Question One

Thursday, 1 November 2018

10:23 PM

Untitled picture.png Machine generated alternative text:
1. 
i) By expanding sin(A + B) + sin(A — B), or otherwise, find 11 = 
sin(5x) cos(x) dr. 
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Machine generated alternative text:
ii) Evaluate the integral 12 
dc.
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Untitled picture.png Machine generated alternative text:
iii) Use appropriate tests to determine whether each of the following series converges or 
diverges 
1 
n(ln n)3' 
2 
2n + 







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Machine generated alternative text:
iv) Let S — 
y IR3 : z a; y 
2 
z 
a) Prove that S is closed under scalar multiplication. 
b) Prove that S is not a subspace of IR3
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Untitled picture.png Machine generated alternative text:
(02 
31) 
v) Let A 
a) Find the eigenvalues and eigenvectors for the matrix A. 
b) Write down an invertible matrix M and a diagonal matrix D such that 
D M-IAM. 
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Machine generated alternative text:
vi) 
Let 
4 
10 ¯ 44 
24 
Using the MAPLE output below, find a basis for ker(A). 
> with (LinearA1gebra) : 
10 ¯ 44 
> ReducedRowEche10nForm (A) ; 
1 
o 
o 
o 
4 
o 
1 
o 
o 
24 
o 
o 
1 
o
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