STUPIDSID
1 (a)
Prove that
\[\int_0^{\infty{}}e^{-t}\dfrac{{sin}^2t}{t}dt= \dfrac{1}{4} log 5\]
\[\int_0^{\infty{}}e^{-t}\dfrac{{sin}^2t}{t}dt= \dfrac{1}{4} log 5\]
5 M
1 (b)
Is the matrix orthogonal ? If not then can it be converted to an orthogonal matrix :-
\[A=\left[\begin{array}{ccc}-8 & 1 & 4 \\4 & 4 & 7 \\1 & -8 & 4\end{array}\right]\]
\[A=\left[\begin{array}{ccc}-8 & 1 & 4 \\4 & 4 & 7 \\1 & -8 & 4\end{array}\right]\]
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 \[ \begin {align*}
f(x)&=\sin kx, &0 \le x <a \\
&=0, &x>a
\end{align*}
\]
6 M
2 (b)
Find the Matrix A if
\[\left[\begin{array}{cc}2 & 1 \\3 & 2\end{array}\right]\ A\ \left[\begin{array}{cc}-3 & 2 \\5 & -3\end{array}\right]=\ \left[\begin{array}{cc}-2 & 4 \\3 & -1\end{array}\right]\]
\[\left[\begin{array}{cc}2 & 1 \\3 & 2\end{array}\right]\ A\ \left[\begin{array}{cc}-3 & 2 \\5 & -3\end{array}\right]=\ \left[\begin{array}{cc}-2 & 4 \\3 & -1\end{array}\right]\]
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 :-
\[\left[\begin{array}{cccc}2 & 3 & -1 & -1 \\1 & -1 & -2 & -4 \\3 & 1 & 3 & -2 \\6 & 3 & 0 & -7\end{array}\right]\]
\[\left[\begin{array}{cccc}2 & 3 & -1 & -1 \\1 & -1 & -2 & -4 \\3 & 1 & 3 & -2 \\6 & 3 & 0 & -7\end{array}\right]\]
6 M
3 (b)
Find the inverse Laplace transform of ?
\[ \left(i\right)\ \frac{e^{-2s}}{s^2+8s+25}
\\ \left(ii\right)\ \frac{e^{-3s}}{{\left(s+4\right)}^3}\]
\[ \left(i\right)\ \frac{e^{-2s}}{s^2+8s+25}
\\ \left(ii\right)\ \frac{e^{-3s}}{{\left(s+4\right)}^3}\]
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
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.
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, \[ {\ \frac{\pi^2}{8} = \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + ....} \]
\[ {\ f(x) = x + \frac{\pi{}}{2}, -\pi{} < x < 0} \\ {= \frac{\pi{}}{2}-x\ 0 < x < \pi{}} \]
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 :-
\[f(s)\ =\frac{{\left(s+3\right)}^2}{{\left(s^2+6s+5\right)}^2}\]
\[f(s)\ =\frac{{\left(s+3\right)}^2}{{\left(s^2+6s+5\right)}^2}\]
6 M
5 (c)
For what value of λ the equations 3x-2y+ λ z=1, 2x+y+z=2, x+2y- λz= -1 will have no unique solution ? Will the equations have any solution for this value of λ ?
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
\[ {\ f(x) = x 0 < x < \frac{\pi{}}{2}}\\{= \pi{} - x \frac{\pi{}}{2}< x < \pi{}}
\\ Hence \ find \ the \ sum \ of \ \sum_{2n-1}^{\infty{}}\ \frac{1}{n^4}\]
\[ {\ f(x) = x 0 < x < \frac{\pi{}}{2}}\\{= \pi{} - x \frac{\pi{}}{2}< x < \pi{}}
\\ Hence \ find \ the \ sum \ of \ \sum_{2n-1}^{\infty{}}\ \frac{1}{n^4}\]
8 M
7 (a)
Find the Fourier transform of-
f(x) = (1-x2), |x|<|
= 0 , |x|>|,
then \[f\left(s\right)=\ -2\sqrt{\frac{2}{\pi{}}}\left[\frac{scoss-sins}{s^3}\right]\]
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.
\[A=\ \left[\begin{array}{ccc}1 & 2 & 3 & 2 \\2 & 3 & 5 & 1 \\1 & 3 & 4 & 5\end{array}\right]\]
is reduced to normal form. Also find its rank.
8 M
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