STUPIDSID
1 (a)
Determine whether the function f(z)=cosh z is analytic or not. If so, find the derivative.
5 M
1 (b)
Obtain the Laurent's expansion for the function \[ f(z)=\dfrac {e^{2z}}{(z-1)^3} about z=1 \]
5 M
1 (c)
Find the inverse Laplace transform of -
\[ \dfrac {S \ e^{-2S}}{S^2 -6S+25} \]
\[ \dfrac {S \ e^{-2S}}{S^2 -6S+25} \]
5 M
1 (d)
\[ If \ A=\begin{bmatrix}0 &1/\sqrt{2} &1/\sqrt{2} \\ \\ \sqrt{2}/\sqrt{3}&1/\sqrt{6} &1/\sqrt{6} \\ \\ 1/\sqrt{3}&1/\sqrt{3} &1/\sqrt{3} \end{bmatrix} \ find \ A^{-1} \]
5 M
2 (a)
\[ Evaluate \ \int_{c} (z^2 +3z) dz \] along the circle |z|=2 from (2,0) to (0,2).
6 M
2 (b)
\[ Evaluate \ \int_{0}^{\infty}\dfrac {t^2 \sin 3t}{e^{2t}}dt \]
6 M
2 (c)
Determine the value of λ for which the following system of equations possesses a non-trivial solution and obtain these solutions for each value of λ.
\[ 3x_{1}+x_{2}-\lambda x_{3}=0\\4x_{1}-2x_{2}-3x_{3}=0\\2\lambda x_{1}+4x_{2}+\lambda x_{3}=0 \]
\[ 3x_{1}+x_{2}-\lambda x_{3}=0\\4x_{1}-2x_{2}-3x_{3}=0\\2\lambda x_{1}+4x_{2}+\lambda x_{3}=0 \]
8 M
3 (a)
\[ Show \ that \ L\left \{erf \ \sqrt{t}\right \}=\dfrac {1}{S\sqrt{S+1}} \ hence \ deduce \ L\left \{t\cdot erf (2\sqrt{t}) \right \} \]
6 M
3 (b)
Reduce to normal form and find the rank of :
\begin{bmatrix} 1 & 3& 5& 7\\ 4& 6& 8& 10\\ 15& 27& 39& 51\\ 6& 12& 18& 24 \end{bmatrix}
\begin{bmatrix} 1 & 3& 5& 7\\ 4& 6& 8& 10\\ 15& 27& 39& 51\\ 6& 12& 18& 24 \end{bmatrix}
6 M
3 (c)
\[ Evaluate \ \int_c \dfrac {z^2}{z^4 -1}dz \ and \ \int_{c} \dfrac {dz}{z^{3}(z+4)} \ \] where C is the circle |z|=2
8 M
4 (a)
Find the residue of the function\[ f(z)= \dfrac {\sin \pi z^2 + \cos \pi z^2}{(z-1)(z-2)^{2}} \] at their poles.
6 M
4 (b)
Show that under the transformer\[ W=\dfrac {3-z}{z-2} \] transforms the circle with centre \[ \left ( \dfrac {5}{2},0 \right ) and \ radius \dfrac {1}{2} \] in the z-plane into imaginary axis in the W-plane.
6 M
4 (c)
\[ Solve \ y^n(t)+9y(t)=18t \ if \ y(0)=1, y\left (\dfrac {\pi}{2} \right )=0 \]
8 M
5 (a)
Find the orthogonal trajectory of the family of curves given by -
ex cos y-xy=c
ex cos y-xy=c
6 M
5 (b)
Is the system of vectors \[ X_{1}= [2 2 1]^T, X_{2}=[1 3 1]^T, X_{3}=[1 2 2]^T \ linearly \ dependent? \]
6 M
5 (c)
\[ Evaluate \ \int^{2\pi}_{0}\dfrac {\sin^{2}\theta}{5-4 \cos \theta}d\theta\ \]
8 M
6 (a)
Obtain the bilinear transformation that maps the points z=0, -i, I onto w=i, 1, 0
6 M
6 (b)
Find the Laplace Transformation of the periodic function
\[ f(t)=\left\{\begin{matrix} t & 0<t<\pi\\ \pi -t& \pi<t<2\pi \end{matrix}\right. \]
\[ f(t)=\left\{\begin{matrix} t & 0<t<\pi\\ \pi -t& \pi<t<2\pi \end{matrix}\right. \]
6 M
6 (c)
Prove that \[ u(x,y)=x^2 - y^2 \ and \ v(x,y)=\dfrac {-y}{x^{2}+y^{2}} \] are both harmonic functions, but u+iv is not analytic
8 M
7 (a)
Find the inverse Laplace Transform of\[ \dfrac {S^{2}+S}{(S^{2}+1)(S^{2}+2S+2)} \]using convolution theroem.
6 M
7 (b)
Determine the analytic function f(z)=u+iv in terms of z, when it is given that 3u+2v=y2-x2+16xy
6 M
7 (c)
Find the characteristics equation of the symmetric matrix -
\[ A=\begin{bmatrix}2 &-1 &1 \\ -1&2 &-1 \\ 1&-1 &2 \end{bmatrix} \]
Verify Cayley Hamilton theorem for A and A-1
\[ A=\begin{bmatrix}2 &-1 &1 \\ -1&2 &-1 \\ 1&-1 &2 \end{bmatrix} \]
Verify Cayley Hamilton theorem for A and A-1
8 M
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