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MU Computer Engineering (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)=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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