Composition of permutations a what is the order of

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Unformatted text preview: of each: (i) κ5, the complete graph of order 5, (3) (ii) a bipartite graph κ3,3. (2) (b) Show that κ3,3 has a Hamiltonian cycle, giving appropriate reasons. (3) (Total 8 marks) 56. The following floor plan shows the ground level of a new home. Is it possible to enter the house through the front door and exit through the rear door, going through each internal doorway exactly once? Give a reason for your answer. Front door Rear door (Total 7 marks) 43 57. (a) Prove that if two graphs are isomorphic, they have the same degree sequence. (3) (b) Are the following graphs isomorphic? Justify your answer. (3) (Total 6 marks) 58. Apply Prim’s algorithm to the weighted graph given below to obtain the minimal spanning tree starting with the vertex A. B g6 i5 j9 C a7 G k 13 f 10 A h5 b9 c 16 F D e3 d5 E Find the weight of the minimal spanning tree. (Total 8 marks) 59. A supplier of copper wire looks for flaws before despatching it to customers. It is known that the number of flaws follow a Poisson probability distribution with a mean of 2.3 flaws per metre. (a) Determine the probability that there are exactly 2 flaws in 1 metre of the wire. (3) (b) Determine the probability that there is at least one flaw in 2 metres of the wire. (3) (Total 6 marks) 44 60. A market research company has been asked to find an estimate of the mean hourly wage rate for a group of skilled workers. It is known that the population standard deviation of the hourly wage of workers is $4.00. Using a confidence interval for the mean, determine how large a sample is required to yield a probability of 95% that the estimate of the mean hourly wage is within $0.25 of the actual mean. (Total 10 marks) 61. A car manufacturer wants to know if there is a relationship between the cost of a new vehicle and the average number of complaints. The company checks its records of complaints and collects the following data from a random sample of 1000 vehicles. Costs Number of complaints 5 or less ≥ $30 001 $15 001–$30 000 ≤ $15 000 6 to 10 11 or more 100 150 50 90 260 250 10 50 40 At the 5% level of significance, is there a relationship between the cost of a new vehicle and the number of complaints? (Total 12 marks) 62. Test the convergence or divergence of the following infinite series, indicating the tests used to arrive at your conclusion: ∞ (a) ∑ k3+ 1 k =1 k (3) ∞ (b) ∑ k (1n1 k ) k =2 3 (4) ∞ (c) ∑ (−1) k =1 k +1 k k2 +1 (5) (Total 12 marks) 45 63. An arithmetic series has five terms. The first term is 2 and the last term is 32. Find the sum of the series. Working: Answers: ....…………………………………….......... (Total 4 marks) 64. The diagram shows the parabola y = (7 – x)(l + x). The points A and C are the x-intercepts and the point B is the maximum point. y B A 0 C x 46 Find the coordinates of A, B and C. Working: Answers: ....…………………………………….......... (Total 4 marks) 65. For the events A and B, p(A) = 0.6, p(B) = 0.8 and p(...
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