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Winter 2014 Exam : Mathematics III(Sant Gadge Baba Amravati University (SGBAU), Amravati)

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Rahul Nimkar
Jawahar Navodaya Vidyalaya (JNV)

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AN-1815 Faculty of Engineering & Technology Third Semester B.E. (Computer Science & Engg.) (CGS) Examination MATHEMATICS III Paper-3 KS 01/3 KE 01/3 IT 01 Sections A & B [Maximum Marks : 80 Time : Three Hours] INSTRUCTIONS TO CANDIDATES (1) All questions carry marks as indicated. (2) Answer THREE questions from Section A and THREE questions from Section B. (3) Due credit will be given to neatness and adequate dimensions. (4) Assume suitable data wherever necessary. (5) Use of calculator is permitted. (6) Use pen of Blue/Black ink/refill only for writing the answer book. (7) Figures to the right indicate full marks. SECTION A 1. (a) Solve : 6-'3( _ dy 6y = ex cosh 2x . dx 2 dx tJYQ- -5430 I 6 (Contd.) 10. (a) Find the bilinear transformation which maps the points z = 1, i, - 1 into the points w = i, 0, - i. Hence find (b) Solve : (2x + 3) dy 2 2 dy - 2(2x + 3) -12y = 6x . dx dx the image of I z <1. 6 7 (b) Determine the analytic function : OR 2. w = u + iv, if v = log(x 2 + y2) + x - 2y. (a) Solve : A2 X2 u dx` 11. (a) Find the divergence and curl of : xi+yj+zk _ v2 2 2 X + y +Z 6 -12y = x 3 .1ogx + 2x Y ddx (b) 2 dx Show that the vector : A = (x + 3y)i + (y - 3z)j + (x - 2z)i 7 y = cosec x . 4 V (b) Apply the method of variation of parameters to solve : d2y 7 3 is solenoidal. 3. (a) Find the Laplace transform of : (c) Evaluate fp o n ds, where s is the surface of the 4 te-2t sin t dt . plane 2x + y + 2z = 6 in the first octant and P = (x + y)2 i - 2x11 + 2yzk (b) Find the inverse Laplace transform of log 7 OR s+1 s-1 12. (a) Evaluate the line integral : 4 (r) lice j[(x 2 + xy)dx + (x 2 + y 2 )dy] , where C is the convolution theorem to find : L-1 [ 6 I s(s 2 + 4)1 . -y+ z at (b) Find the directional derivative of D = 4e2" the point (1, 1, -1) in the direction towards the 7 point (- 3, 5, 6). OR UYQ-5430 square formed by the lines y = 1 and x = 1. 7 2 (Contd.) UYQ-5430 5 3950 4. (a) Find the inverse Laplace transform of : 21s - 33 7 (s +1) (s - 2) 3 (b) Use Laplace transform method to solve the equation : d2x 2 dx + 2 + 5x = e-t sint, when x(0) = 0 and dt dt x'(0) = 1. 5. (a) 7 Solve : Y.+2 (ii) un 2 - +1 2y. = 5n 2u. + u = n2.2.. (b) Find the z-transform of sin (5k + 3). 4 4 5 OR 6. (a) Solve : yx+3 + 12y, = 4 - x. 3Yx+2 4 (b) Find the inverse z-transform of : 4z . ifizi>121. z-2 (c) Solve the difference equation : 6 Yk -- 2 Yk + 1 Yk z-transform. UYQ-5430 4 3 0, y(0) = 0, y(1) = 1 by 5 (Contd.) SECTION B 7. (a) Solve the following equations : y Z 4 p + xzq = y2 . + 1 ) = a2 . z 2 (p2 4 (b) Find the Fourier sine transform of f(x) x.5 OR 8. (a) Solve the following equations : (i) z = px + qy + sin (p + q) z 2 (p2 q2) = x2 + 2 y. 4 4 (b) Find the Fourier transform of : I. 1 x2 f(x) = { if ix 0 if lxl>1 5 9. (a) Show that the function u(x, y) = 4xy 3x + 2 is harmonic. Construct the corresponding analytic function f(z) = u(x, y) + i v(x, y). Express f(z) in terms of complex variable z. 6 (b) Expand the following function in Laurent's series : 1 , for 1<jzi<3. f(z) = 2 z 4z+3 7 OR UYQ-5430 4 (Contd.)

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