STUPIDSID
1 (a)
Find Laplace Transform of \[ \dfrac {\sin t}{t} \]
5 M
1 (b)
Prove that f(z)=sinhz is analytic and find its derivative.
5 M
1 (c)
Find Fourier Series for f(x)=9-x2 over (-3,3).
5 M
1 (d)
Find Z[f(k)*g(k)] if \[ f(k)= \dfrac {1}{3^k}\cdot g(k)= \dfrac {1}{5^k} \]
5 M
2 (a)
Prove that F =yexy cos zi+°xtxy cos x j ?xy sin r k ° is irrotational and Scalar Potential for F . Hence evaluate \[ \int_c \overline{F * d\overline{r}}\] along the curve C joining the points (0,0,0) and (-1,2,?).
6 M
2 (b)
Find the Fourier series for \[ f(x)= \dfrac {\pi -x}{2}, 0\le x \le 2x. \]
6 M
2 (c)
Find inverse Laplace Transform of \[ i) \dfrac {s+29}{(s+4)(s^2+9)} \\ ii) \dfrac {e^{-2s}}{s^2+8s+25} \]
8 M
3 (a)
Find the Analytic function \[ f(z)=u+iv \ if \ u+v=\dfrac {x}{x^2 + y^2} \]
6 M
3 (b)
Find Inverse Z transform of \[ \dfrac {1}{(z-1/2)(z-1/3} -1/3<|z|<1/2 \]
6 M
3 (c)
Solve the Differential equation \[ \dfrac{d^2 y}{dt^2}+ y =i, y(0)=1, y'(0)=0 \] using Laplace Transform.
8 M
4 (a)
Find the Orthogonal Trajectory of 3x2y-y3=k.
6 M
4 (b)
Using Green's theorem evaluate \[ \int_c (xy+y^2) dx+ x^2 dy\cdot C \] is closed path formed by y=x.y=x2
6 M
4 (c)
Find Fourier Integral of \[ f(x) = \left\{\begin{matrix}
\sin x &0 \le x \le \pi \\0
&x> \pi
\end{matrix}\right. \] Hence show that \[ \int^\infty_c \dfrac {\cot (l \pi /2)}{1-\lambda^2}d \lambda = \dfrac {\pi}{2} \]
8 M
5 (a)
Find Inverse Laplace Transform using Convolution theorem \[ \dfrac{3}{(s^4+8s^2+16)} \]
6 M
5 (b)
Find the Bilinear Transformation that maps the points z=1,i,-1 into w=i,0,-i.
6 M
5 (c)
Evaluate \[ \int_c \overline{F} * d \overline{r} \] where C is the boundary of the plane 2x+y+r=2 cut off by co-ordinates planes and F=(x+y)f+(y+z)j-xk.
8 M
6 (a)
Find the Directional derivative of ?=x2+y2+z2 in the direction of the line \[ \dfrac{x}{3} = \dfrac {y}{4} = \dfrac {z}{5} \] at (1,2,3,).
6 M
6 (b)
Find complex Form of Fourier Series for e2x; 0
6 M
6 (c)
Find Half Range Cosine Series for \[ f(x) = \left\{\begin{matrix} kx;= &;0\le x \le 1/2 \\k(1-x)
&; 1/2 \le x \le 1 \end{matrix}\right. \] hence that \[ \dfrac{1}{1^2}+ \dfrac {1}{3^2}+ \dfrac {1}{5^2}+ \cdots \ \cdots \]
8 M
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