STUPIDSID
1 (a)
Find Laplace of {t5 cosht}.
5 M
1 (b)
Find Fourier series for f(x)=t-x2 in (-1, 1).
5 M
1 (c)
Find a, b, c, d, e, if,
f(z)=(ax4+bx2y2+cy4+dx2-2y2)+i (4x3-exy3+4xy) is analytic.
f(z)=(ax4+bx2y2+cy4+dx2-2y2)+i (4x3-exy3+4xy) is analytic.
5 M
1 (d)
Prove that \( \Delta = \left ( \dfrac {1}{r} \right ) = \dfrac {r}{r^3} \)
5 M
2 (a)
If f(z)= u + iv is analytic and \( u+v= \dfrac{2\sin 2x}{e^{2y}+ e^{-2y}-2 \cos 2x} \) Find f(z).
6 M
2 (b)
Find inverse Z-transform of \( f(z)= \dfrac {z+2}{z^2 - 2z+1} \) for |z|>1.
6 M
2 (c)
Find Fourier series for \( f(x) = \sqrt{1-\cos x }\text{ in } (0, 2\pi) \) Hence, deduce that \( \dfrac {1}{2} = \sum^\infty_1 \dfrac {1}{4n^2 - 1} \)
8 M
3 (a)
Find \( L^{-1} \left { \dfrac {1}{(s-3)+(s+3)} \right } \) 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 \( \int_c \overline {F} \cdot \overline{dr} \text { where } \overline {F} = (x2-y2)i+(x+y)j \) 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 \( \begin {align*}
f(x)&= x+ \dfrac {\pi}{2} \text { where } -\pi < x < 0 \\
&= \dfrac{\pi}{2} - x \text { where }0< x<\pi
\end{align*} \)
Hence, deduce that \( i) \ \dfrac {\pi^2} {8} = \dfrac {1}{1^2} + \dfrac {1}{3^2}+ \dfrac {1}{5^2}+ \cdots \ \cdots \\ ii) \ \dfrac {\pi^4}{96} = \dfrac {1}{1^4} + \dfrac {1}{3^4} + \dfrac {1}{5^4} + \cdots \ \cdots \)
Hence, deduce that \( i) \ \dfrac {\pi^2} {8} = \dfrac {1}{1^2} + \dfrac {1}{3^2}+ \dfrac {1}{5^2}+ \cdots \ \cdots \\ ii) \ \dfrac {\pi^4}{96} = \dfrac {1}{1^4} + \dfrac {1}{3^4} + \dfrac {1}{5^4} + \cdots \ \cdots \)
8 M
5 (a)
Using Gauss Divergence theorem to evaluate \( \iint_s \ \overline{N} \cdot \overline {F}ds \text{ where } \overline {F} = 4xi - 2y^2 j+ z^k \) 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)=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 points 1, i, -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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