STUPIDSID
1 (a)
Find the Laplace Transform of sint cos2t cosht.
5 M
1 (b)
Find the Fourier series expansion of f(x)=x2 (-?, ?)
5 M
1 (c)
Find the z-transform of \[ \left ( \dfrac {1}{3} \right )^{1K1} \]
5 M
1 (d)
Find the directional derivative of 4xz2+x2yz at (1, -2, -1) in the direction of 2i-j-2k
5 M
2 (a)
Find an analytic function f(z) whose real part is ex (xcosy-ysiny)
6 M
2 (b)
Find inverse Laplace Transform by using convolution theorem \[ \dfrac {1}{(s-3)(s+4)^2} \]
6 M
2 (c)
Prove that \[ \overline{F} = (6xy^2 - 2z^3) \overline{i} + (6x^2 y +2yz) \overline{j}+ (y^2 - 6z^2 x) \overline {k} \] is a conservative field. Find the scalar potential ? such that ??=F. Hence find the workdone by F in displacing a particle from A(1,0,2) to B(0,1,1) along AB.
8 M
3 (a)
Find the inverse z-transform of \[ f(z)= \dfrac {z^3}{(z-3)(z-2)^2} \]
i) 2<|z|<3 ii) |z|>3
i) 2<|z|<3 ii) |z|>3
6 M
3 (b)
Find the image of the real axis under the transformation \[ w= \dfrac {2}{z+i} \]
6 M
3 (c)
Obtain the Fourier series expansion of \[ \begin {align*}f(x)&=\pi x;0\le x \le 1 \\ &= \pi (2-x); 1 \le x \le 2 \end{align*} \] Here deduce That \[ \dfrac {1}{1^2} + \dfrac {1}{3^2}+ \cdots \ \cdots = \dfrac {\pi^2}{8} \]
8 M
4 (a)
Find the Laplace Transform of \[ \begin {align*}f(t) & = E; 0 \le t \le p/2 \\ & = E; p/2 \le t \le p, \end{align*} f(t+p)= f(t) \]
6 M
4 (b)
Using Green's theorem evaluate \[ \int_c \dfrac {1}{y} dx + \dfrac {1}{x} dy where c is the boundary of the region bounded by x=1, x=4, y=1, y=?x
6 M
4 (c)
Find the Fourier integral for \[ \begin {align*} f(x) & = 1-x^2. 0 \le x \le 1 \\ & = 0 \ \ \ x>1 \end{align*} \] Hence Evaluate \[ \int^\infty_0 \dfrac {\lambda \cos \lambda - \sin \lambda}{\lambda^3} \cos \left ( \dfrac {\lambda}{2} \right )d \lambda \]
8 M
5 (a)
If F =x2 i + (x-y)j+ (y+z)k moves a particular from A(1,0,1) to B(2,1,2) along line AB. Find the work done.
6 M
5 (b)
Find the complex form of Fourier series f(x)= sinhax(-l,l).
6 M
5 (c)
Solve the differential equation using Laplace Transform. (D2+2D+5)y=e-t sint y(0)=0 y'(0)=1
8 M
6 (a)
\[ If \ int^\infty_{0} e^{-2t} sn(t+\alpha)\cos (t-\alpha) dt = \dfrac {3}{8} \] find the value of ?.
6 M
6 (b)
\[ \iint_s (y^2 z^2 \overline{i} + z^2 x^2 \overline {j}+ z^2 y^2 \overline{k})\cdot \overline n ds \] where is the hemisphere x2+y2+z2=1 above xy-plane and bounded by this plane.
6 M
6 (c)
Find Half range sine series for f(x)=lx-x2 (0, l) Hence prove that \[ \dfrac {1}{1^6}+ \dfrac {1}{3^6}+ \cdots \cdots = \dfrac {\pi ^6}{960} \]
8 M
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