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)= 4^kU(K), g(k)= 5^kU(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