AS & A Level Further Mathematics (9231) — October–November 2024, Paper 13

The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel to the x-axis, scale factor k (k != 0), followed by a shear, x-axis fixed,…

Prove by mathematical induction that, for all positive integers n, d^n/dx^n (tan^(-1) x) = Pn(x)(1 + x^2)^(-n), where Pn(x) is a polynomial of degree n - 1.

The quartic equation x^4 + 2x^3 - 1 = 0 has roots alpha, beta, gamma, delta. Find a quartic equation whose roots are alpha^4, beta^4, gamma^4, delta^4 and state the value of…

Use the method of differences to find sum{r=1}^{n} 5k/((5r + k)(5r + 5 + k)) in terms of n and k, where k is a positive constant.

Show that the curve with Cartesian equation (x^2 + y^2)^2 = 6xy has polar equation r^2 = 3 sin 2theta.

The curve C has equation y = (4x^2 + x + 1)/(2x^2 - 7x + 3). Find the equations of the asymptotes of C.

The lines l1 and l2 have equations r = i + 3j - 2k + lambda(2i + j + k) and r = i - 2j + 9k + mu(i - 4j + 2k) respectively. The plane Pi1 contains l1 and is parallel to l2. Find…