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. Ink Drawings Ink Drawings Ink Drawings Ink Drawings                  
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 +        Ink Drawings Ink Drawings Ink Drawings Ink Drawings  Ink Drawings Ink Drawings Ink Drawings 
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. Ink Drawings Ink Drawings Ink Drawings             Ink Drawings Ink Drawings Ink Drawings      
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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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