STUPIDSID
1 (a)
\[Find\ L\left\{t\ e^{31}\sin t\right\}\]
5 M
1 (b)
Show that every square matrix can be uniquely expressed as the sum of a Hermitian and skew-Hermitian matrix.
5 M
1 (c)
Find Z-transform and region of convergence of f(k)=3k, k≥0.
5 M
1 (d)
Find the Fourier expansion of f(x)= x2 where -π ≤x ≤π.
5 M
2 (a)
Prove that following matrix is orthogonal and hence find its inverse.
\[A=\frac{1}{9}\left[\begin{array}{ccc}-8 & 4 & 1 \\1 & 4 & -8 \\4 & 7 & 4\end{array}\right]\]
\[A=\frac{1}{9}\left[\begin{array}{ccc}-8 & 4 & 1 \\1 & 4 & -8 \\4 & 7 & 4\end{array}\right]\]
6 M
2 (b)
\[Find\ L^{-1}\left\{\frac{s+2}{{\left(s^2+4s+5\right)}^2}\right\}\]
6 M
2 (c)
Obtain the Fourier expansion of \[
f\left(x\right)={\left(\frac{\pi{}-x}{2}\right)}^2
\] in the internal and 0≤x≤2π and f(x+2π)=f(x). Also deduce that,
\[ \left(i\right)\ \frac{{\pi{}}^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{} \]
\[ \left(ii\right)\frac{{\pi{}}^4}{90}=\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{} \]
\[ \left(i\right)\ \frac{{\pi{}}^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{} \]
\[ \left(ii\right)\frac{{\pi{}}^4}{90}=\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{} \]
8 M
3 (a)
Investing for what values of λ and μ the equations.
x+y+z=6
x+2y+3z=10
x+2y+λz=μ have,
(i) No Solution
(ii) a unique solution
(iii) Infinite no. of solutions.
x+y+z=6
x+2y+3z=10
x+2y+λz=μ have,
(i) No Solution
(ii) a unique solution
(iii) Infinite no. of solutions.
6 M
3 (b)
Obtain complex form of Fourier series for f(x)=eax (-π,π) where is not integer.
6 M
3 (c)
Solve (D2 - D - 2) y=20 sin 2t with y(0)=1, y'(0)=2.
8 M
4 (a)
Find Laplace transform of
\[ f\left(t\right)=a\sin{pt\ 0<t\leq{}\frac{\pi{}}{p}} \\ f\left(t\right)=0\frac{\pi{}}{P}< t\leq{}\frac{2\pi{}}{P} \\ and\ \ f\left(t\right)=\left(t+\frac{2\pi{}}{P}\right) \]
\[ f\left(t\right)=a\sin{pt\ 0<t\leq{}\frac{\pi{}}{p}} \\ f\left(t\right)=0\frac{\pi{}}{P}< t\leq{}\frac{2\pi{}}{P} \\ and\ \ f\left(t\right)=\left(t+\frac{2\pi{}}{P}\right) \]
6 M
4 (b)
Find the inverse Z-transform for
\[f\left(z\right)=\frac{1}{\left(z-3\right)\left(z-2\right)}\]
for 2<|z|<3.
\[f\left(z\right)=\frac{1}{\left(z-3\right)\left(z-2\right)}\]
for 2<|z|<3.
6 M
4 (c)
Find inverse Laplace transform of
\[ \left(i\right)\ \frac{e^{4-3s}}{{\left(s+4\right)}^{\frac{5}{2}}}\\ (ii)\tan^{-1}\frac{2}{s} \]
\[ \left(i\right)\ \frac{e^{4-3s}}{{\left(s+4\right)}^{\frac{5}{2}}}\\ (ii)\tan^{-1}\frac{2}{s} \]
8 M
5 (a)
Examine whether the following vectors are linearly independent or dependent [2,1,1], [1,3,1], [1,2,-1]
6 M
5 (b)
Using Convolution theorem prove that
\[ l^{-1}\left[\frac{1}{s}\ ln\ \left(\frac{s+1}{s+2}\right)\right]=\ \int_0^t\left(\frac{e^{-2u}-e^{-u}}{u}\right)du\ \]
\[ l^{-1}\left[\frac{1}{s}\ ln\ \left(\frac{s+1}{s+2}\right)\right]=\ \int_0^t\left(\frac{e^{-2u}-e^{-u}}{u}\right)du\ \]
6 M
5 (c)
Using Fourier cosine Integral prove that
\[e^{-x}\cos{x=\frac{1}{\pi{}}\int_0^{\infty{}}\frac{w^2+2}{w^4+4}\ \cos{wx\ dw}}\]
\[e^{-x}\cos{x=\frac{1}{\pi{}}\int_0^{\infty{}}\frac{w^2+2}{w^4+4}\ \cos{wx\ dw}}\]
8 M
6 (a)
Find the Fourier Transform 0+f(x)=e-1x1
6 M
6 (b)
Find z[f(x)] where
\[
f\left(k\right)=\cos{\left(\frac{k\pi{}}{u}+a\right)}
\] where k≥0.
6 M
6 (c)
Find Fourier expansion of f(x)=2x - x2 where 0 ≤ x ≤ 3 and period is 3.
8 M
7 (a)
Reduce the following matrix to noraml form and find its rank.
\[A=\left[\begin{array}{ccc}1 & -1 & 3 & 6 \\1 & 3 & -3 & -4 \\5 & 3 & 3 & 11\end{array}\right]\]
\[A=\left[\begin{array}{ccc}1 & -1 & 3 & 6 \\1 & 3 & -3 & -4 \\5 & 3 & 3 & 11\end{array}\right]\]
6 M
7 (b)
\[Evalute\ \int_0^{\infty{}}\frac{\cos{6t-\cos{4t}}}{t}dt\]
6 M
7 (c)
Show that the set of function
\[\sin{\left(\frac{\pi{}x}{2L}\right)},\sin{\left(\frac{3\pi{}x}{2L}\right)},\sin{\left(\frac{5\pi{}x}{2L}\right)},\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\]
is orthogonal over..(0,L) Hence construct corresponding orthonormal set.
\[\sin{\left(\frac{\pi{}x}{2L}\right)},\sin{\left(\frac{3\pi{}x}{2L}\right)},\sin{\left(\frac{5\pi{}x}{2L}\right)},\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\]
is orthogonal over..(0,L) Hence construct corresponding orthonormal set.
8 M
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