STUPIDSID
1 (a)
Discuss the analyticity of the function \[f(z)=\dfrac {1}{z-1}\]
5 M
1 (b)
Obtain Laurent's expansion for the function \[f(z)=\dfrac {1}{z^2 \sin h \ z}\]
5 M
1 (c)
\[ Prove \ that \ L\left \{ \sqrt[2]{\dfrac {t}{P}}\right \}=\dfrac {1}{s^{2/3}}\ and \ hence \ show \ that \ L \left \{ \dfrac {1}{\sqrt{\pi t}} \right \}=\dfrac {1}{\sqrt{s}} \]
5 M
1 (d)
The matrix A is given by \[ A=\begin{bmatrix}1 &2 &-3 \\ 0&3 &2 \\ 0&0 &-2 \end{bmatrix}\] Find the eigen values of \[3A^3+5A^2-6A+2I-6SA^{-1} \]
5 M
2 (a)
Evaluate \[\int^{4+2i}_{0} \bar{z}\ dz \]along the path of the line from 0 to I and then to 4+2i.
6 M
2 (b)
Evaluate the integral \[ \int^{\infty}_{t=0}\int^{t}_{u=0}\dfrac {e^t \sin u}{u}du \ dt \]
6 M
2 (c)
Discuss for the values of k, the following system of equations possesses trivial and non-trivial solutions -
\[ 2x+3ky+(3k+4)z=0 \\
x+(k+4)y+(4k+2)z=0\\
x+2(k+1)y+(3k+4)z=0 \]
\[ 2x+3ky+(3k+4)z=0 \\
x+(k+4)y+(4k+2)z=0\\
x+2(k+1)y+(3k+4)z=0 \]
8 M
3 (a)
\[ Show \ that \ L^{-1}\left \{ \dfrac {1}{s} \cos \left ( \dfrac {1}{s} \right )\right \}=1 \dfrac {t^2}{(2!)^2}+\dfrac {t^4}{(4!)^2}-\dfrac {t^6}{(6!)^2}+.....\]
6 M
3 (b)
\[ if \ A(\alpha)= \begin{bmatrix}\cos \alpha &-\sin \alpha &0 \\ \sin \alpha&\cos \alpha & 0\\ 0&0 &1 \end{bmatrix} \ prove \ that [A(\alpha)]^{-1}= A(-\alpha)\]
6 M
3 (c)
Evaluate \[ \int_{c}\dfrac {\cos \pi z}{z^2-1}dz \] where c is
(i) a rectangle with vertices at 2 ± i and -2 ± i.
(ii) a square with vertices at ± i and 2± i.
(i) a rectangle with vertices at 2 ± i and -2 ± i.
(ii) a square with vertices at ± i and 2± i.
8 M
4 (a)
Prove that the sum of the residues of the function \[ f(z)=\dfrac {e^{z}}{z^2+a^2}\ is \ \dfrac {\sin a}{a} \]
6 M
4 (b)
Prove that is circle |z|=1 in the z-plane mapped onto the cardiode in the w-plane under the transformation w=z2+2z
6 M
4 (c)
Obtain the Laplace transformation of \[ \left \{ t\cdot erf \left ( 3\sqrt{t} \right ) \right \} \]
8 M
5 (a)
Find the orthogonal trajectories of u=constant where
\[ u=x^2-y^2 +5x+y-\dfrac {y}{x^2 +y^2} \]
\[ u=x^2-y^2 +5x+y-\dfrac {y}{x^2 +y^2} \]
6 M
5 (b)
Examine the linear depedence of the vector [1 0 2 1], [3 1 2 1], [4 6 2 -4] and [-6 0 -3 -4] and find the relation between them if possible.
6 M
5 (c)
\[ Evaluate \ \int^{2 \pi}_{0}\dfrac {d\theta}{3-2 \cos \theta + \sin \theta }\]
8 M
6 (a)
Find the bilinear transformation which maps the points z=2, 1, 0 onto w=1, 0, i.
6 M
6 (b)
\[ If \ f(t)=\left\{\begin{matrix}3t &&0<t<2 \\ 6 &&2<t<4 \end{matrix}\right.\] where f(t) has period 4.
(i) Draw graph of f(t)
(ii) Find L { f(t) }
(i) Draw graph of f(t)
(ii) Find L { f(t) }
6 M
6 (c)
Show that \[ u=\left ( \gamma + \dfrac {a^2}{\gamma} \right )\cos \theta \] is harmonic, find v(γ, θ) so that u+iv is analytic.
8 M
7 (a)
Obtain -
\[ (i) \ L^{-1}\left \{ \dfrac {3s-8}{s^{2}+4}- \dfrac {4s-24}{s^{2}-16}\right \}\\(ii) \ L^{-1} \left \{ \dfrac {3s-2}{s^{5/2}}- \dfrac {7}{3s+2} \right \} \]
\[ (i) \ L^{-1}\left \{ \dfrac {3s-8}{s^{2}+4}- \dfrac {4s-24}{s^{2}-16}\right \}\\(ii) \ L^{-1} \left \{ \dfrac {3s-2}{s^{5/2}}- \dfrac {7}{3s+2} \right \} \]
6 M
7 (b)
If f(z)=u+iv is an analytic function of z=x+iy and \[ u-v=\dfrac {e^y -\cos x + \sin x}{\cos hy-\cos x}, \] find f(z) subject to the condition \[ f\left (\dfrac {\pi}{2} \right )=\dfrac {3-i}{2} \]
6 M
7 (c)
Show that the matrix \[ A=\begin{bmatrix}9 &-1 &9 \\ 3&-1 &3 \\ -7&1 &-7 \end{bmatrix} \] is diagonalizable. Find the transforming matrix and the diagonal form.
8 M
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