School of Distance Education

Vector Calculus 1

UNIVERSITY OF CALICUTSCHOOL OF DISTANCE EDUCATIONB.Sc. Mathematics

(2011 Admn.)

V Semester Core Course

VECTOR CALCULUS

Question Bank & Answer Key

1. The Components of the vector a with initial point p : (6,2,-1) and terminal point ( 7, -1, 2)

are

a) (1,-3,3) b) (1,3,-3) c) (1,3,3) d) (1,-3,-3)

2. The length of the vector a with initial point p: (3,2,5) and terminal point (5,1,3) is

a) 3 b) 4 c) 5 d) 6

3. The angle between the vectors a = [1,2, 3 ] and b = [0, -2,1] is

a) cos √ b) cos √c) cos √ d) cos √

4. The normal vector to the line x-2y+2 = 0 is

a) [-1,-2] b) [1,-2]

c) [-1,2] d) [1,2]

5. The straight line through the point (1,3) in the x y plane and perpendicular to the straight line-2y+2 = 0 is

a) 3 -y=2 b) + y=1

c) 2 +y=5 d) 2 -y=5

6. The unit vector perpendicular to the plane 4x+2y+4z = -7 is

a) [ , , ] b) [ , , ]

c) [ , , ] d) [ , , ]

7. If a = [1,1, 0] and b=[3,0,0] in the right-handed co-ordinates, then a x b is

a) [0,0,3] b) [0,0,-3]

c) [3,0,0] d) [-3,0,0]

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8. The volume of the parallelopiped whose co-terminal edges representing the vectors 3i+4j,2i+3j+4k and 5k is

a) 3 b) 4 c)5 d)6

9.The volume of the tetrahedron with co-terminal edges representing the vectors i+ j, i-j and 2k is

a) b) c) d)

10.if the vectors a= 2i-j+k, b=i+2j-3k and c=3i+ j+5k, are coplanar, then the value of is

a) -8 b) 6 c)2 d)-4

11.The equation of the plane determined by the points(2,-1,1) , (3,2,-1) and (-1,3,2) is

a) 11 +5y+13z=30 b)11 -5y-13z=30

c) 11 +5y-13z=-30 d)11 -5y+13z=30

12.The parametric equations for the line through (-3,2,-3) and (1,-1,4) are

a) =1+4t, y=-1-3t, z=4+7t b) =2+4t, y=-2-3t, z=-4+7t

c) =3+4t, y=8-3t, z=5+7t d) =1-4t, y=-1+3t, z=-4-7t

13.The distance between the point (1,1,5) and the line LX = 1+t, y=3-t , z=2t is

a) 2 b) √3 c)√7 d) √514.The equation of the plane through the point (-3,0,7) perpendicular to the vector 5i+2j-k is

a)5x+2y-z=-22 b) 5x-2y+z=-22 c) 5x+2y+z=22 d) 5x+2y-z=22

15.The point of intersection of the line x= +2t, y=-2t, z=1+t and the plane 3x+2y+6z=6 is

a) (1,1,2) b) (2,0,1) c) ( ,2,0) d) (0,1,3)

16. The spherical co-ordinate equation for the cone z= + is

a) ф =∏ b) ф =∏ /4 c) ф =∏ /2 d) none of these

17.if r(t) = sin + y+3k, then is

a) sin +3k b)cos + j+3k c) cos - j d) sin - j

18. A particle moves along the curve= 3 , = -2 , z= then the acceleration at = 1 is

a) 6i+2j+6k b) 6i+3k c) 6i+6k d)6i+2j+3k

19. If F(t) = (t- ) i + 2 j – 3k, then ∫ ( ) dt is

a) 5i+6j-3k b)8i+3j+k c) i+ j -3k d) i+ j-3k

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20. The length of one turn of the helix r(t) = (cos ) +tk is

a) 4 √2 b) 2 √2 c) √2 d) 3 √221. The unit tangent vector at a point to the curve r= a cos + a sint

a) -sin - cos b) sin + cosc) cos - sin d)-sin +cos

22. The radius of curvature of + = 2 at (3,4) is

a) b) c) d)

23. The value of ∫ 2 is

a) 2r+c b) c) +c d) none of these

24. The domain of the function f(x,y,z) = xyln(z)

a) Entire Space b) {(x,y,z) : xyz≠0}

c) half space z>0 d) half space z<0

25. The range of the function f( x,y) = cos is

a) [0,1] b) [-1,1] c) [0,&] d) [-1,1]

26. Which of the following holds for the function f(x,y) = ?

a) lim( , )→( , ) ( , ) exists b) lim( , )→( , ) ( , ) doesn’t exists

c) lim( , )→( , ) ( , ) = 0 d) none of these

27. Which of the following holds for the function f(x,y) = ?

a) f is continuous everywhere b) f is continuous nowhere

c) f is continuous on {(x,y)∈R2: x≠y} d) f is continuous on {(x’y) ∈ R2:x=y}

28. Let f(x,y) = x –y and g(z,y) = be two continuous functions. Then the composition function

g(f(x,y)) = is

a) discontinuous b) continuous

c) Continuous at origin d) None of these.

