AS & A Level Further Mathematics (9231) — May–June 2023, Paper 23

Find the Maclaurin series for sin^(-1) x up to and including the term in x^3.

The variables x and y are related by the differential equation 6 d^2x/dt^2 + 5 dx/dt + x = t^2 + 10t + 13. Find the general solution for x in terms of t.

By considering the binomial expansions of (z + 1/z)^4 and (z - 1/z)^4, where z = cos(theta) + i sin(theta), use de Moivre's theorem to show that cot^4(theta) = (cos 4theta + a cos…

The curve C has equation 4y^3 + (x + y)^6 = 109. Show that, at the point (-4, 3) on C, dy/dx = 1/17.

Starting from the definitions of cosh and sinh in terms of exponentials, prove that 2 cosh^2(x) = cosh(2x) + 1.

The diagram shows the curve with equation y = (1 - x)^2 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,…

The integral In, where n is an integer, is defined by In = integral from 0 to 4/3 of (1 + x^2)^(n/2) dx. Find the exact value of I(-1) giving your answer in the form ln(a), where…

The matrix A is given by A = [ [a, -6a, 2a+2], [0, 1-a, 0], [0, 2-a, -1] ], where a is a constant with a != 0 and a != 1. Show that the equation A (x, y, z)^T = (1, 2, 3)^T has a…