That the seed grows calculate the probability that it

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Unformatted text preview: r the set S = {1, 3, 4, 9, 10, 12} on which the operation ∗ is defined as multiplication modulo 13. (a) Write down the operation table for S under ∗. (4) (b) Assuming multiplication modulo 13 is associative, show that (S , ∗) is a commutative group. (4) 492 (c) State the order of each element. (3) (d) Find all the subgroups of (S, ∗) (3) (Total 14 marks) 735. Let (H, ×) be a group and let a be one of its elements such that a × a = a. Show that a must be the identity element of the group. (Total 4 marks) 736. (a) Explain what is meant by a cyclic group. (2) Let (G, #) be a finite group such that its order p is a prime number. (b) Show that (G, #) is cyclic. (5) (Total 7 marks) 737. Let G be a weighted graph with 6 vertices L, M, N, P, Q, and R. The weight of the edges joining the vertices is given in the table below: L M N P Q R L – 4 3 5 1 4 M 4 – 4 3 2 7 N 3 4 – 2 4 3 P 5 3 2 – 3 4 Q 1 2 4 3 – 5 R 4 7 3 4 5 – For example the weight of the edge joining the vertices L and N is 3. (a) Use Prim’s algorithm to draw a minimum spanning tree starting at M. (5) 493 (b) What is the total weight of the tree? (1) (Total 6 marks) 738. Let G1 and G2 be two graphs whose adjacency matrices are represented below. G2 G1 A B C D E F (a) A 0 2 0 2 0 0 B 2 0 1 1 0 1 C 0 1 0 1 2 1 D 2 1 1 0 2 0 E 0 0 2 2 0 2 F 0 1 1 0 2 0 a 0 1 3 0 1 2 a b c d e f b 1 0 1 3 2 0 c 3 1 0 2 1 3 d 0 3 2 0 2 0 e 1 2 1 2 0 1 f 2 0 3 0 1 0 Which one of them does not have an Eulerian trail? Give a reason for your answer. (2) (b) Find an Eulerian trail for the other graph. (4) (Total 6 marks) 739. Let p, q and r ∈ + with p and q relatively prime. Show that r ≡ p (mod q) and r ≡ q (mod p) if and only if r ≡ p + q (mod pq). (Total 7 marks) 740. Find the range of values of x for which the following series is convergent. ∞ xn ∑ n +1 n =0 (Total 7 marks) 494 ∞ 741. (a) Show that the series 2π ∑ sin n n =1 2 is convergent. (3) ∞ Let S = n =1 (b) 2π ∑ sin n 2 . Show that for positive integers n ≥ 2, 1 1 1 < −. 2 n −1 n n (1) (c) Hence or otherwise show that 1 ≤ S < 2π. (4) (Total 8 marks) 495 a b 742. Let A = c 0 and B = (a) 0 . Giving your answers in terms of a, b, c, d and e, e write down A + B; (b) 1 d find AB. ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..................................................................................................................................... ..........................................................................
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