MORE IN Applied Mathematics 3
MU Information Technology (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)=sinhz 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 $f(k)= \dfrac {1}{3^k}\cdot g(k)= \dfrac {1}{5^k}$
5 M

2 (a) Prove that F =yexy cos zi+°xtxy cos x j ?xy sin r k ° is irrotational and Scalar Potential for F . Hence evaluate $\int_c \overline{F * d\overline{r}}$ along the curve C joining the points (0,0,0) and (-1,2,?).
6 M
2 (b) Find the Fourier series for $f(x)= \dfrac {\pi -x}{2}, 0\le x \le 2x.$
6 M
2 (c) Find inverse Laplace Transform of $i) \dfrac {s+29}{(s+4)(s^2+9)} \\ ii) \dfrac {e^{-2s}}{s^2+8s+25}$
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 $\dfrac{d^2 y}{dt^2}+ y =i, y(0)=1, y'(0)=0$ using Laplace Transform.
8 M

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

5 (a) Find Inverse Laplace Transform using Convolution theorem $\dfrac{3}{(s^4+8s^2+16)}$
6 M
5 (b) Find the Bilinear Transformation that maps the points z=1,i,-1 into w=i,0,-i.
6 M
5 (c) Evaluate $\int_c \overline{F} * d \overline{r}$ where C is the boundary of the plane 2x+y+r=2 cut off by co-ordinates planes and F=(x+y)f+(y+z)j-xk.
8 M

6 (a) Find the Directional derivative of ?=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 $f(x) = \left\{\begin{matrix} kx;= &;0\le x \le 1/2 \\k(1-x) &; 1/2 \le x \le 1 \end{matrix}\right.$ hence that $\dfrac{1}{1^2}+ \dfrac {1}{3^2}+ \dfrac {1}{5^2}+ \cdots \ \cdots$
8 M

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