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Unformatted text preview: on the interval [0,1]. The random variable Z is given by 12 Z= ∑X n −6 n =1 (a) Show that E(Z) = 0 and Var(Z) = 1. (6) (b) Jim states that Z is approximately N(0,1) distributed. Justify this statement. (2) 544 (c) Jim writes a computer program to generate 500 values of Z. He obtains the following table from his results. Range of values of Z Frequency (–∞,–2) 16 [–2,–1) 66 [–1,0) 180 [0,1) 155 [1,2) 65 [2,∞) 18 (i) Use a chi-squared goodness of fit test to investigate whether or not, at the 5 % level of significance, the N(0, 1) distribution can be used to model these results. (ii) In this situation, state briefly what is meant by (a) a Type I error; (b) a Type II error. (16) (Total 24 marks) 794. Using deMorgan’s laws, prove that A ∆ B = (A ∪ B) ∩ (A ∩ B)′. (Total 6 marks) 795. The binary operation a * b is defined by a * b = (a) ab , where a, b ∈ a+b + . Prove that * is associative. (7) (b) Show that this binary operation does not have an identity element. (4) (Total 11 marks) 545 796. (a) Consider the functions f and g, defined by f: → g= × (i) where f(n) = 5n + 4, → × where g(x, y) = (x + 2y, 3x – 5y) Explain whether the function f is (a) (b) (ii) injective; surjective. Explain whether the function g is (a) (b) (iii) injective; surjective. Find the inverse of g. (13) (b) Consider any functions f : A → B and g : B → C. Given that is g ° f surjective, show that g is surjective. (3) (Total 16 marks) x + 2 x 797. Let the matrix T be defined by x − 5 − x such that det T = 1. (a) (i) Show that the equation for x is 2x2 – 3x – 9 = 0. (ii) The solutions of this equation are a and b, where a > b. Find a and b. (5) 546 (b) Let A be the matrix where x = 3 (i) Find A2. (ii) Assuming that matrix multiplication is associative, find the smallest group of 2 × 2 matrices which contains A, showing clearly that this is a group. (6) (Total 11 marks) 798. The group (G, ×) has a subgroup (H, ×). The relation R is defined on G (x R y) ⇔ (x–1 y ∈ H), for x, y ∈ G. (a) Show that R is an equivalence relation. (8) (b) Given that G = {e, p, p2, q, pq, p2q}, where e is the identity element, p3 = q2 = e, and qp = p2q, prove that qp2 = pq. (3) (c) Given also that H = {e, p2q}, find the equivalence class with respect to R which contains pq. (5) (Total 16 marks) 1 1 799. Calculate lim − . x →0 x sin x (Total 6 marks) ∞ 800. Use the integral test to show that the series 1 ∑n n =1 p , is convergent for p > 1. (Total 6 marks) 547 801. (a) (i) Find the first four derivatives with respect to x of y = ln(1 + sin x). (ii) Hence, show that the Maclaurin series, up to the term in x4, for y is y=x– 12 13 14 x + x − x +… 2 6 12 (10) (b) Deduce the Maclaurin series, up to and including the term in x4, for (i) y = ln(1 – sin x); (ii) y = ln cos x; (iii) y = tan x. (10) (c) tan( x 2 ) . Hence calculate lim x →0 ln cos x (4) (Total 24 marks) 802. Consider the differential equation (a) xy dx + = 1, where x < 2 and y = 1 when x = 0. dy 4 − x 2 Use Euler’s method with h = 0.25, to find an approximate value of y when x = 1, giving your answer to two decimal places. (10) (b) (i) By first finding a...
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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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