# 8

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online