STUPIDSID
1 (a)
Find L.T. of \[f\left(t\right)=\ f\left(x\right)= \Bigg\{\begin{align*}{}1,\ \ \ \ \ 0<t<a \\-1,\ \ \ \ a<t<2a\end{align*} \]
And f(t) = f(t+2a)
And f(t) = f(t+2a)
5 M
1 (b)
Find the Fourier series of f(x) = cos μx in (-π,π), where μ is not an integer.
5 M
1 (c)
Find the value of P for which the following matrix A will be of rank one, rank two, rank three where
\[\left[A=\begin{array}{ccc}3 & p & p \\p & 3 & p \\p & p & 3\end{array}\right]\]
\[\left[A=\begin{array}{ccc}3 & p & p \\p & 3 & p \\p & p & 3\end{array}\right]\]
5 M
1 (d)
Find Z-transform of {k2 - 2k + 3}k ≥ 0
5 M
2 (a)
Solve by using L.T.
\[ \dfrac {dy}{dt}+2y+\int^{t}_{0}y \ dt=\sin t \ when \ y \ (0)=1\]
\[ \dfrac {dy}{dt}+2y+\int^{t}_{0}y \ dt=\sin t \ when \ y \ (0)=1\]
8 M
2 (b)
Find Fourier series for f(x) = x + x2 in (-π , π) Hence deduce that
\[ \dfrac{1}{1^2}+ \dfrac{1}{3^2}+ \dfrac{1}{5^2} +\frac{1}{7^2} +....= \dfrac{{\pi}^2}{8} \]
\[ \dfrac{1}{1^2}+ \dfrac{1}{3^2}+ \dfrac{1}{5^2} +\frac{1}{7^2} +....= \dfrac{{\pi}^2}{8} \]
6 M
2 (c)
Show that vectors X1, X2, X3 are linearly independent and vector X4 depends upon them where
\[X_1=\ \left(1,2,4\right)\]\[X_2=\left(2,-1,3\right)\]\[X_3=\left(0,1,2\right)\ and \\ X_4=\left(-3,7,2\right)\]
\[X_1=\ \left(1,2,4\right)\]\[X_2=\left(2,-1,3\right)\]\[X_3=\left(0,1,2\right)\ and \\ X_4=\left(-3,7,2\right)\]
6 M
3 (a)
Find the Fourier integral representation of the function \[f\left(x\right)=\left\{\begin{array}{l}1-x^2,\ \ \ & \ \ when\ \vert{}x\vert{}\ & \leq{}1 \\ x,\ \ \ & \ \ when\ \vert{}x\vert{}\ & \ >{}1\end{array}\right.\]
And hence evaluate \[ \int_0^{\infty{}}\left[\frac{x\cos x-\sin x}{x^3}\right]\cos{\frac{x}{2}}\ dx\ \]
And hence evaluate \[ \int_0^{\infty{}}\left[\frac{x\cos x-\sin x}{x^3}\right]\cos{\frac{x}{2}}\ dx\ \]
8 M
3 (b)
Find matrix A if adj \[\left[A=\begin{array}{ccc}-2 & 1 & 3 \\-2 & -3 & 11 \\2 & 1 & -5\end{array}\right]\]
6 M
3 (c)
\[ Find L \ \left\{\frac{1-cost}{t^2}\right\} \]
6 M
4 (a)
\[{(i) Find L^{-1}\left\{{tan}^{-1}{\left(s+2\right)}^2\right\} } \\ {(ii) Find L\left\{t^2H\left(t-2\right)-cos h t\ \delta{}\left(t-4\right)\right\}} \]
6 M
4 (b)
Find inverse Z-transform of \[\frac{z\left(z+1\right)}{\left(z-1\right)\left(z^2+z+1\right)}\]
6 M
4 (c)
Find the nm singular matrices P and Q such that PAQ is in the normal form and hence find rank of A and rank of (PAQ) where
\[A=\left[\ \begin{array}{ccc}3 & 2 & -1 & 5 \\5 & 1 & 4 & -2 \\1 & -4 & 11 & 19\end{array}\right]\]
\[A=\left[\ \begin{array}{ccc}3 & 2 & -1 & 5 \\5 & 1 & 4 & -2 \\1 & -4 & 11 & 19\end{array}\right]\]
6 M
5 (a)
Find half range cosine series for
\[ f\left(x\right)=\left \{\begin{array}{l}x,\ \ \ & 0 < x < \frac{\pi{}}{2} \\\pi{}-x,\ \ \ & \frac{\pi{}}{2}<x<\pi{}\end{array}\right.\]
hence find the sum \[ \sum^{\infty}_{n=1} \dfrac {1}{n^4} \]
\[ f\left(x\right)=\left \{\begin{array}{l}x,\ \ \ & 0 < x < \frac{\pi{}}{2} \\\pi{}-x,\ \ \ & \frac{\pi{}}{2}<x<\pi{}\end{array}\right.\]
hence find the sum \[ \sum^{\infty}_{n=1} \dfrac {1}{n^4} \]
8 M
5 (b)
Discuss the consistancy of the following system of equation and if consistant solve them
3x+3y+2z=1
x+2y+=4
10y+3z= -2
2x-3y-z=5
3x+3y+2z=1
x+2y+=4
10y+3z= -2
2x-3y-z=5
6 M
5 (c)
Evaluate by using L.T. \[\int_0^{\infty{}}{t^3e}^{-t}\ \ sin\ t\ dt\]
6 M
6 (a) (i)
(i) Find complex form of Fourier series for f(x) = cosh3x + sinh3x in (-π,π)
4 M
6 (a) (ii)
(ii) Show that the functions \[{\left\{\sin{\left(2n-1\right)}\right\}}_{n=0}^{\infty{}}\] are orthogonal on \[\left[0,\frac{\pi{}}{2}\right]\]hence construct an orthonormal set of functions from this.
4 M
6 (b)
Apply Gauss- Seidal itterative method to solve the equations upto three itteratism
3x+20y-z=-18
2x-3y+20z=25
20x+y-2z=17
3x+20y-z=-18
2x-3y+20z=25
20x+y-2z=17
6 M
6 (c)
Find Z-transform of \[\{k^2\ a^{k-1}\ \cup{}\ \left(k-1\right)\}\]
6 M
7 (a) (i)
By using cinvolution theorem finf
\[L^{-1}\left\{\frac{1}{{\left(s-4\right)}^4(s+3)}\right\}\]
\[L^{-1}\left\{\frac{1}{{\left(s-4\right)}^4(s+3)}\right\}\]
5 M
7 (a) (ii)
Find : - L(sin2t cost cosh2t)
3 M
7 (b)
Find inverse Z-transform of \[\frac{2z^2-10z+13}{{\left(z-3\right)}^2\left(z-2\right)}\]
if R.O.C. is 2|z|<3.
if R.O.C. is 2|z|<3.
6 M
7 (c)
Find Fourier sin a integral of f(x) where
\[ f(x)=\left\{\begin{matrix} x & ; &0<x <1 \\ 2-x&; &1<x<2 \\ 0&; & x>2 \end{matrix}\right.\]
\[ f(x)=\left\{\begin{matrix} x & ; &0<x <1 \\ 2-x&; &1<x<2 \\ 0&; & x>2 \end{matrix}\right.\]
6 M
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