AS & A Level Further Mathematics (9231) — May–June 2025, Paper 12
7 questions · every question answerable online with instant point-by-point AI marking against the official mark scheme.
Start with Question 1 →Use standard results from the list of formulae (MF19) to show that sum{r=1}^{n} (2 - 3r)(5 - 3r) = an^3 + bn^2 + cn, where a, b and c are integers to be determined.
Answer this question and get it marked →The cubic equation x^3 + 2x + 1 = 0 has roots alpha, beta, gamma. Find a cubic equation whose roots are alpha^3 - 1, beta^3 - 1, gamma^3 - 1.
Answer this question and get it marked →The sequence u1, u2, u3, ... is such that u1 = 5 and u{n+1} = 6un + 5 for n = 1. Prove by induction that un = 6^n - 1 for all positive integers n.
Answer this question and get it marked →The matrix M is given by M = ( 1 2 ; 0 1 )( cos(theta) -sin(theta) ; sin(theta) cos(theta) ), where 0 < theta < 2pi. (Here ( 1 2 ; 0 1 ) is the 2x2 matrix with first row (1, 2)…
Answer this question and get it marked →The curve C has polar equation r = thetae^{(1/8)theta}, for 0 <= theta <= 2pi. Sketch C.
Answer this question and get it marked →The points A, B, C have position vectors i - 2k, i + 2j + 2k, 2i - j - k, respectively. Find the equation of the plane ABC, giving your answer in the form ax + by + cz = d.
Answer this question and get it marked →The curve C has equation y = (2x^2 - 5x)/(2x^2 - 7x - 4). Find the equations of the asymptotes of C.
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