29. If f(x,y) = sin , then the value of at (3, ) is

a) 0 b) 1

c) -1 d) 2

30. If f(x,y) = , then, the value of ʄ is

a) ( ) ) b) ( ) )

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c) d) ( ) )31. The plane x=2 intersects the paraboloid z= + in parabola. Then the slope of the tangent

to the parabola at (5,2,4) is

a) 6 b) 4 c) 5 d) 1

32. If f(x,y)= then is

a) b) c) d) NOT

33. If ( , ) = + , thenƒ

a) b) 0 c) 1 d) NOT

34. If z=tan ( ), then + is

a) ( ) b) ( ) c) ( ) d) 0

35. If u= log (tan + tan +tan ), then ∑sin 2 is

a) 1 b) c) 3 d) 2

36. Let ᴡ (p,v, , ,g)=PV+ then the partial derivative W is given by:

a) b) pv + c) d) NOT

37. The linearization of ( , ) = + +1 at the point (1,1) is

a) 4 − + 7 b) 2 + 2 − 7c) 2 − 2 − 1 d) 2 − 2 + 1

38 If w= + , = = = then /t =3

a D b) 1 c) 2 d) 3

39. The derivative of ( , ) = + cos ( ) at the point (2,0) in the direction A = 3 − 4 is

a. 1 b) 0 c) -1 d) 2

40. The equation for the tangent to the ellipse + =2 at the point (3,2) is

a) 3 + 8 = 25 b) 8 + 3 + 25c) 3 − 8 = 25 d) 8 − 3 + 25

41. The unit normal to the surface y+2 = 4 at the point (2 -2, 3) is

a) + = b) + 2 + 5c) – − 7 d) − −

42. The equation for the tangent plane to the surface 2 + 3 − 4 = 7 at (1, -1, 2)

a) + − = 3 b) 2 + 5 − 6 =

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c) 7 − 3 + 8 = 26 d) 7 − 3 + 8 = 5

43. For the ( , , ) = − , ( , , ) = z the value of ∇ ( / ) is

a) - -( )

K b) − − K

c) − – k d) NOT

44. The function ( , ) = has a

a) local maximum b) local minimum

c) both local maximum & minimum d) no local extreme values

45. The Centre of curvature at the point (2,2) to the curve = 4a) (2,2) b) (2,4) c) (4,2) d) (4,4)

46. The absolute maximum value of ( , ) = 2 + 2 + 2 − − on the triangular plate inthe guardant bounded by the lines = 0 = 9 −

a) 4 b) 2 c) 3 d) 61

47. The point ( , , ) closest to the origin on the plane 2 + − − 5 = 0 is.

a) ( , , ) b) ( , , )

c) ( , , ) d) ( , , )

48. The minimum value that the function ( , ) = takes on the ellipse + = 1 is

a) 2 b) -2 c) 4 d) -4

49. The maximum value that the function( , ) =3 + 4 takes on the circle + =1 is

a) 5 b) 2 c) 3 d) None of these

50. The plane + + = 1 cuts the cylinder + = 1 in an ellipse. The points on theellipse that lies closest to the origin are

a) (1,0,0) and (0,0,1) b) (0,1,0) and (0,0,1)c) (1,0,0) and (0,1,0) d) (1,0,0) and (0,1,1)

51. Which among the following is the value of ∫ ∫ ( − ) ?a) 4 b) c) d)

52. What is the value of ʃ ʃ over the first guadrant of the circle + = ?a) b) c) d)53. The value of the integral ∫ ∫ isa) 0 b) 1 c) -1 d)54. The area enclosed between = 5, = 10 and = and = 5 + isa) 5 b) 6 c) 25 d) 36

