Of the ball after t seconds is given by the equation

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Unformatted text preview: e reflection of the vertices of the square in the line L1; K is a clockwise rotation about O of π. (a) Write down the table of operations for the set S = {U, H, V, K} under °, the composition of these geometric transformations. (4) (b) Assuming that ° is associative, prove that (S, °) forms a group. (4) 228 Consider the set C = {1 , –1 , i, –i} and the binary operation ◊ defined on C, where ◊ is the multiplication of complex numbers. (c) Find the operation table for the group (C, ◊) . (3) (d) Determine whether the groups (S, °) and (C, ◊) are isomorphic. Give reasons for your answer. (4) (Total 15 marks) 332. Let (G, *) be a group where * is a binary operation on G. The identity element in G is e, such that G ≠ {e} . The group G is cyclic, and its only subgroups are {e} and G. Prove that G is a finite cyclic group of prime order. (Total 6 marks) 333. For any positive integers a and b, let gcd (a, b) and lcm (a , b) denote the greatest common divisor and the least common multiple of a and b, respectively. Prove that a × b = (gcd(a, b) × (1cm(a, b)). (Ttotal 5 marks) 334. (a) If G is a connected simple planar graph with v vertices (v ≥ 3) and e edges, prove that e ≤ 3v – 6. (5) (b) Hence prove that κ5 is not a planar graph. (3) (Total 8 marks) 229 335. (a) Find Maclaurin's series expansion for f(x) = ln (1 + x), for 0 < x < 1. (4) (b) Rn is the error term in approximating f(x) by taking the sum of the first (n + 1) terms of its Maclaurin's series. Prove Rn ≤ 1 , (0 ≤ x < 1) n +1 (2) (Total 6 marks) 336. Test the convergence or divergence of the following series ∞ (a) 1 ∑ sin n ; n=1 (5) ∞ (b) cos nπ n1.4 ∑ (n + 10) n =1 (5) (Total 10 marks) 230 337. The first three terms of an arithmetic sequence are 7, 9.5, 12. (a) What is the 41st term of the sequence? (b) What is the sum of the first 101 terms of the sequence? Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks) 231 338. The diagram below shows a line passing through the points (1,3) and (6,5). y (6,5) (1,3) x 0 Find a vector equation for the line, giving your answer in the form x a = +t y b c , where t is any real number. d Working: Answers: ………………………………………….. (Total 4 marks) 232 339. The diagram shows parts of the graphs of y = x2 and y = 5 – 3(x – 4)2. y y = x2 8 6 y = 5 – 3(x–4) 2 4 2 –2 0 2 4 6 x 233 The graph of y = x2 may be transformed into the graph of y = 5 – 3(x – 4)2 by these transformations. A reflection in the line y = 0 a vertical stretch with scale factor k a horizontal translation of p units a vertical translation of q units. followed by followed by followed by Write down the value of (a) k; (b) p; (c) q. Working: Answers: (a) ………………………………………….. (b) .................................................................. (c) ………………………………...
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