 MORE IN Applied Mathematics 3
MU Computer Engineering (Semester 3)
Applied Mathematics 3
May 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 Transform of $\dfrac {\sin t} {t}$
5 M
1 (b) Prove that f(z)=sinh z is analytic and find its derivative.
5 M
1 (c) Find Fourier Series for f(x)=9-x2 over (-3,3).
5 M
1 (d) Find Z{f(k)*g(k)} if $f(k) = \dfrac {1}{3^k}, \ g(k)=\dfrac {1}{5^k}$
5 M

2 (a) Prove that F =yexy cos z i + xexy cos z j-exy sin z k is irrotational. Find Scalar potential for F Hence evaluate $\int_c \overline {F}\cdot d\overline {r}$ along the curve C joining the points (0, 0, 0) and (-1, 2, π).
6 M
2 (b) Find the Fourier series for $f(x) = \dfrac {\pi - x}{2}; 0\le x \le 2\pi$
6 M
2 (c) Find Inverse Laplace Transform $i) \ \dfrac {s+29}{(s+4)(s^2+9)} \\ ii) \ \dfrac {e^{-2x}}{s^2+8s+25}$
8 M

3 (a) Find the Analytic function f(z)=u+iv if $u+v= \dfrac {x}{x^2 + y^2} .$
6 M
3 (b) Find inverse Z transform of $\dfrac {1}{(z-1/2)(z-1/3), 1/3<|z|<1/2$
6 M
3 (c) Solve the Differential Equation $\dfrac {d^2y}{dt^2} + y=t, \ y(0)=1, y'(0)=0,$ using Laplace Transform.
8 M

4 (a) Find the Orthogonal Trajectory of 3x2y-y3=k.
6 M
4 (b) Using Greens theorem evaluate $\int_c (xy+y^2) dx+x2 dy,$ C is closed path formed by y=x, y=x2.
6 M
4 (c) "Find Fourier Integral$f(x)= \begin{Bmatrix} \sin x &0 \le x \le \pi \\0 & x> \pi \end{matrix}$ Hence show that $\int^\infty_0 \dfrac {\cos (\lambda \pi /2)}{1-\lambda^2} d\lambda = \dfrac {\pi}{2}$"
8 M

5 (a) Find Inverse Laplace Transform using Convolution theorem $\dfrac {s} {(s^4 + 8s^2 + 16) }$
6 M
5 (b) Find the Bilinear Transformation that maps the point z=1, i, -1 into w=i, 0, -i.
6 M
5 (c) Evaluate $\int_C \overline {F} \cdot d\overline{r}$ where C is the boundary of the plane 2x+y+z=2 cut off by co-ordinate planes and F=(x+y)i + (y+z)j-xk.
8 M

6 (a) Find the directional derivative of ϕ=x2+y2+z2 in the direction of the line $\dfrac {x}{3} = \dfrac {y}{4} = \dfrac {z} {5} \ at \ (1, 2, 3). 6 M 6 (b) Find Complex Form of Fourier Series for e2x; 0 6 M 6 (c) Find Half Range Cosine Series for \[ f(x) = \left\{\begin{matrix} kx &; 0\le x \le l/2 \\k(l-x) & ; l/2 \le x \le l \end{matrix}\right.$ hence find $\dfrac {1}{l^2} + \dfrac {1}{3^2} + \dfrac {1}{5^2} + \cdots \ \cdots$
8 M

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