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55. The area enclosed by the ellipse + = 1 isa) ⫪ b) 2⫪ c) ⫪ d) None of these.56. The volume of the solid enclosed by the sphere of radius a is(a) 4πa3 (b) (c) d)3πa357. The volume enclosed by the co-ordinate planes and the portion of the planex + y + z = 1 in the first octant is(a) ½ (b) 1/3 (c) 1/6 (d) ¼58. The value of ∬ + where R is the semi circular region bounded by the x- axis andthe curve y = √1 − is(a) (1 − ) (b) (1 + ) (c) ( − 1) (d) ( − 1)59. Which among the following is the value of ∫ ∫ ∫√/ ?(a) (26 + log 27) (b) (27 − log 26)(c) (27 + log 26) (d) (26 − log 27)60. The volume of the region D enclosed by the surfaces z = + 3 &z = 8 − − is(a) 8π √2 (b) 4π √2 (c) 2π √2 (d) π √261. Let V be the volume bounded by the planes x = 0, y = 0, z = 0 and x + y + z = 1.Then what is the value of ∭ ( ) ?(a) + log √2 (b) − log √2 (c) + log √2 (d) − log √262. The centroid solid of the (with density given by δ = 1) enclosed by the cylinderx2 +y2 = 4, bounded above by the paraboloid z = x2 +y2 and below by the xy planeis(a) lies inside the solid (b) lies outside the solid(c)lies on the solid (d) None of these63. The volume of the upper region D cut form the solid sphere ≤ 1 by the cone∅ =π/3 is(a) π/2 (b) π/4 (c) π/6 (d) π/364. A solid of constant density δ = 1 occupies the upper region D cut from the solid ≤ 1 bythe cone ∅ =π/3. The solid’s moment of inertia about the z - axis is given by.(a) π/12 (b) π/6 (c) π/8 (c)π/465. The volue of ∫ ∫ + ( − 2 ) is(a) 4/3 (b) 8/11 (c) 7/5 (d)2/9

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66.Consider the transformation x = u cosv, y = u sin v the jacobian for the transformation is(a) v (b)uv (c) u +v (d) u67.What is the value obtained by integrating the function f(x,y,z) =x-3y2z over the linesegment joining the origin and the point (1,1,1)?(a) 1 (b)0 (c)1/2 (d)None of these68. A coil spring lies along the helix r(t) = (cos4t)I +(sin4t)j +k, 0 ≤ t ≤2π. The spring’sdensity is a constant, δ = 1. Then the radius of gyration of the spring about the z-axis is(a)1 (b) 2 (c)3 (d)469. A slender metal arch denser at the bottom than top, lies along the semi circley2 +z2 = 1, z ≥ 0, in the yz-plane. If the density at the point (x, y, z) on the arch isδ(x, y, z) = 2-z, then the centre of mass of the arch is(a) (0,0,0) (b)(0,0.57,0) (c) (0,0, 0.57) (d) (0.57, 0, 0)70. The gradient field of f(x, y, z) – xyz is(a) yzi + xzj + xyk (b)xyi + xzj + yzk(c) xzi + yzj + xyk (d) None of these71. The unit normal to the surface x2y + 2xz + 4 at the point (2, -2. 3) is(a) + − (b) + +(c) + + (d) − −72. If ∇∅ = (y + y2 +z2)i +(x + z + 2xy)j +(y +2xz)k and ∅(1,1,1) = 3, then what is ∅ ?(a) xz + xy + yz2 -1 (b) xz + yz + xz2(c) xy2 +xz2 -1 (d) xy + xy2 + xz2 + yz -173. Which among the following is the work done in moving a particle once round a circle C inthe xy-plane. Given the circle has centre at the origin and radius 3 and the force field isgiven by F = (2x – y + z)i + (x + y – z2 )j + (3x – 2y + 4z)k.(a) 8π (b) 80π (c) 88π (d) 18π74. If F = (3x2 + 6y)j – 14yzj + 20xz2k, then the value of ∫ . where c is a curve from(0,0,0) to (1,1,1) with parametric from x = t, y = t2 , z = t3 is(a) 13 (b) 7 (c) 5 (d) 1175. A fluid’s velocity field is F = xi + zj + yk. Then the flow along the helix r(t) = (cost)i +(sin t)j + t k, 0 ≤ t ≤ π/2 is(a) - (b) (c) + (d) -76. The circulation of the field F = (x – y)i + xj around the circle r(t) = (cost)i + (sin t)j,0 ≤ t ≤ 2π is(a) π (b) 3π (c) (d) 2π

