STUPIDSID
1 (a)
Find Laplace of {t5 cosht}
5 M
1 (b)
Find Fourier series for f(x)=1-x2 in (-1, 1)
5 M
1 (c)
Find a, b, c, d, e if, \[ f(z)= (ax^4 + bx^2y^2+ cy^4+dx^2 - 2y^2)+ (4x^3y - exy^3 + 4xy) \] is analytic.
5 M
1 (d)
Prove that \[ \nabla \left ( \dfrac {1}{r} \right ) = \dfrac {f}{r^3} \]
5 M
2 (a)
If f(z)=u+iv is analytic and \[ u+v = \dfrac {2 \sin 2x}{e^{2y}+e^{-2y}-2\cos 2x}, \text {find f(z)} \]
6 M
2 (b)
Find inverse Z-transform of \[ f(z)= \dfrac {z+2}{z^2 -27 +1 } \ \text {for} \ |z|>1 \]
6 M
2 (c)
Find Fourier series for \[ f(x)= \sqrt{1 \ominus \cos x} \ \text {in} \ (0,2\pi) \] Hence, deduce that \[ \displaystyle \dfrac {1}{2} = \sum^{\infty}_{1} \dfrac {1}{-4n^2 - 1} \]
8 M
3 (a)
Find \[ L^{-1} \left \{ \dfrac {1}{(s-2)(s+3)} \right \} \] using Convolution theorem.
6 M
3 (b)
prove that f1(x)=1, f2(x)=x, f3(x)=(3x2-1)/2 are orthogonal over (-1,1)
6 M
3 (c)
Verify Green's theorem for \[ \displaystyle \int_c \overline {F}\cdot \overline {dr}\ \text{where} \ \overline {F}= (x^2 - y^2)i + (x+y)j \] and c is the triangle with vertices (0,0), (1,1), (2,1).
8 M
4 (a)
Find Laplace Transform of f(t)=|sinpt|, t≥0.
6 M
4 (b)
Show that F= (ysinz-sinx)i + (xsinz+2yz)j+(xycosz+y2) k is irrotational. Hence, find its scalar potential.
6 M
4 (c)
Obtain Fourier expansion of \[ \begin {align*} f(x) &=x+\dfrac {\pi}{2} \ \text {where} \ -\pi
8 M
5 (a)
Using Gauss Divergence theorem to evaluate \[ \iint_s \overline{N}\cdot \overline{F}ds \text{where} \overline {F}=4xi-2y^2j+z^2k\] and S is the region bounded by x2+y2=4, z=0, z=3
6 M
5 (b)
Find Z{2k cos (3k+2)}, k≥0
6 M
5 (c)
Solve (D2+2D+5)y=e-t sint, with y(0) and y'(0)=1.
8 M
6 (a)
Find \[ L^{-1}\left \{ \tan ^{-1} \left ( \dfrac {2}{s^2} \right ) \right \} \]
6 M
6 (b)
Find the bilinear transformation which maps the points 2, i, -2 onto point 1, j, -1 by using cross-ratio property.
6 M
6 (c)
Find Fourier Sine integral representation for \[f(x) = \dfrac {e^{-ax}}{x} \]
8 M
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