MU Electronics Engineering (Semester 3)
Applied Mathematics - 3
May 2012
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary


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 M
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 M
1 (c) Obtain complex form of Fourier series for f(x) = eax in (-l ,l).
5 M
1 (d) Find the Z-transform of f(k) =ak, k?0.
5 M

2 (a) Find the Fourier sine transform of f(x) if \[ \begin {align*} f(x)&=\sin kx, &0 \le x a\end{align*} \]
6 M
2 (b) Find the Matrix A if
\[\left[\begin{array}{cc}2 & 1 \\3 & 2\end{array}\right]\ A\ \left[\begin{array}{cc}-3 & 2 \\5 & -3\end{array}\right]=\ \left[\begin{array}{cc}-2 & 4 \\3 & -1\end{array}\right]\]
6 M
2 (c) (D2- 3D+2) y=4 e21, with y(0) = -3, y'(0)=5 solve using Laplace transform.
8 M

3 (a) Reduce the matrix to normal form and find its rank :-
\[\left[\begin{array}{cccc}2 & 3 & -1 & -1 \\1 & -1 & -2 & -4 \\3 & 1 & 3 & -2 \\6 & 3 & 0 & -7\end{array}\right]\]
6 M
3 (b) Find the inverse Laplace transform of ?
\[ \left(i\right)\ \frac{e^{-2s}}{s^2+8s+25}
\\ \left(ii\right)\ \frac{e^{-3s}}{{\left(s+4\right)}^3}\]
6 M
3 (c) \[ \left.\begin{matrix}f(x)&= \pi x 0\leq x \leq 1 \\ f(x)&=\pi (2-x)1 \leq x \leq 2\end{matrix}\right\} with \ period \ 2 \]
Find the Fourier series expansion
8 M

4 (a) 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).
6 M
4 (b) If f(k)= 4kU(K), g(k)= 5kU(k), then find the z-transform of f(k) x g(k).
6 M
4 (c) Solve the following equations by Gauss-Seidel Method.
28x+4y-z=32
2x+17y+4z=35
x+3y+10z=24.
8 M

5 (a) Obtain Fourier series for
\[ {\ f(x) = x + \frac{\pi{}}{2}, -\pi{} < x < 0} \\ {= \frac{\pi{}}{2}-x\ 0 < x < \pi{}} \]
Hence deduce that, \[ {\ \frac{\pi^2}{8} = \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + ....} \]
6 M
5 (b) State Convolution theorem and hence find inverse Laplace transform of the function using the same :-
\[f(s)\ =\frac{{\left(s+3\right)}^2}{{\left(s^2+6s+5\right)}^2}\]
6 M
5 (c) 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 ? ?
8 M

6 (a) Show that every square matrix A can be uniquely expressed as P+iQ when P and Q are Hermitian matrices
6 M
6 (b) If L[f(t)] = f(s), then prove that L[ tn f(t)] = (-1)n dn/dsn f(s), Hence find the Laplace transform of f(t) = t cos2t
6 M
6 (c) Obtain the half rang sine series for f(x) when
\[ {\ f(x) = x 0 < x < \frac{\pi{}}{2}}\\{= \pi{} - x \frac{\pi{}}{2}< x < \pi{}}
\\ Hence \ find \ the \ sum \ of \ \sum_{2n-1}^{\infty{}}\ \frac{1}{n^4}\]
8 M

7 (a) Find the Fourier transform of-
f(x) = (1-x2), |x|<|
= 0 , |x|>|,
then \[f\left(s\right)=\ -2\sqrt{\frac{2}{\pi{}}}\left[\frac{scoss-sins}{s^3}\right]\]
6 M
7 (b) Find the inverse z transform of \[ F\left(z\right)=\ \frac{z}{\left(z-1\right)\left(z-2\right)},\ \left\vert{}z\right\vert{}>2\]
6 M
7 (c) Find the non-singular matrices P and Q such that -
\[A=\ \left[\begin{array}{ccc}1 & 2 & 3 & 2 \\2 & 3 & 5 & 1 \\1 & 3 & 4 & 5\end{array}\right]\]
is reduced to normal form. Also find its rank.
8 M



More question papers from Applied Mathematics - 3
SPONSORED ADVERTISEMENTS