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(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 xintercepts 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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 Fall '13
 Apple
 The Land

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