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(7)
(Total 7 marks) 48. A new blood test has been shown to be effective in the early detection of a disease. The
probability that the blood test correctly identifies someone with this disease is 0.99, and the
probability that the blood test correctly identifies someone without that disease is 0.95. The
incidence of this disease in the general population is 0.0001.
A doctor administered the blood test to a patient and the test result indicated that this patient
had the disease. What is the probability that the patient has the disease?
(Total 6 marks) 49. The quality control department of a company making computer chips knows that 2% of the
chips are defective. Use the normal approximation to the binomial probability distribution, with
a continuity correction, to find the probability that, in a batch containing 1000 chips, between
20 and 30 chips (inclusive) are defective.
(Total 7 marks) 50. Consider the function f : x a x – x2 for – 1 ≤ x ≤ k, where 1 < k ≤ 3.
(a) Sketch the graph of the function f.
(3) (b) Find the total finite area enclosed by the graph of f, the xaxis and the line x = k.
(4)
(Total 7 marks) 40 51. Give exact answers in this part of the question.
The temperature g(t) at time t of a given point of a heated iron rod is given by
g(t) = 1nt ,
t
(a) where t > 0. Find the interval where g′ (t) > 0.
(4) (b) Find the interval where g″ (t) > 0 and the interval where g″ (t) < 0.
(5) (c) Find the value of t where the graph of g(t) has a point of inflexion.
(3) (d) Let t* be a value of t for which g′ (t*) = 0 and g″ (t*) < 0. Find t*.
(3) (e) Find the point where the normal to the graph of g(t) at the point
(t*, g(t*)) meets the taxis.
(3)
(Total 18 marks) 52. Let S be the group of permutations of {1, 2, 3} under the composition of permutations.
(a) What is the order of the group S?
(2) (b) Let p0, py, p2, be three elements of S, as follows:
1 2 3 1 2 3
1 2 3
p0 = 1 2 3 , p1 = 2 3 1 , p2 = 3 1 2 . List the other elements of S and show that S is not an Abelian group.
(4) (c) Find a subgroup of S of order 3.
(2)
(Total 8 marks) 41 53. (a) a b
Let A be the set of all 2 × 2 matrices of the form − b a , where a and b are real numbers, and a2 + b2 ≠ 0. Prove that A is a group under matrix
multiplication.
(10) (b) 1
Show that the set: M = 0 0 1 0 – 1 0 − 1 0 , , , 1 0 − 1 0 1 0 − 1 forms a group under matrix multiplication.
(5) (c) Can M have a subgroup of order 3? Justify your answer.
(2)
(Total 17 marks) 54. (a) Define an isomorphism between two groups (G, o) and (H, •).
(2) (b) Let e and e′ be the identity elements of groups G and H respectively.
Let f be an isomorphism between these two groups. Prove that f(e) = e′.
(4) (c) Prove that an isomorphism maps a finite cyclic group onto another
finite cyclic group.
(4)
(Total 8 marks) 42 55. Let κn be the complete graph of order n and κm,n be a bipartite graph of orders m and n.
(a) Explain the following, giving one example...
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This note was uploaded on 06/24/2013 for the course HISTORY exam taught by Professor Apple during the Fall '13 term at Berlin Brothersvalley Shs.
 Fall '13
 Apple
 The Land

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