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

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Total No. of Questions : 4] P480 [Total No. of Pages : 3 [3717] - 202 S.Y. B.Sc. MATHEMATICS Paper - II (A) and II (B) Vector Calculus (Sem. - II) (New Course) (Paper - II(A) Time : 2 Hours] [Max. Marks : 40 Instructions to the candidates : 1) All questions are compulsory. (In selected paper). 2) Figures to the right indicate full marks. Q1) Answer the following questions. [10] a) If u = ( t 2 1) j + cos t k, b) 23 2 For the curve r ( t ) = ti + t j + t k , find a unit tangent vector at t = 1. 3 c) If d) If f ( x , y, z ) = xi + yj + zk , find e) Find a if u = ( x + 3 y )i + ( y 2 z ) j + ( x + ( z )) k is solevoidal. f) g) = sin ti + e t j ; find ltim u 0 . ( x , y, z ) = 2 xz 4 x 2 y find at the point (2, 2, 1). f using the definition. x In what direction from the point (2, 1, 1) is the directional dirivative of ( x , y, z ) = x 2 yz3 a maximum? Find the maximum value. If f ( x , y ) = cos xy i + (3xy 2x )j (3x + 2y)k, find 2 2 f at the point xy (1, 0). 2 h) If u (t ) = ti t j + ( t 1) k , ( t ) = 2t i + 6tk ; evaluate u dt . 2 2 0 P.T.O. 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 xdy ydx . C Q2) Attempt any two of the following: a) [10] If u , v are differentiable functions of t then show that d dv du (u v ) = u + v . dt dt dt b) Find the equation of the normal plane at t = 2 for the curve r ( t ) = 2t 2 i + ( t 2 4 t ) j + (3t 5)k c) Find the directionald erivative 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) If u a nd a re respectively vector and scalar functions of x, y, z possessing first order partial derivatives then prove that curl b) [10] ( u ) = grad u + curl u Find eigen space corresponding to largest eigen value of the matrix. 2 1 1 A = 0 3 1 2 1 3 c) Test the consistency and solve the system, if consistent. 2x + 2y + 2z = 0 [3717] - 601 2 2x + 5y + 2z = 1 8x + y + 4z = 1 Q4) Attempt any one of the following: a) [10] Define characteristic polynomial of n n matrix A. State and prove Cayley - Hamilton theorem and verify it for 3 2 A= . 1 0 b) i) ii) Transform the basis set S = {(1, 1, 1), (1, 1, 0), (1, 0, 0)} of Euclidean inner product space R3 to the orthonormal basis set, by using Gram - Schmidt process. [6] 4 2 For the matrix = , verify that A . adj (A) = |A| . I2. [4] 7 8 Y [3717] - 601 3

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