MORE IN Applied Mathematics 3
MU Information Technology (Semester 3)
Applied Mathematics 3
December 2014
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 the Laplace Transform of sint cos2t cosht.
5 M
1 (b) Find the Fourier series expansion of f(x)=x2 (-?, ?)
5 M
1 (c) Find the z-transform of $\left ( \dfrac {1}{3} \right )^{1K1}$
5 M
1 (d) Find the directional derivative of 4xz2+x2yz at (1, -2, -1) in the direction of 2i-j-2k
5 M

2 (a) Find an analytic function f(z) whose real part is ex (xcosy-ysiny)
6 M
2 (b) Find inverse Laplace Transform by using convolution theorem $\dfrac {1}{(s-3)(s+4)^2}$
6 M
2 (c) Prove that $\overline{F} = (6xy^2 - 2z^3) \overline{i} + (6x^2 y +2yz) \overline{j}+ (y^2 - 6z^2 x) \overline {k}$ is a conservative field. Find the scalar potential ? such that ??=F. Hence find the workdone by F in displacing a particle from A(1,0,2) to B(0,1,1) along AB.
8 M

3 (a) Find the inverse z-transform of $f(z)= \dfrac {z^3}{(z-3)(z-2)^2}$
i) 2<|z|<3 ii) |z|>3
6 M
3 (b) Find the image of the real axis under the transformation $w= \dfrac {2}{z+i}$
6 M
3 (c) Obtain the Fourier series expansion of \begin {align*}f(x)&=\pi x;0\le x \le 1 \\ &= \pi (2-x); 1 \le x \le 2 \end{align*} Here deduce That $\dfrac {1}{1^2} + \dfrac {1}{3^2}+ \cdots \ \cdots = \dfrac {\pi^2}{8}$
8 M

4 (a) Find the Laplace Transform of \begin {align*}f(t) & = E; 0 \le t \le p/2 \\ & = E; p/2 \le t \le p, \end{align*} f(t+p)= f(t)
6 M
4 (b) Using Green's theorem evaluate \int_c \dfrac {1}{y} dx + \dfrac {1}{x} dy where c is the boundary of the region bounded by x=1, x=4, y=1, y=?x 6 M 4 (c) Find the Fourier integral for \[ \begin {align*} f(x) & = 1-x^2. 0 \le x \le 1 \\ & = 0 \ \ \ x>1 \end{align*} Hence Evaluate $\int^\infty_0 \dfrac {\lambda \cos \lambda - \sin \lambda}{\lambda^3} \cos \left ( \dfrac {\lambda}{2} \right )d \lambda$
8 M

5 (a) If F =x2 i + (x-y)j+ (y+z)k moves a particular from A(1,0,1) to B(2,1,2) along line AB. Find the work done.
6 M
5 (b) Find the complex form of Fourier series f(x)= sinhax(-l,l).
6 M
5 (c) Solve the differential equation using Laplace Transform. (D2+2D+5)y=e-t sint y(0)=0 y'(0)=1
8 M

6 (a) $If \ int^\infty_{0} e^{-2t} sn(t+\alpha)\cos (t-\alpha) dt = \dfrac {3}{8}$ find the value of ?.
6 M
6 (b) $\iint_s (y^2 z^2 \overline{i} + z^2 x^2 \overline {j}+ z^2 y^2 \overline{k})\cdot \overline n ds$ where is the hemisphere x2+y2+z2=1 above xy-plane and bounded by this plane.
6 M
6 (c) Find Half range sine series for f(x)=lx-x2 (0, l) Hence prove that $\dfrac {1}{1^6}+ \dfrac {1}{3^6}+ \cdots \cdots = \dfrac {\pi ^6}{960}$
8 M

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