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Unformatted text preview: a , a ≠ 0, b > 0, c > 0. b + e – cx ( ac 2 e – cx e – cx – b (b + e ) – cx 3 ). (4) (b) Find the coordinates of the point on the curve where f″ (x) = 0. (2) (c) Show that this is a point of inflexion. (2) (Total 8 marks) 413 615. A random variable X is normally distributed with mean µ and standard deviation σ, such that P (X > 50.32) = 0.119, and P(X < 43.56) = 0.305. (a) Find µ and σ. (5) (b) Hence find P(|X – µ| < 5). (2) (Total 7 marks) 616. Consider the following system of equations where b is a constant. 3x + y + z = 1 2x + y – z = 4 5x + y + bz = 1 (a) Solve for z in terms of b. (4) (b) Hence write down, with a reason, the range of values of b for which this system of equations has a unique solution. (2) (Total 6 marks) 617. The random variable X has a Poisson distribution with mean λ. (a) Given that P (X = 4) = P (X = 2) + P (X = 3), find the value of λ. (3) (b) Given that λ = 3.2, find the value of (i) P(X ≥ 2); (ii) P(X ≤ 3 X ≥ 2). (5) (Total 8 marks) 414 618. A medical statistician is studying the weights, x kg, of new-born babies in a hospital. She finds that, in one month, 15 babies were born. For these babies, ∑x = 55.5 and ∑x2 = 215.8. Assuming that weights of babies are normally distributed, calculate a 99 % confidence interval for the mean weight of babies born in this hospital. (Total 5 marks) 619. Eggs at a farm are sold in boxes of six. Each egg is either brown or white. The owner believes that the number of brown eggs in a box can be modelled by a binomial distribution. He examines 100 boxes and obtains the following data. Number of brown eggs in a box 0 10 1 29 2 31 3 18 4 8 5 3 6 (a) Frequency 1 (i) Calculate the mean number of brown eggs in a box. (ii) Hence estimate p, the probability that a randomly chosen egg is brown. (2) (b) By calculating an appropriate χ2 statistic, test, at the 5 % significance level, whether or not the binomial distribution gives a good fit to these data. (8) (Total 10 marks) 620. (a) Use a Venn diagram to show that (A ∪ B)′ = A′ ∩ B′. (2) 415 (b) Prove that [(A′ ∪ B) ∩ (A ∪ B′)]′ = (A ∩ B)′ ∩ (A ∪ B). (4) (Total 6 marks) 621. Let S = {f, g, h, j} be the set of functions defined by f(x) = x, g(x) = –x, h(x) = (a) 1 1 , j(x) = – , where x ≠ 0. x x Construct the operation table for the group {S,°}, where ° is the composition of functions. (3) (b) The following are the operation tables for the groups {0, 1, 2, 3} under addition modulo 4, and {1, 2, 3, 4} under multiplication modulo 5. + 0 1 2 3 × 1 2 3 4 0 0 1 2 3 1 1 2 3 4 1 1 2 3 0 2 2 4 1 3 2 2 3 0 1 3 3 1 4 2 3 3 0 1 2 4 4 3 2 1 By comparing the elements in the two tables given plus the table constructed in part (a), find which groups are isomorphic. Give reasons for your answers. State clearly the corresponding elements. (6) (Total 9 marks) 622. (a) The binary operation # is defined on the set of real numbers by a # b = a + b +1. Show that the binary operatio...
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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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