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77. The flux of F = (x – y) i + xj across the circle x2 +y2 = 1 in the xy – plane is π. The rest flowacross the curve is(a) outward (b) inward (c) no flow (d)None of these78. The work done by the conservative field ∇ (xyz) along any smooth curve c joining thepoint (-1, 3, 9) to 1, 6, -14) is(a) 2 (b) 3 (c) 1 (d) 079. Let F = (2x – 3)i – xj + (cos z)k, F is(a) always consecutive (b) not consecutive(c) may be consecutive (d) may not be consecutive80. The differential form ydx + xdy + 4dz is(a) exact (b) not exact (c) may be exact (d) note of these81. The circulation density or curl of a vector field F = Mi +Nj at the poing (x,y) is(a) + (b) − (c) − (d) −82. The divergence of F(x,y) = (x2 – 2y)i + (xy – y2)j is(a) x + 2y (b) 3x + y (c) 2x – 3y (d) 3x – 2y83. The curl of the vector field F (x,y) = (x2 – 2y)i + (xy – y2) j is(a) y +2 (b) x + 2 (c) xy + 1 (d) x + y84. The area of the cap cut from the hemisphere x2 + y2 + z2 = 2, z ≥ 0, by the cylinderx2 + y2 = 1 is(a) 2π (2 + √2) (b) π (2 - √2) (c) 2π (2 - √2) (d) 2π (π + √π)85. A parameterization of the sphere x2 + y2 + z2 = a2 is given by(a) r (∅, θ) = (a cos ∅ cosθ)i + (a sin ∅ sin θ)j + (a sin θ)k, 0 ≤ ∅ ≤ π, 0 ≤ θ ≤ 2π(b) r (∅, θ) = (a sin ∅ cosθ)i + (a sin ∅ sin θ)j + (a cos ∅)k, 0 ≤ ∅ ≤ π, 0 ≤ θ ≤ 2π(c) r (∅, θ) = (a sin θ cos∅)i + (a sin ∅ sin θ)j + (a sin ∅)k, 0 ≤ ∅ ≤ 2π, 0 ≤ θ ≤ 2π(d) None of these86. The surface area of the cone z = + , 0 ≤ z ≤ 1 is(a) √ (b) π √2 (c) π √3 (d) π √587. The circulation of the field F=( − ) + 4 + k) around the curve C in which the

plane z=2 meets the cone z= + , counterclokwise as viewed from above is

a) ⫪ b) 2 ⫪ c) 4⫪ d) √⫪88. curl (grad ) is

a) 1 b) 0 c) d)

89. If r = + + and |r|=r, then grad r is

a)ɤ

b)ɤ

c)ɤ

d) None of these

90. A vector is called sdenoidal if its

a) divergence is zero b) curl is zero c) gedceint is zero d) None of these

91. The net outward flux of th field F = ( + +zk), = + + across the boundary of

the region D: + + ≤ is

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a) 2 ⫪ b) 0 c) 4 ⫪ d) ⫪92. If ^ is the unit vector in the direction of r and r = |r|, then di v( ^) is

a) b) r c) zr d)

93. A vector F is irrotational if

a) ∇F= 0 b) ∇ F= 0 c) ∇ x F = 0 d) None of these

94. A vector a is called orthogonal to a vector b if

a) a.b = 0 b) a x b = 0 c) a + b = 0 d) a-b = o

95. Vector product is

a) Commutative b) anticommutative

c) associative d) not distributive wet vector addition.

96. Scalar triple product of three coplanar vectors is

a) less than 0 b) greater than 0

c) equal to 0 d) None of these

97. The necessary and sufficient condition for the vector function F(t) to have constant magnitudeis

a) = 0 b) F. = 0 c) F x = 0 d) None of these

98. If F,G are differentiable vector functions and ∅is a differentiable scalar function. Then

a) curl (F x G) = (grad ∅ ) x F + ∅ curl (F)

b) div ( F X G) = -F curl G + g curl F

c) div ( F x G) = ( G, ∇) F – ( ∇) G + Fdiv G- G div F

d) curl ( F x G) = F curl G - G curl F

99. The unit vector along the vector u = 2 + 3 − is

a) √ b) c) )

100. The inequality |a.b| ≤|a| |b| is called

a) Parallelogram identity b) Triangle ineuqlity

c) Schwarz inequality d) None of these.

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Vector Calculus 10

ANSWER KEY

©Reserved

1 a 26 b 51 b 76 d

2 a 27 d 52 d 77 a

3 b 28 b 53 b 78 b

4 b 29 b 54 c 79 b

5 c 30 d 55 a 80 a

6 d 31 b 56 c 81 c

7 b 32 c 57 c 82 d

8 c 33 c 58 d 83 a

9 a 34 d 59 d 84 c

10 d 35 d 60 a 85 b

11 a 36 c 61 c 86 b

12 a 37 b 62 b 87 c

13 d 38 b 63 d 88 b

14 a 39 c 64 a 89 c

15 c 40 a 65 d 90 a

16 b 41 d 66 d 91 c

17 c 42 c 67 b 92 d

18 a 43 a 68 a 93 c

19 c 44 d 69 c 94 a

20 b 45 d 70 a 95 b

21 d 46 a 71 c 96 c

22 a 47 c 72 d 97 b

23 c 48 b 73 d 98 b

24 c 49 d 74 c 99 a

25 d 50 c 75 a 100 c