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Unformatted text preview: e discarded at a loss of $0.85 per bolt.
Bolts whose diameters measure over 1.83 mm are sold at a reduced profit of $0.50 per
bolt.
Find the expected profit for manufacturer B.
(6)
(Total 16 marks) 585 856. The diagram shows a trapezium OABC in which OA is parallel to CB. O is the centre of a
circle radius r cm. A, B and C are on its circumference. Angle OCB = θ. O C r A B Let T denote the area of the trapezium OABC. (a) 2
Show that T = r (sin θ + sin 2θ).
2 (4) For a fixed value of r, the value of T varies as the value of θ varies.
(b) Show that T takes its maximum value when θ satisfies the equation
4cos2θ + cosθ – 2 = 0, and verify that this value of T is a maximum.
(5) (c) Given that the perimeter of the trapezium is 75 cm, find the maximum value of T.
(6)
(Total 15 marks) 857. Let X be a random variable with a Poisson distribution, such that P(X > 2) = 0.404.
Find P(X < 2).
(Total 4 marks) 586 858. An urn contains a large number of black and white balls. It is claimed that twothirds 2 of 3
the balls are white. To check this claim, five balls are taken at random from the urn and the
number of white balls recorded. This experiment is repeated 243 times, with the following
results:
Number of white balls in the sample 0 1 2 3 4 5 Number of times this sample occurred 8 9 52 78 70 26 At the 5 % level of significance can the claim be accepted?
(Total 8 marks) 859. The random variable X is normally distributed with mean µ. A random sample of 12
observations is taken on X, and it is found that
12 ∑ (x i − x ) 2 = 99 . i =1 (a) Determine a 95 % confidence interval for µ .
(5) (b) Another confidence interval [60.31, 65.69] is calculated for this sample.
Find the confidence level for this interval.
(4)
(Total 9 marks) 860. Define the operation # on the sets A and B by A # B = A′ ∪ B′.
Show algebraically that
(a) A# A = A′ ;
(1) (b) (A#A)#(B#B) = A ∪ B;
(2) (c) (A#B)#(A#B) = A ∩ B.
(3)
(Total 6 marks) 587 861. Let S = {integers greater than 1}. The relation R is defined on S by
m R n ⇔ gcd(m, n) > 1, for m, n ∈ S.
(a) Show that R is reflexive.
(1) (b) Show that R is symmetric.
(2) (c) Show using a counter example that R is not transitive.
(3)
(Total 6 marks) 862. Let T = {all real numbers except 1}. The operation * is defined on T by
a * b = ab – a – b + 2, for a b ∈ T.
(a) Show that T is closed under the operation *.
(5) You may now assume that T is a group under *.
(b) Find the identity element of T under *.
(3) (c) (i) Prove by mathematical induction that
n times
64748
a ∗ a ∗ ... ∗ a = (a – 1)n + 1, n ∈ + . 5 times (Note that a * a * ... * a = a * a * a * a * a).
(ii) Hence show that there is exactly one element in T which has finite order, apart
from the identity element. Find this element and its order.
(10)
(Total 18 marks) 588 863. (a) Use the Euclidean algorithm to find the greatest common divisor, d, of 272 and 656.
(3) (b) Hence express d in the form 272a + 656b, where a, b ∈ .
(3)
(Total 6 marks) 864. A graph G has adjacency matrix given by ABCDEF
A 0 B 1 C 0
D 1
E 0 F 0 1 0 1 0 0 0 1 0 2 0
1 0 1 0 0 0 1 0 1 1 2 0 1 0 1
0 0 1 1 0 (a) Draw the graph G. (b) Explain why G has a Eulerian circuit. Find such a circuit. (c) Find a Hamiltonian path.
(Total 6 marks) 589 865. The following diagram shows a weighted graph.
Q 12 10 R S
5 4
2 P 5 4 3 4 7 T
10 W V 10 2 U Use Dijkstra’s Algorithm to find the length of the shortest path between the vertices P and T.
Show all the steps used by the algorithm and write down the shortest path.
(Total 8 marks) ∞ 866. Find the radius of convergence of the series ( 2k − 2)! ∑ k!(k − 1)!x k . k =1 (Total 5 marks) 590...
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 Fall '13
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