231) — May–June 2025, Paper 21

Find the roots of the equation z^3 = 27 - 27i, giving your answers in the form r e^(itheta), where r 0 and -pi < theta <= pi.

Let In = integral from 0 to 1 of (1 - x)^n sinh(x) dx, where n is a non-negative integer. Show that, for n = 2, In = -1 + n(n - 1) I(n-2).

By considering the binomial expansion of (z - 1/z)^5, where z = cos(theta) + i sin(theta), use de Moivre's theorem to show that cosec^5(theta) = a / (sin(5theta) + b sin(3theta) +…

The diagram shows the curve with equation y = (1/sqrt(x)) e^(sqrt(x)) for x = 1, together with a set of n - 1 rectangles of unit width. The rectangles are drawn between x = 1 and…

Find the particular solution of the differential equation 6 (d^2 x / dt^2) + 3 (dx/dt) + 6x = e^(-t), given that, when t = 0, x = dx/dt = 0.

Starting from the definitions of tanh and sech in terms of exponentials, prove that 1 - tanh^2(u) = sech^2(u).

Find the solution of the differential equation dy/dx - ((x + 5)/(x^2 + 10x + 61)) y = 1, given that y = 0 when x = 3. Give your answer in an exact form.

It is given that lambda is an eigenvalue of the non-singular square matrix A, with corresponding eigenvector e. Show that e is an eigenvector of A^3 with corresponding eigenvalue…