OCR 2023 GCE FURTHER MATHEMATICS B MEI Y435/01: EXTRA PURE A LEVEL QUESTION PAPER & MARK SCHEME (MERGED)

A surface is defined in 3-D by z = 3x 3 + 6xy + y 2 . Determine the coordinates of any stationary points on the surface. [7] 2 A sequence is defined by the recurrence relation 4t n+1 -t n = 15n+17 for n H 1, with t 1 = 2. (a) Solve the recurrence relation to find the particular solution for t n . [7] Another sequence is defined by the recurrence relation (n+1)u -u 2 = 2n - 1 for n H 1, with u1 = 2. n+1 n n 2 (b) (i) Explain why the recurrence relation for un cannot be solved using standard techniques for non-homogeneous first order recurrence relations. [1] (ii) Verify that the particular solution to this recurrence relation is given by un = an+ b where a and b are constants whose values are to be determined. n [5] t A third sequence is defined by vn = n n for n H 1. (c) Determine n li " m3 vn . [2] 3 A surface, S, is defined by g(x, y, z) = 0 where g(x, y, z) = 2x 3 -x 2 y+2xy2 +27z. The normal to S at the point a1, 1, - 1 k and the tangent plane to S at the point (3, 3, - 3) intersect at P. Determine the position vector of P. [8] 3 © OCR 2023 Y435/01 Jun23 4 4 The set G is given by G = {M: M is a real 2 # 2 matrix and detM = 1}. (a) Show that G forms a group under matrix multiplication, #. You may assume that matrix multiplication is associative. [5] J 1 0 N (b) The matrix An is defined by An = K n L O for any