6 marks 2 3 641 the points a and b have

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Unformatted text preview: Total 19 marks) 673. Carlos drives to work every morning. He records the times taken, in minutes, to complete the journey over a 10-day period. The times are as follows: 32.6 30.9 35.8 34.3 36.3 31.9 33.2 32.7 31.3 32.8 Assuming that these times form a random sample from a normal population, calculate (a) unbiased estimates of the mean and variance of this population; (3) (b) a 90 % confidence interval for the mean. (3) (Total 6 marks) 452 674. Students studying mathematics at a certain college took an examination at the end of the first term. Their performance was classified as “Distinction”, “Pass” or “Fail”. The numbers of male and female students in each category are given in the table below. Distinction Pass Fail Male 26 75 12 Female 28 42 10 The head of the mathematics department wishes to investigate whether or not there is any association between the classification obtained in the examination and gender. (a) State the null hypothesis. (1) (b) Calculate the six expected frequencies under the null hypothesis. (4) (c) Calculate the value of χ2. (2) (d) Write down the number of degrees of freedom and state your conclusion at the 5% level of significance. (3) (Total 10 marks) 675. The random variable X has a Poisson distribution with mean λ. Let p be the probability that X takes the value 1 or 2. (a) Write down an expression for p. (1) (b) Sketch the graph of p for 0 ≤ λ ≤ 4. (1) 453 (c) Find the exact value of λ for which p is a maximum. (5) (Total 7 marks) 676. The difference, A – B, of two sets A and B is defined as the set of all elements of A which do not belong to B. (a) Show by means of a Venn diagram that A – B = A ∩ B′. (1) (b) Using set algebra, prove that A – (B ∪ C) = (A – B) ∩ (A – C). (4) (Total 5 marks) 677. The relation R is defined on the non-negative integers a, b such that aRb if and only if 7a ≡ 7b (modulo 10). (a) Show that R is an equivalence relation. (4) (b) By considering powers of 7, identify the equivalence classes. (4) (c) Find the value of 7503 (modulo 10). (1) (Total 9 marks) 678. Let S be the set of all (2 × 2) non-singular matrices each of whose elements is either 0 or 1. Two matrices belonging to S are 1 1 0 1 1 0 and 0 1 . (a) Write down the other four members of S. (4) 454 (b) You are given that S forms a group under matrix multiplication, when the elements of the matrix product are calculated modulo 2. (i) Find the order of all the members of S whose determinant is negative. (ii) Hence find a subgroup of S of order 3. (6) (Total 10 marks) 679. The group (G, *) is defined on the set {e, a, b, c}, where e denotes the identity element. Prove that a * b = b * a. (Total 6 marks) 680. (a) Define the following terms. (i) A bipartite graph. (ii) An isomorphism between two graphs, M and N. (4) (b) Prove that an isomorphism between two graphs maps a cycle of one graph into a cycle of the other graph. (3) (c) The graphs G, H and J are drawn below. G H J (i) Giving a reason, determine whether or not G is a bipartite g...
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