Applied Mathematics - 3 - Dec 2013

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 real and imaginary parts of an analytic function F(z)=u+iv are harmonic function.(5 marks) 1 (b) Find the fourier series for f(x)=|sin x| in (-?,?).(5 marks) 1 (c) Find the Laplace Transform of $$\int_0^t\ {ue}^{-3u}\sin{4udu}$$(5 marks) 1 (d) $$If\ \ \vec{F}=xye^{2z}i+xy^2coszj\ \ \ +x^2cosxyk.Then\ find\ div\ \bar{F\ }and\curl\ \bar{F.}$$$$$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(5 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (a)\lt/b\gt Using laplace transofrm solve- \ltbr\gt(D\ltsup\gt2\lt/sup\gt + 3D + 2) y= e\ltsup\gt-2t\lt/sup\gt. Sin t where y(0) = 0 and y' (0) =0.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (b)\lt/b\gt Find the directional derivative of d=x\ltsup\gt2\lt/sup\gt y cos?z at (1,2, ?/2)in the direction of t=2i + 3j + 2k. \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (c) \lt/b\gt Find the Fourier expansion of $$f(x)=\sqrt{1-cosx}\ in\left(0,2\pi{}\right).Hence\ prove\frac{1}{2}=\sum_{n=1}^{\infty{}}\frac{1}{4n^2-1}.$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (a)\lt/b\gt Prove that $$J_{\frac{3}{2}}=\sqrt{\frac{2}{\pi{}x}}.\left(\frac{\sinx}{x}-\cosx\right)$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (b)\lt/b\gt Evaluate by Green's theorem $$\oint_c\left(x^2ydx+y^3dy\right)$$ where c is closed path formed by y=x,y=x\ltsup\gt2\lt/sup\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (c) \lt/b\gt i) Find the Laplace Transform of $$\frac{\cos{bt-\cos{at}}}{t}$$\ltbr\gt ii) Find the Laplace Transform of $$\frac{d}{dt}\left[\frac{sint}{t}\right].$$\ltbr\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a)\lt/b\gt Show that the set of functions {sin??x,sin??3x?} OR {sin??(2n+1)x:n=0,1,2,3} is orthogonal over [0,??2],Hence construct orthonormal set of functions.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (b)\lt/b\gt Find the imaginary part whose real part is u= x\ltsup\gt3\lt/sup\gt - 3xy\ltsup\gt2\lt/sup\gt + 3x\ltsup\gt2\lt/sup\gt + 1\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c) \lt/b\gt Find Inverse Laplace Transform of?\ltbr\gt$$i)log\left(\frac{s^2+4}{s^2+9}\right)$$\ltbr\gt$$ii)\frac{s}{\left(s^2+4\right)\left(s^2+9\right)}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt Obtain half range sine series for f(x)=x\ltsup\gt2\lt/sup\gt in 0\ltx\lt3.\lt a=""\gt\ltbr\gt\ltbr\gt \lt/x\lt3.\lt\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt A Vector field F is given by $$\bar{F}=\left(x^2-yz\right)\hat{i}+\left(y^2-zx\right)\hat{j}+\left(z^2-xy\right)\hat{k}$$ is irroational and hence find scalar point function ? such that F = ? ?\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c) \lt/b\gt Show that the function V=e\ltsup\gtx\lt/sup\gt (xsiny+ycosy) satisfies Laplace equation and find its corresponding analytic function \lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt By using stoke's theorem ,evaluate \ltbr\gt$$\oint_c\left[\left(x^2+y^2\right)\hat{i}+\left(x^2-y^2\right)\hat{j}\right]\bullet{}d\bar{r}$$\ltbr\gtwhere c is the boundary of a region enclosed by circles x\ltsup\gt2\lt/sup\gt + y\ltsup\gt2\lt/sup\gt =4, x\ltsup\gt2 + y\ltsup\gt2\lt/sup\gt = 16.\lt/sup\gt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt Show that under the transformation w= 5-4z/4z-2 the circle |z|=1 in the z plane is transformed into a circle of unity in w-plane.\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c) \lt/b\gt Prove that $$\intJ_3\left(x\right)dx=\ -\frac{2J_1(x)}{x}-J_2(x)$$(8 marks)