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# quizbmarkscheme - Math2061 Vector Calculus Quiz Version B...

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Math2061 Vector Calculus Quiz Version B Solutions 1. (a) 2 marks for correct sketch r(t)=7i +6j + t(-3i -5j), (2 marks) Also correct is x =7 -3t,y =6 -5t, 0 = t = 1 (1 mark) (b) 2 marks for correct sketch x = t,y = t3 (1 mark) -2 = t = 1 (c) 2 marks for correct sketch x = 4 cos t,y = 4 sin t (1

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mark) 0 = t = 2# (1 mark) 2. (a) F nds = ((3x + z)i + y2j + (2z + x)k) (xi + yj)ds . CC = ((3x + z)i + y2j) (xi + yj)ds C @ =( @ (3x + z)+ (y2))dA (1 mark) = (3+2y)dA (1 mark) R 2# 1 = ( (3+2r sin # rdr) d# (1 mark)
00 2# 1 2 = ((3r +2rsin # dr) d# 00 2# 23 2r = ([ 3r + sin #]10) d# 02 3 2# 2 =( 3 + sin #) d# 02 3 =3# (1 mark) (b) We have z = 0. (1 mark) The circle is parametrised by

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x = 2 cos #, y = 2 sin #,0 = # = 2#. (1 mark) 2 (y+ y)ds (1 mark) C 2# = (4 sin2 # + 2 sin #)ds (1 mark) 0 1 = [2(# - sin 2#) + 2(-cos #)]2# 20 =4# (1 mark) 3. (a) r(t)= t2i + (sin t)j + etk (1 mark) 1

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(b) r ' (t)=2ti + (cos t)j + etk (1 mark) = j + k (1 mark) (c) r '' (t)=2i +(-sin t)j + etk (1 mark) =2i-sin 3j + e3k (1 mark) (d) F dr C 5 ( dx dy dz = (3xzi -y2j +(x -z2)k) i + j + k)dt (1 mark) 0 dt dt dt 5
= (3t2eti -(sin2t)j +(t2 -e2t)k (2ti + (cost)j + etk)dt (1 mark) 0 = -05(6t3et -sin2tcost + t2et -e3t)dt (1 mark) t - sin3 tt - 13t]5 = [(6t3 - 18t2 + 36t - 36)e +(t2 - 2t + 2)ee using repeated 3 30 integration by parts for 6t3et and t2et (1 mark) sin3

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quizbmarkscheme - Math2061 Vector Calculus Quiz Version B...

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