Applied Mathematics - 3 - Dec 2014

Electronics & Telecomm. (Semester 3)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) Prove that f(z)=x²-y^{+2ixy is analytic and find f'(z).}(5 marks) 1 (b) Find the Fourier series expansion for f(x)=|x|, in (-π, π).(5 marks) 1 (c) Using laplace transform solve the following differential equation with given condition $$ \dfrac {d^2y}{dt^2} +y=t, $$ given taht y(0)=1&y'(0)=0.(5 marks) 1 (d) If $$ \overline A = \nabla (xy+ yz+ zx), \ find \ \nabla\cdot \overline A \ and \ \nabla \times \overline A $$(5 marks) 2 (a) $$ if \ L[J_0 (t) ] = \dfrac {1}{\sqrt{s^2+1}} $$ prove $$ \int^\infty_0 e^{-6t} t J_0 (4t) dt = 3/500 $$(6 marks) 2 (b) Find the directional derivative of &straighthi;=x⁴+y⁴+z⁴ at (1,-2,1) in the direction of AB where B is (2,6,-1). Also find the maximum directional of ϕ at (1,-2,1).(6 marks) 2 (c) Find the Fourier series expansion for f(x)=4-x², in (0,2) Hence deduce that $$ \dfrac {\pi^2}{6}=\dfrac {1}{1^2} + \dfrac {1}{2^2}+ \dfrac {1}{3^2} \cdots \ \cdots $$(8 marks) 3 (a) Prove that $$ J_{1/2}(x) = \sqrt{\dfrac {2}{\pi x}} \sin x $$(6 marks) 3 (b) Using Green's theorm evaluate $$ int_c (2x^2-y^2)dx + (x^2+y^2) dy where 'c' is the boundary of the surface enclosed by the line x=0, y=0, x=2, y=2.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (c)\lt/b\gt i) Find Laplace Transform of $$ e^{-\pi} \int^t_c u \sin 3u \ du
ii) Find Laplace Transform of $$ \dfrac {d}{dt} \left ( \dfrac {1-\cos 2t}{t} \right )\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a)\lt/b\gt Obtain complex form of Fourier series for the function f(x)=sin ax in (-?, ?), where a is not an integer.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (b)\lt/b\gt Find the analytic function whose imaginary part is $$ v=\dfrac {x}{x^2+y^2} + \cos h \ y\cdot \cos x$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c)\lt/b\gt Find inverse Laplace Transform of following $$ i) \ \log \left [\dfrac {s^2+a^2}{\sqrt{s+b}} \right ] \ ii) \ \dfrac {1}{s^3 (s-1)} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt Obtain half-range cosine series for f(x)=x(2-x) in 0\ltx\lt2.\lt a=""\gt\ltbr\gt\ltbr\gt \lt/x\lt2.\lt\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt Prove that $$ \overline{F} = \dfrac {\overline r}{r^3} $$ is both irrotational and solenoidal.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c)\lt/b\gt Show that the function u=sin x cosh y+2 cos x sinh y+ x\ltsup\gt2\lt/sup\gt-y\ltsup\gt2\lt/sup\gt+4xy satisfies Laplace's equation and find it corresponding analytic function.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt Evaluate by Stoke's theorem $$ \int_c (xy \ dx + x y^2 \ dy) $$ where C is the square in the xy-plane with vertices (1,0), (0,1), (-1,0) and (0,-1).\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt Find the bilinear transformation, which maps the points z=1,1,? onto the points w=-i, -1, i.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c)\lt/b\gt Show that the general solution of $$ \dfrac {d^2 y }{d x^2} + 4x^2 y=0 \ is \ y=\sqrt{z} [A J_{1/4} (x^2) + B \ J_{-1/4} (x^2)] $$ where A and B are constants.(8 marks)