AS & A Level Further Mathematics (9231) — October–November 2024, Paper 12

The sequence u1, u2, u3, ... is such that u1 = 4 and u(n+1) = 3un - 2 for n = 1. Prove by induction that un = 3^n + 1 for all positive integers n.

The line l1 has equation r = i + 3j - k + lambda(i - j - 4k). The plane Pi contains l1 and is parallel to the vector 2i + 5j - 4k. Find the equation of Pi, giving your answer in…

It is given that alpha + beta + gamma + delta = 2, alpha^2 + beta^2 + gamma^2 + delta^2 = 3, alpha^3 + beta^3 + gamma^3 + delta^3 = 4. Find the value of alphabeta + alphagamma +…

The matrices A, B and C are given by A = (1, 2, 3; 2, 1, 3; 3, 2, 5) (the 3x3 matrix with rows [1, 2, 3], [2, 1, 3], [3, 2, 5]), B = (0, -2; -1, 3; 0, 0) (the 3x2 matrix with rows…

It is given that Sn = sum from r = 1 to n of ur, where ur = x^(f(r)) - x^(f(r+1)) and x 0. Find Sn in terms of n, x and the function f.

The curve C has equation y = (x^2 + 3)/(x^2 + 1). Show that C has no vertical asymptotes and state the equation of the horizontal asymptote.

The curve C1 has polar equation r = a(cos(theta) + sin(theta)) for -(1/4)pi <= theta <= (3/4)pi, where a is a positive constant. Find a Cartesian equation for C1 and show that it…