STUPIDSID
1 (a)
Find Laplace transform of sin √t
5 M
1 (b)
If \[ A=\begin{bmatrix}123 &231 &312 \\ 231&312 &123 \\ 312&123 &231 \end{bmatrix} \] P.T. (i) one of the characteristics roots of is 666
(ii) given A is non-singular, one of the characteristics roots A is negative.
(ii) given A is non-singular, one of the characteristics roots A is negative.
5 M
1 (c)
Evaluate \[ \int ^{1+2i}_{0}z^{2}dz \] along the curve 2x2=y
5 M
1 (d)
Find the image of the circle with centre at (0,3) and radius 3 in the z-plane into the w-plane inder the transformation \[ w=\dfrac {1}{2} \]
5 M
2 (a)
Evaluate \[ \int^{\infty}_{0}t\left (\dfrac {\sin t}{e^{t}} \right )^{2}dt \]
7 M
2 (b)
Find non-singular matrices P and Q such that PAQ is in normal form. Also find of A and A-1 where
\begin{bmatrix}1 &2 &-2 \\ -1&3 &0 \\ 0&-2 &1 \end{bmatrix}
\begin{bmatrix}1 &2 &-2 \\ -1&3 &0 \\ 0&-2 &1 \end{bmatrix}
7 M
2 (c)
Find the imaginary part of the analytic function whose real part is e2x(x cos 2y-y sin 2y)
6 M
3 (a)
Find inverse Laplace transform of \[ \dfrac {5s^{2}-15s-11}{(s+1)(s-2)^{2}} \]
7 M
3 (b)
Find the characteristics equation of the matrix A and hence find the matrix represented by
\[ A^{8}-5A^{7}+7A^{6}-3A^{5}+A^{4}-5A^{3}+8A^{2}-2A+1 \\where \ A=\begin{bmatrix}2 &1 &1 \\ 0&1 &0 \\ 1&1 &2 \end{bmatrix}\]
\[ A^{8}-5A^{7}+7A^{6}-3A^{5}+A^{4}-5A^{3}+8A^{2}-2A+1 \\where \ A=\begin{bmatrix}2 &1 &1 \\ 0&1 &0 \\ 1&1 &2 \end{bmatrix}\]
7 M
3 (c)
Find the bilinear transformation which maps the points 0, 1 ∞ onto the points w=-5, -1, 3.
6 M
4 (a)
Evaluate \[ \int^{\pi}_{0}\dfrac {d\theta}{3+2\cos \theta} \]
7 M
4 (b)
Find inverse Laplace transform by convolution thm, of \[ \dfrac {1}{(s+3)(s^{2}+2s+2)}. \]
7 M
4 (c)
Investigation for what values of λ and μ, the equation 2x+3y+5z=9, 7x+3y-2z=8, 2x+3y+λz=μ have (i) no solution, (ii) unique solution (iii) infinite no. of solutions.
6 M
5 (a)
Find the inverse of the matrix
\[ S= \begin{bmatrix}0 &1 &1 \\ 1&0 &1 \\ 1&1 &0 \end{bmatrix} \ and \ if \ A=\dfrac {1}{2}\begin{bmatrix}4 &-1 &1 \\ -2&3 &-1 \\ 2&1 &5 \end{bmatrix} \] S.T. SAS-1 is a diagonal matrix.
\[ S= \begin{bmatrix}0 &1 &1 \\ 1&0 &1 \\ 1&1 &0 \end{bmatrix} \ and \ if \ A=\dfrac {1}{2}\begin{bmatrix}4 &-1 &1 \\ -2&3 &-1 \\ 2&1 &5 \end{bmatrix} \] S.T. SAS-1 is a diagonal matrix.
7 M
5 (b)
Evaluate \[ \int_{c}\dfrac {z^{2}+4}{(z-2)(z+3i)}dz \] where C is
(i) |z+1|=2
(ii) |z-2|=2
(i) |z+1|=2
(ii) |z-2|=2
7 M
5 (c)
Obtain Taylor's and Laurent's expansion of \[ f(z)= \dfrac {z-1}{z^{2}-2z-3} \] indicating regions of convergece.
6 M
6 (a)
Find Laplace transform of \[ \dfrac {e^{-at}-\cos at}{t} \ hence \ evaluate \ \int^{\infty}_{0}\dfrac {e^{-1}-\cos t}{te^{4t}}dt \]
7 M
6 (b)
Find Eigen value Eigen vectors of \[ A^3 +1 \ where \ A=\begin{bmatrix}2 &2 & 1\\ 1&3 &1 \\ 1&2 &2 \end{bmatrix} \]
7 M
6 (c)
Evaluate using residue theorem \[ \int_c \dfrac {(z+4)^{2}}{z^4+5z^3+6z^2}dz \] where C is is |z|=1
6 M
7 (a)
Solve the foll: equation using Laplace transform,
\[ \dfrac {dy}{dt}+2y+\int^{t}_{0}ydt=\sin t \ given \ y(0)=1 \]
\[ \dfrac {dy}{dt}+2y+\int^{t}_{0}ydt=\sin t \ given \ y(0)=1 \]
7 M
7 (b)
Express the foll: matrix A as P+iQ where P and Q are both hermitian given: \[ A= \begin{bmatrix}2 &3-i &2+i \\ i&0 &1-i \\ 1+2i&1 &3i \end{bmatrix} \]
7 M
7 (c)
Find sum of residue at singular points of
\[ f(z)=\dfrac {z}{az^2+bz+c} \]
\[ f(z)=\dfrac {z}{az^2+bz+c} \]
6 M
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