Question Paper: Applied Mathematics 3 : Question Paper May 2015 - Information Technology (Semester 3) | Mumbai University (MU)
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## 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 =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 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 e2x; 0<x&lt;2.&lt; a="">

</x&lt;2.&lt;&gt;<>(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)