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Unformatted text preview: The International Medical Association claims that the proportion of people who do not suffer from colds during the winter months is higher for those who received flu immunisation injections than those who do not have flu immunisation injections. The following table gives data collected from a random sample of 100 people. No flu immunisation injections Flu immunisation injections Colds 35 5 No colds 45 15 Does the claim appear to be justified at the 5 % level of significance? (Total 8 marks) 82 118. Discuss the convergence or divergence of the following series: ∞ (a) 2 ∑ n 32 n +1 n n =0 (4) ∞ (b) k +1 ∑ nn + k , k is a positive integer. n=0 k (5) (Total 9 marks) 119. (a) 2 − x 2 . Show that Consider the function f(x) = f″ (x) = 2 ( x − 2) 2 − x 2 2 (3) (b) The third and fourth derivatives of f(x) with respect to × are given by: f′″ (x) = − 6x ( x 2 − 2) 2 2 − x 2 and f (4)(x) = 12(2 x 2 + 1) ( x 2 − 2) 2 2 − x 2 . Use Maclaurin's series to expand f(x), and to show that 2 − x2 = 2 2 1 − x + kx 4 . Find the value of k. 4 (5) 83 1 (c) Hence or otherwise show that the value of ∫x 2 2 – x 2 dx is 0 approximately 0.39438. (d) Use the substitution x = 2 sin θ to show that the exact value of (4) 1 ∫x 2 0 2 – x 2 d x is π 8 . Hence find an approximation of π to 4 decimal places. (5) (Total 17 marks) 120. In an arithmetic sequence, the first term is 5 and the fourth term is 40. Find the second term. Working: Answers: ....…………………………………….......... (Total 4 marks) 121. Two functions f, g are defined as follows: f : x → 3x + 5 g : x → 2(1 – x) 84 Find (a) f –1(2); (b) (g o f)(–4). Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks) 122. In a survey, 100 students were asked ‘do you prefer to watch television or play sport?’ Of the 46 boys in the survey, 33 said they would choose sport, while 29 girls made this choice. Boys Girls Sport 33 29 Total 46 Total Television 100 By completing this table or otherwise, find the probability that 85 (a) a student selected at random prefers to watch television; (b) a student prefers to watch television, given that the student is a boy. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks) 123. The vectors u, v are given by u = 3i + 5j, v = i – 2j. Find scalars a, b such that a(u + v) – 8i + (6 – 2) j. Working: Answers: ....…………………………………….......... (Total 4 marks) 86 124. If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for (a) log2 5; (b) loga 20. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks) 125. Solve the equation 3 cos x = 5 sin x, for x in the interval 0° ≤...
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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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