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Pune University - SY BSc (Sem - II) MATHEMATICS - II (A), April 2010

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Total No. of Questions : 4] [Total No. of Pages : 2 P480 [3717]-202 S.Y. B.Sc. MATHEMATICS Paper - II (A) : Vector Calculus (Sem. - II) (New Course) Time : 2 Hours] [Max. Marks : 40 Instructions : 1) All questions are compulsory. 2) Figures to the right indicate full marks. Q1) Answer the following questions : [10] a) If u = (t 2 1) j + cos t k , v = sin t i + et j; Find lim u v . t 0 b) For the curve r (t ) = ti + t 2 j + t 3k , find a unit tangent vector at t = 1. c) If ( x, y, z ) = 2 xz 4 x 2 y , find at the point (2, -2, -1). d) If e) Find if u = (x + 3 y ) i + ( y 2 z ) j + (x + z ) k is solenoidal. f) In what direction from the point (2, 1, -1) is the directional derivative of a maximum? Find the maximum value. 2 3 ((x,,y,,zz)) = x 2 yz 3 + zk f x y = xi + yj , find f using the definition. x g) 2 f If f ( x, y ) = cos xy i + 3xy 2 x j (3x + 2 y ) k , find at the point (1, 0). x y h) If u (t ) = ti t 2 j + (t 1 ) k , v (t ) = 2 t 2 i + 6 tk ; ( 2 ) 2 evaluate u v dt . 0 i) Define an irrotational vector field. j) Using Green's theorem, show that the area bounded by a simple closed 1 curve C is given by, 2 x dy y dx . C P.T.O. 1 Q2) Attempt any two of the following: [10] a) If u , v are differentiable functions of t then show that b) Find the equation of the normal plane at t = 2 for the curve r (t ) = 2t 2i + (t 2 4t ) j + (3t 5) k . c) Find the directional derivative of (x, y, z ) = xy + yz + zx at (1, -1, 1) along the line joining the points (1, 1, 1) and (2, -2, 2). Q3) Attempt any two of the following. a) [10] If u and a re respectively vector and scalar functions of x, y, z possessing first order partial derivatives then prove that curl ( u ) = grad u + curl u . b) If r = xi + yj + zk , r = r , find grad rn. Hence or otherwise find r . c) Find the scalar potential if 1 d grad = (x + 2 y + 4dui + (2 x 3 y z ) j + (4 x y + 2 z ) k (u v ) = u dv +. z ) v dt dt dt Q4) Attempt any one of the following. a) State and prove Green's theorem in plane. b) i) [10] Evaluate (x i + y 2 2 ) j + z 2k n ds where S is the surface of the ellipsoid S x2 y2 z2 + + =1 . a2 b2 c2 ii) Evaluate f dr by using Stoke's theorem, where f = xyi + xy 2 j C taken round the square C with vertices (1, 0), (-1, 0), (0, 1), (0, -1). [3717]-202(Paper IIA) 2 -2-

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Additional Info : S.Y. B.Sc. Mathematics (Semister - II) : Vector Calculus Paper - II (A) (New Course), Pune University
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