Page 60 - Engineering Mathematics Workbook_Final

Vector Calculus
) )
31. Let C be the boundary of the triangle  (  sin x i + 2y j − z (1 sin2x k  nds = ____
$
$
$
2 $
−
formed by the points (1,0,0), (0,1,0) and S
(0,0,1). Then the value of the line 
integral (a) 1 (b) 2
 − 2ydx + (3x − 4y 2 ) dy + ( z + 3y ) dz
2
C (c)  (d) 2
is _______

(a) 0 (b) 1 35. The value of  − ydx + xdy where C
2
2
C x + y
(c) 2 (d) 4
1
is z = is _____
32. Let,
−
B = (  , , y z ): , , y z  R & x + y + z   4 (a) 2 (b) 2
2
2
2
x
x
=
+
. Let, r xi y j + 3k be vector (c) 0 (d) none
valued function defined on B. If 36. The line integral of F = 3i x j + yk
+
r = x + y + z then the value of on the circle x + 2 y = 2 2
2
2
2
2
0
    r r dv is ______. a , z =
2
S described in clockwise is _______
37. For the vector field
(a) 16 (b) 32
F = − 2 y 2 i + 2 x 2 j + sin y cos zk
3
2
(c) 64 (d) 128 x + y x + y
, the value of  F  dr where C is
33. The maximum magnitude of the C
directional derivative for the surface closed contour in xy plane consisting of
2
x + xy yz = 9 at the point (1,2,3) is parabolas y =  ( x + ) 1 and straight
+
2
along the direction __________. lines x =  1.
+
+
2
(a) i + j k (b) 2i + 2 j k 38. Let D be the triangular region in R
bounded by the y-axis, the line y = 1
=
(c) i + 2 j + 3k (d) i − 2 j + 3k and the line y x . Let C be the
boundary curve of the region and C be
34. Let S be the surface bounding the region oriented counter clockwise. Then the
x + 2 y  2 1, x , y  and n is value of the line integral
0
$
0
2
2
1
outward drawn unit normal to S. z   ( x + 2sin x cos ) x dx + (4x + y 2 ) dy
C
Then is
(a) 1 (b) 2

(c) 4 (d) 8

58