Applied Mathematics - 3 - May 2012

Electronics Engineering (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.

This Qs paper appeared for Applied Mathematics - 3 of Electronics & Telecomm. (Semester 3)

1 (a) Prove that
$$\int_0^{\infty{}}e^{-t}\dfrac{{sin}^2t}{t}dt= \dfrac{1}{4} log 5$$(5 marks) 1 (b) Is the matrix orthogonal ? If not then can it be converted to an orthogonal matrix :-
$$A=\left[\begin{array}{ccc}-8 & 1 & 4 \\4 & 4 & 7 \\1 & -8 & 4\end{array}\right]$$(5 marks) 1 (c) Obtain complex form of Fourier series for f(x) = e^ax in (-l ,l).(5 marks) 1 (d) Find the Z-transform of f(k) =a^k, k?0.(5 marks) 2 (a) Find the Fourier sine transform of f(x) if $$ \begin {align*} f(x)&=\sin kx, &0 \le x \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (b)\lt/b\gt Find the Matrix A if \ltbr\gt $$\left[\begin{array}{cc}2 & 1 \3 & 2\end{array}\right]\ A\ \left[\begin{array}{cc}-3 & 2 \5 & -3\end{array}\right]=\ \left[\begin{array}{cc}-2 & 4 \3 & -1\end{array}\right]$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt2 (c) \lt/b\gt (D\ltsup\gt2\lt/sup\gt- 3D+2) y=4 e\ltsup\gt21\lt/sup\gt, with y(0) = -3, y'(0)=5 solve using Laplace transform. \lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (a)\lt/b\gt Reduce the matrix to normal form and find its rank :- \ltbr\gt $$\left[\begin{array}{cccc}2 & 3 & -1 & -1 \1 & -1 & -2 & -4 \3 & 1 & 3 & -2 \6 & 3 & 0 & -7\end{array}\right]$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (b)\lt/b\gt Find the inverse Laplace transform of ? \ltbr\gt $$ \left(i\right)\ \frac{e^{-2s}}{s^2+8s+25}
\ \left(ii\right)\ \frac{e^{-3s}}{{\left(s+4\right)}^3}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt3 (c) \lt/b\gt $$ \left.\begin{matrix}f(x)&= \pi x 0\leq x \leq 1 \ f(x)&=\pi (2-x)1 \leq x \leq 2\end{matrix}\right} with \ period \ 2 $$\ltbr\gt Find the Fourier series expansion \lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a) \lt/b\gt Show that the set of functions $$ {\ \left(\frac{\pi{}x}{2l}\right), sin\left(\frac{3\pi{}x}{2l}\right),\ \sin\left(\frac{5\pi{}x}{2l}\right),.....}$$is orthogonal over (0,l).\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (b)\lt/b\gt If f(k)= 4\ltsup\gtk\lt/sup\gtU(K), g(k)= 5\ltsup\gtk\lt/sup\gtU(k), then find the z-transform of f(k) x g(k).\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (c) \lt/b\gt Solve the following equations by Gauss-Seidel Method. \ltbr\gt 28x+4y-z=32\ltbr\gt 2x+17y+4z=35\ltbr\gt x+3y+10z=24.\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a) \lt/b\gt Obtain Fourier series for \ltbr\gt $$ {\ f(x) = x + \frac{\pi{}}{2}, -\pi{} < x < 0} \ {= \frac{\pi{}}{2}-x\ 0 < x < \pi{}} $$ \ltbr\gt Hence deduce that, $$ {\ \frac{\pi^2}{8} = \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + ....} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt State Convolution theorem and hence find inverse Laplace transform of the function using the same :-\ltbr\gt $$f(s)\ =\frac{{\left(s+3\right)}^2}{{\left(s^2+6s+5\right)}^2}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c) \lt/b\gt For what value of ? the equations 3x-2y+ ? z=1, 2x+y+z=2, x+2y- ?z= -1 will have no unique solution ? Will the equations have any solution for this value of ? ? \lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt Show that every square matrix A can be uniquely expressed as P+iQ when P and Q are Hermitian matrices \lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt If L[f(t)] = f(s), then prove that L[ t\ltsup\gtn\lt/sup\gt f(t)] = (-1)\ltsup\gtn\lt/sup\gt d\ltsup\gtn\lt/sup\gt/ds\ltsup\gtn\lt/sup\gt f(s), Hence find the Laplace transform of f(t) = t cos\ltsup\gt2\lt/sup\gtt\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c) \lt/b\gt Obtain the half rang sine series for f(x) when \ltbr\gt $$ {\ f(x) = x 0 < x < \frac{\pi{}}{2}}\{= \pi{} - x \frac{\pi{}}{2}< x < \pi{}}
\ Hence \ find \ the \ sum \ of \ \sum_{2n-1}^{\infty{}}\ \frac{1}{n^4}$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(8 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (a)\lt/b\gt Find the Fourier transform of- \ltbr\gt f(x) = (1-x\ltsup\gt2\lt/sup\gt), |x|<| \ltbr\gt = 0 , |x|>|, \ltbr\gt then $$f\left(s\right)=\ -2\sqrt{\frac{2}{\pi{}}}\left[\frac{scoss-sins}{s^3}\right]$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (b)\lt/b\gt Find the inverse z transform of $$ F\left(z\right)=\ \frac{z}{\left(z-1\right)\left(z-2\right)},\ \left\vert{}z\right\vert{}>2$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (c) \lt/b\gt Find the non-singular matrices P and Q such that -\ltbr\gt $$A=\ \left[\begin{array}{ccc}1 & 2 & 3 & 2 \2 & 3 & 5 & 1 \1 & 3 & 4 & 5\end{array}\right]$$
is reduced to normal form. Also find its rank. (8 marks)