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[q 1] A theorem that relates surface integral with the volume is called (A) Stokes theorem (B) Gauss divergence theorem (C) Carnot's theorem (D) Maximum power transfer theorem [q 2] Given a vector field A=2rcosø r in cylindrical coordinates. For the contour as shown, (A) 1 (B) 1 - (π/2) (C) 1+(π/2) (D) - 1 [q 3] The curl of vector field A =ρz sin ϕ uρ + 3ρz^2 **cos uρ at point (5, 90°,
- is** (A) 0 (B) 12 uθ (C) 6 ur (D) 5 uφ [q 4] vectorsR= (-5 ar+10 aϕ+ 3 az) & S = (ar+2 aθ- 6az) are (A) Parallel
Vector objective
03 August 2015
(A) Parallel (B) perpendicular (C) at angle 88^0 (D) unrelated [q 5] Given A=20 r+30+ 20 øat (1,π/2.πσ) in spherical coordinates, the component of Aperpendicular to surface = is (A) 20 r +20 ø (B) 20 σ (C) 20 σ (D) 30 θ [q 6] The gradient of field f = y^2 x + xyz is (A) y (y+ z) ux + x (2y + z) uy+ x y uz (B) y (2x+z)) ux + x (x + z) uy + x y uz (C) y2 ux + 2 y x uy + x y uz (D) y (2y + z) ux + x (2y + z) uy+ x y uz [q 7] B= ∇ x A because (A) Div B = (B) ∇ x B= 0 (C) ∇. ∇ x B = (D) none of these [q 8] The gradient of field f = y^2 x + xyz is (A) y + z ) u x + x (2 y + z ) u y + xy u z (B) v (2 x + z ) u x + x ( x> + z ) u y + xy u z (C) v u x + 2 yx u y + xy u z (D) y (2 y + z ) u x + x ( 2y + z ) u y + xy u z
(A) Curl of A is zero (B) Potential difference between two points is zero (C) It is gradient of a scalar potential (D) The work done in a closed path inside the field is zero [q 13] Which of the following is zero? (A) grad div A (B) div grad V (C) div curl A (D) curl curl A [q 14] If D = xy ux + yz uy + zx uz, then the value of ∮ A. ds is, where S is the surface of the cube defined by 0 x 1, 0 y 1, 0 z 1 (A) 0. (B) 3 (C) 0 (D) 1. [q 15] If r = x i + y j +z k & r =|r|, the divergence of vector A = rnr is (A) (n+1) r n-^1 (B) 3n r n-^1 (C) (3n+1) r n (D) (nr+3) r n/ [q 16] If r=x i +y j + z k is the position vector of the point (x, y, z) & r=|r|. What is the value of ∇ r (A) r/ r (B) r /r (C) r r (D) 0 The velocity vector for a moving particle at time t is given by
[q 17] The velocity vector for a moving particle at time t is given by If at time t = 3π/2 the particle was at the point (3, - 6 π, 0), what was the x-coordinate of the particle at time t = 0? (A) 4 (B) 0 (C) 2 π (D) 2 [q 18] Given an irrigational vector field F = (k 1 xy+k 2 z^3 ) ax + (3x^2 - k 3 z) ay+ (3xz^2 - y) az Find divergence of F at (1, 1,-2) (GATE 1998) (A) 0.162πm^3 (B) 0.162 m^3 (C) 0.162π^2 m^3 (D) 16.2π m^3 [q 19] Consider points P and Q in the x-y plane, with P= (1,0) and Q=(0,1). The line integral 2&nint;QP (xdx + ydy) along the semicircle with the line segment PQ as its diameter(GATE-2008) (A) Is - 1 (B) Is 0 (C) Is 1 (D) Depends on the direction (clockwise or anti-clockwise) of the semicircle [q 20] If D = xy ux + yz uy + zx uz, then the value of is, where S is the surface of the cube defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 (A) 0.
