STUPIDSID
1 (a)
Show that \[ L\left [ \dfrac {\cos \sqrt{t}}{\sqrt{t}} \right ]= \sqrt {\dfrac {\pi}{s}} \ e^{-\frac {1}{4s}} \]
5 M
1 (b)
Use adjoint method to find inverse of -
\[ A= \begin{bmatrix}7 &-3 &-3 \\ -1&1 &0 \\ -1&0 &1 \end{bmatrix} \]
\[ A= \begin{bmatrix}7 &-3 &-3 \\ -1&1 &0 \\ -1&0 &1 \end{bmatrix} \]
5 M
1 (c)
Evaluate \[ \int^{1+i}_{0} (x^{2}-iy) dz \] along the path
(i) y=x
(ii) y=x2
(i) y=x
(ii) y=x2
5 M
1 (d)
IF f(z)=u+iv is analytic and u-v=ex (cos y -sin y) then find f(z) in terms of z.
5 M
2 (a)
Evaluate \[ \int^{\infty}_{0} \dfrac {t^2 \sin 3t}{e^{2t}} dt \]
6 M
2 (b)
Find non-singular matrices P and Q such that PAQ is in normal form. Also find rank of A where
\[ A \begin{bmatrix}1 &2 &3 &2 \\ 2&3 &5 &1 \\ 1&3 &4 &5 \end{bmatrix} \]
\[ A \begin{bmatrix}1 &2 &3 &2 \\ 2&3 &5 &1 \\ 1&3 &4 &5 \end{bmatrix} \]
6 M
2 (c)
Find the bilinear transformation which maps the points 2, I, -2 onto points 1, I, -1. Also the fixed points of this bilinear transformation.
8 M
3 (a)
\[ \begin {align*} Find \ &(i) \ L^{-1} \left [ \dfrac {s+2}{s^{2}+4s+7} \right ] \\ \\ &(ii) \ L^{-1}\left [log \dfrac {s^{2}+a^{2}}{\sqrt{s+b}} \right ]\\ \end{align*} \]
6 M
3 (b)
\[ Compute \ A^{7}-4A^6-20A^5-34A^4-4A^3-20A^2-33A+I \\ where \ A =\begin{bmatrix}1 &3 &7 \\ 4&2 &3 \\ 1&2 &1 \end{bmatrix} \]
6 M
3 (c)
Prove that the circle |z-3|=5 is mapped onto the circle \[ \left |w+\dfrac {3}{16} \right | =\dfrac {5}{16} \] under the transformation \[ w=\dfrac {1}{z} \]
8 M
4 (a)
Evaluate \[ \int^{2\pi}_{0}\dfrac {\cos 3 \theta}{5+4 \cos \theta} d \theta \]
6 M
4 (b)
Test for consistency and solve -
\[ 2x_{1}-3x_{2}+5x_{3}=1 \\3x_{1}+x_{2}-x_{3}=2\\x_{1}+4x_{2}-6x_{3}=1\]
\[ 2x_{1}-3x_{2}+5x_{3}=1 \\3x_{1}+x_{2}-x_{3}=2\\x_{1}+4x_{2}-6x_{3}=1\]
6 M
4 (c)
Find \[ L^{-1}\left [ \dfrac {s}{(s^{2}+a^{2})(s^{2}+b^{2})} \right ] \] by using convolution theorem.
8 M
5 (a)
Express the Hermitian matrix \[ A= \begin{bmatrix}3 &2-i &1+2i \\ 2+i&2 &3-2i \\ 1-2i&3+2i &0 \end{bmatrix}\] as P+iQ where P is real symmetric and Q is real skew-symmetric.
6 M
5 (b)
Evaluate \[ \int_{C} \dfrac {\sin \pi z^2+\cos \pi z^{2}}{z^{2}+3z+2}dz \] where C is (i) |z|=1 (ii) |z|=2
6 M
5 (c)
Find all possible Laurent's expansion of the function \[ f(z)=\dfrac {7z-2}{z(z-2)(z+1)} \] about z=-1
8 M
6 (a)
Find \[ L\left [ \dfrac {d}{dt}\left ( \dfrac {1-\cos 2t}{t} \right ) \right ] \]
6 M
6 (b)
Find Eigen value and Eigen vector for -
\[ A= \begin{bmatrix}4 &6 &6 \\ 1&3 &2 \\ -1&-5 &-2 \end{bmatrix}\]
\[ A= \begin{bmatrix}4 &6 &6 \\ 1&3 &2 \\ -1&-5 &-2 \end{bmatrix}\]
6 M
6 (c)
Evaluate using residue theorem \[ \int_c \dfrac {4z^2 +1}{(2z-3)(z+2)^2}dz \] where C is |z|=4
8 M
7 (a)
State and prove Cauchy's integral theorem.
6 M
7 (b)
\[ if \ A=\begin{bmatrix}1/3 &2/3 &a \\ 2/3&1/3 &b \\ 2/3&-2/3 &c \end{bmatrix} \] is orthogonal matrix then find a,b,c. Also find A-1.
6 M
7 (c)
Solve (D2-D-2) y=20 sin 2 t with y(0)=1 and y'(0)=2 by using Laplace transform.
8 M
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