AS & A Level Further Mathematics (9231) — October–November 2024, Paper 22

Find the value of the integral from 6 to 7 of 1/sqrt((x - 5)^2 - 1) dx, giving your answer in the form ln(a + sqrt(b)), where a and b are integers to be determined.

The curve C has equation 4y^2 + 4 ln(xy) = 1. Show that, at the point (2, 1/2) on C, dy/dx = -1/6.

The curve C has parametric equations x = (1/2)e^(2t) - (1/3)t^3 - 1/2, y = 2 e^t (t - 1), for 0 <= t <= 1. Find the exact length of C.

Use de Moivre's theorem to show that cot(6 theta) = (cot^6(theta) - 15 cot^4(theta) + 15 cot^2(theta) - 1) / (6 cot^5(theta) - 20 cot^3(theta) + 6 cot(theta)).

Find the particular solution of the differential equation 3 d^2y/dx^2 + 2 dy/dx + y = x^2, given that, when x = 0, y = dy/dx = 0.

The diagram shows the curve with equation y = e^(1 - x) for 0 <= x <= 1, together with a set of n rectangles of width 1/n. By considering the sum of the areas of these rectangles,…

Show that d/dx(ln(tanh x)) = 2 cosech(2x).

The matrix A is given by A = ( -2 0 0 ; 0 7 9 ; 4 1 7 ). Show that the characteristic equation of A is lambda^3 - 12 lambda^2 + 12 lambda + 80 = 0 and find the eigenvalues of A.