X b a b are values of x for which fx is not defined a

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Unformatted text preview: ficance determine whether the sample meets the company’s standard for acceptance. (7) 114 (e) Every week, the company randomly selects the test results of 1000 hard discs and checks if these results come from a normal distribution (with a mean of 68 and standard deviation of 3) or not. The following table gives the results, R, for one such test. Results Frequency 56 ≤ R ≤ 59 5 59 ≤ R < 62 17 62 ≤ R < 65 146 65 ≤ R < 68 333 68 ≤ R < 71 360 71 ≤ R < 74 113 R ≥ 74 26 Check, at the 5% level of significance, whether the above data comes from a normal population with a mean of 68 and standard deviation of 3. (8) (Total 30 marks) 167. Let X and Y be two non-empty sets. (a) Define the operation X • Y by X • Y = (X ∩ Y) u (X ′ ∩ F′). Prove that (X • Y)′ = (X ∪ Y) ∩ (X′ ∪ Y′). (3) (b) Let f : → be defined by f(n) = n + 1, for all n ∈ . Determine if f is an injection, a surjection, or a bijection. Give reasons for your answer. (3) (c) Let h : X → Y, and let R be an equivalence relation on Y. y1Ry2 denotes that two elements y1 and y2 of Y are related. Define a relation S on X by the following: For all a, b ∈ X, a S b if and only if h(a) R h(b). Determine if S is an equivalence relation on X. (4) (Total 10 marks) 115 168. (a) Let f1, f2, f3, f4 be functions defined on – {0}, the set of rational numbers excluding 1 1 zero, such that f1(z) = z, f2(z) = –z, f3(z) = , and f4(z) = – , where z ∈ – {0}. z z Let T= {f1, f2, f3, f4}. Define ° as the composition of functions i.e. (f1 ° f2)(z) = fl(f2(z)).Prove that (T, °) is an Abelian group. (6) (b) Let G = {1, 3, 5, 7} and (G, ◊) be the multiplicative group under the binary operation ◊, multiplication modulo 8. Prove that the two groups (T, °) and (G, ◊) are isomorphic. (5) (Total 11 marks) 169. Let a, b and p be elements of a group (H, *) with an identity element e. (a) If element a has order n and element a–l has order m, then prove that m = n. (5) (b) If b=p–1 *a*p, prove, by mathematical induction, that bm = p–1 * am * p, where m = 1, 2,.... (4) (Total 9 marks) 170. Let G be the graph given below: B 9 8 A 8 11 3 9 5 U D 13 6 C 3 E 116 (a) Has G got an Eulerian circuit? Give a reason for your answer. (2) (b) What is the adjacency matrix of the graph G? Determine how many walks of length 2 are there from vertex A to vertex C. (4) (c) Use Kruskal’s algorithm to find the minimum spanning tree for graph G. (5) (Total 11 marks) 171. (a) Find the Maclaurin series of the function g(x) = sin x2 using the series expansion of sin x, ∞ i.e. sin x = ∑ ( −1) n n=0 x 2 n +1 . ( 2n + 1)! (1) (b) Using the Maclaurin series of g(x) = sin x2 evaluate the definite integral 1 ∫ sin x dx 2 0 correct to four decimal places. (5) (Total 6 marks) 172. (a) Use the ratio test to calculate the radius of convergence of the power ∞ ( x − 5) n n =1 series 3 n2 ∑ . (3) (b) Using your result from part (a), determine all points x where the pow...
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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.

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