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MU Information Technology (Semester 3)
Applied Mathematics 3
December 2015
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary


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.
5 M
1 (d) Prove that Δ=(1r)=rr3
5 M

2 (a) If f(z)= u + iv is analytic and u+v=2sin2xe2y+e2y2cos2x Find f(z).
6 M
2 (b) Find inverse Z-transform of f(z)=z+2z22z+1 for |z|>1.
6 M
2 (c) Find Fourier series for f(x)=1cosx in (0,2π) Hence, deduce that 12=114n21
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 f(x)=x+π2 where π<x<0=π2x where 0<x<π
Hence, deduce that i) π28=112+132+152+ ii) π496=114+134+154+ 
8 M

5 (a) Using Gauss Divergence theorem to evaluate 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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