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Unformatted text preview: 73. (a) Sketch the graph of the function 1 C ( x) = cos x + cos 2 x 2 for –2π ≤ x ≤ 2π (5) (b) Prove that the function C(x) is periodic and state its period. (3) (c) For what values of x, –2π ≤ x ≤ 2π, is C(x) a maximum? (2) (d) Let x = x0 be the smallest positive value of x for which C(x) = 0. Find an approximate value of x0 which is correct to two significant figures. (2) 190 (e) (i) Prove that C(x) = C(–x) for all x. (2) (ii) Let x = x1 be that value of x, π < x < 2π, for which C(x) = 0. Find the value of x1 in terms of x0. (2) (Total 16 marks) 274. (a) The function f is defined by f :xa ex – 1 – x (i) Find the minimum value of f. (2) (ii) Prove that ex ≥ 1 + x for all real values of x. (3) (b) Use the principle of mathematical induction to prove that 1 1 1 (1 + 1) ( 1 + ) (1 + ) ....(1 + ) = n + 1 2 3 n for all integers n ≥ 1. (6) (c) Use the results of parts (a) and (b) to prove that e (1 + 11 1 + + ....+ ) 23 n > n. (4) (d) Find a value of n for which 11 1 1 + + + ..... + > 100 23 n (3) (Total 18 marks) 191 275. In a reforested area of pine trees, heights of trees planted in a specific year seem to follow a normal distribution. A sample of 100 such trees is selected to test the validity of this hypothesis. The results of measuring tree heights, to the nearest centimetre, are recorded in the first two columns of the table below. Height of tree Observed frequency Expected frequency 15 ≤ h < 45 6 6 45 ≤ h < 75 11 a 15 ≤ h < 105 15 16 105 ≤ h < 135 20 20 135 ≤ h < 165 18 b 165 ≤ h < 195 14 15 195 ≤ h < 225 10 c 225 ≤ h < 255 6 5 100 100 Total (a) Describe what is meant by (i) a goodness of fit test (a complete explanation required); (3) (ii) the level of significance of a hypothesis test. (1) (b) Find the mean and standard deviation of the sample data in the table above. Show how you arrived at your answers. (4) (c) Most of the expected frequencies have been calculated in the third column. (Frequencies have been rounded to the nearest integer, and frequencies in the first and last classes have been extended to include the rest of the data beyond 15 and 225. Find the values of a, b and c and show how you arrived at your answers. (4) (d) In order to test for the goodness of fit, the test statistic was calculated to be 1.0847. Show how this was done. (3) 192 (e) State your hypotheses, critical number, decision rule and conclusion (using a 5% level of significance). (5) (Total 20 marks) 276. Let S = {(x, y) x, y ∈ }, and let (a, b), (c, d) ∈ S. Define the relation ∆ on S as follows: (a, b) ∆ (c, d) ⇔ a2 + b2 = c2 +d2 (a) Show that ∆ is an equivalence relation. (4) (b) Find all ordered pairs (x, y) where (x, y) ∆ (1, 2). (2) (c) Describe the partition created by this relation on the (x, y) plane. (1) (Total 7 marks) 277. Define the function f: 2 → 2 such that f(x, y) = (2y – x, x + y) (a) Show that f is injective. (4) (b) Show that f is surjective. (3) (c...
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