Applied Mathematics 3 - May 2015

Information Technology (Semester 3)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1 (a) Find Laplace Transform of $$ \dfrac {\sin t}{t} $$(5 marks) 1 (b) Prove that f(z)=sinhz is analytic and find its derivative.(5 marks) 1 (c) Find Fourier Series for f(x)=9-x² over (-3,3).(5 marks) 1 (d) Find Z[f(k)*g(k)] if $$ f(k)= \dfrac {1}{3^k}\cdot g(k)= \dfrac {1}{5^k} $$(5 marks) 2 (a) Prove that F =ye^xy cos zi+°xt^xy 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 marks) 2 (b) Find the Fourier series for $$ f(x)= \dfrac {\pi -x}{2}, 0\le x \le 2x. $$(6 marks) 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 marks) 3 (a) Find the Analytic function $$ f(z)=u+iv \ if \ u+v=\dfrac {x}{x^2 + y^2} $$(6 marks) 3 (b) Find Inverse Z transform of $$ \dfrac {1}{(z-1/2)(z-1/3} -1/3<|z|<1/2 $$(6 marks) 3 (c) Solve the Differential equation $$ \dfrac{d^2 y}{dt^2}+ y =i, y(0)=1, y'(0)=0 $$ using Laplace Transform.(8 marks) 4 (a) Find the Orthogonal Trajectory of 3x²y-y³=k.(6 marks) 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 marks) 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 marks) 5 (a) Find Inverse Laplace Transform using Convolution theorem $$ \dfrac{3}{(s^4+8s^2+16)} $$(6 marks) 5 (b) Find the Bilinear Transformation that maps the points z=1,i,-1 into w=i,0,-i.(6 marks) 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 marks) 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 marks) 6 (b) Find complex Form of Fourier Series for e^2x; 0<x<2.< a="">

</x<2.<></span>(6 marks) 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 marks)