The form z a bi where a and b are real numbers

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Unformatted text preview: s and a standard deviation of 3 minutes. (a) On a specific morning, what is the probability that Roger waits more than 12 minutes? (3) 263 (b) During a particular week (Monday–Friday), what is the probability that (i) his total waiting time does not exceed 65 minutes? (3) (ii) he waits less than 12 minutes on at least three days of the week? (4) (iii) his average daily waiting time is more than 13 minutes? (3) (Total 13 marks) 385. In a computer assembly shop it is claimed that the quality of a part depends on the day it is produced. To test this claim, a random sample of 500 parts is selected, and each part is classified according to the day it was produced and its quality rating. The data is shown in the table below. Day Produced Monday Tuesday Wednesday Thursday Friday Superior 44 74 79 72 31 Good 14 25 27 24 10 Average 15 20 20 23 9 Mediocre 3 5 5 0 0 Quality Test at the 5% level of significance whether the claim is valid. Write down your assumptions and mention any restrictions you may need to impose on the data. (Total 10 marks) 386. (a) 1 Prove that the set of matrices of the form 0 0 where a, b, c ∈ < a 1 0 b c, 1 , is a group under matrix multiplication. (7) 264 (b) Show that this group is Abelian if and only if there exists a real constant k such that c = ka. (4) (Total 11 marks) 387. Let A = {a , b , c , d, e , f}, and R be a relation on A defined by the matrix below. 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 (Note that a '1' in the matrix signifies that the element in the corresponding row is related to the element in the corresponding column, for example dRb because there is a '1' on the intersection of the d-row and the b-column). (a) Assuming that R is transitive, verify that R is an equivalence relation. (3) (b) Give the partition of A corresponding to R. (4) (Total 7 marks) 388. (a) In any group, show that if the elements x, y, and xy have order 2, then xy = yx. (5) (b) Show that the inverse of each element in a group is unique. (3) (c) Let G be a group. Show that the correspondence x ↔ x –1 is an isomorphism from G onto G if and only if G is abelian. (4) (Total 12 marks) 265 389. The diagram shows a weighted graph with vertices A, B , C , D , E and F. 6 B D 3 8 A 1 2 F 2 7 5 C E 6 Use Dijkstra's Algorithm to find the length of the shortest path between the vertices A and F . Show all the steps used by the algorithm and draw the path. (Total 6 marks) 390. Consider the function f ( x) = (a) 1 , where x ∈ xx + . 1 – 1n x Show that the derivative f ′( x) = f ( x) . 2 x (3) (b) Sketch the function f(x), showing clearly the local maximum of the function and its horizontal asymptote. You may use the fact that 1n x = 0. x→ ∞ x lim (5) (c) Find the Taylor expansion of f(x) about x = e, up to the second degree term, and show that this polynomial has the same maximum value as f(x) itself. (5) (Total 13 marks) 1 391. Determine whether the series 1 n –1 1+ n n ∞ ∑ converges. (Total 4 marks) 266 392. From January to September, the mean n...
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