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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 chisquared 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.
 Fall '13
 Apple
 The Land

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