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MU Information Technology (Semester 3)
Applied Mathematics 3
December 2013
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) $Find \ L^{-1}\left\{\frac{e^{\frac{4-3}{s}}}{{\left(s+4\right)}^{\frac{5}{2}}}\right\}$
5 M
1 (b) Find the constant a,b,c,d and e If
$f\left(z\right)=\left(ax^4+bx^2y^2+cy^4+dx^2-2y^2\right)+i\ \left(4x^3y-exy^3+4xy\right)\$ is analytic.
5 M
1 (c) Obtain half range Fourier cosine series for f(x)=sin x, x ∈ (0, π).
5 M
1 (d) If r and r have their usual meaning and a is constant vector, prove that
$abla{}x\left[\frac{a\ x\\bar{r}}{r^n}\right]=\frac{\left(2-n\right)}{r^n}a+\frac{n\left(a\cdot{}\bar{r}\right)\bar{r}}{r^{n+2}}$
5 M

2 (a) Find the analytic function f(c) =u+iv if 3u+2v=y2- x2 + 16 xy.
6 M
2 (b) Find the z-transform of $\left\{a^{\left\vert{}k\right\vert{}}\right\}$ and hence find the z-transform of $\left\{{\left(\frac{1}{2}\right)}^{\left\vert{}k\right\vert{}}\right\}$
6 M
2 (c) Obtain Fourier series expansion for $f\left(x\right)=\sqrt{1-\cos{x,\ }}\ x\in{}\left(0,\ 2\pi{}\right)$ and hence deduce that $\sum_{n=1}^{\infty{}}\frac{1}{4n^2-1}=\frac{1}{2}.$
8 M

3 (a) $\left(i\right)\ \ L^{-1}\left\{\frac{s}{{\left(2\ s+1\right)}^2}\right\}$
$\left(ii\right)\ \ L^{-1}\left\{\log{\begin{array}{l}\frac{s^2+a^2}{\sqrt{s+b}}\\\ \end{array}}\right\}$
6 M
3 (b) Find the orthogonal trajectories of the family of curves e-x cos y+xy=? where ? is the real constant in xy - plane.
6 M
3 (c) Show that $\bar{F}=(y\ e^{xy}\cos{z)i+(x\ e^{xy}\cos{z)\ j-(e^{xy}\sin{z)\ k\ }}}$ is irrotational and find the scalar potential for $\ \bar{F}\ and\ evaluate\ \int_c^{\ }\bar{F}\ \$ dr along the curve joining the points (0,0,0) and (-1,2,π).
8 M

4 (a) Evaluate by Green's theorem ∫ e-x sin y dx+e-x cos y dy where c is the rectangle whose vertices are (0,0) (π,0) (π, π/2) and (0, π/2)
6 M
4 (b) Find the half range sine series for the function.
$f\left(x\right)=\frac{2\ k\ x}{l},\ \ 0\leq{}x\leq{}\frac{l}{2}$
$f\left(x\right)=\frac{2k}{l}\left(l-x\right),\ \frac{l}{2}\leq{}z\leq{}l$
6 M
4 (c) Find the inverse z-transform of $\frac{1}{\left(z-3\right)\left(z-2\right)}$
(i) |z|<2
(ii) 2<|z|<3
(iii) |z|>3.
8 M

5 (a) Solve using Laplace transform.
$\frac{d^2y}{dx^2}+4\frac{dy}{dx}+3y=e^{-x},\ y\left(0\right)=1,\ y^{'}\left(0\right)=1.$
6 M
5 (b) Express f(x)= π/2 e-x cos x for x>0 as Fourier sine integral and show that
$\int_0^{\infty{}}\frac{w^3\sin{wx}}{w^4+4}\ dw=\frac{\pi{}}{2}\ e^{-x}\cos{x.}$
6 M
5 (c) $Evaluate\ \iint_s^{\ }F\cdot{}nds,\ where\ \bar{F}=xi-yi+\left(z^2-1\right)k\ \$ and s s=is the cylinder formed by the surface z=0, z=1, x2+y2=4, using the Gauss-Divergence theorem.
8 M

6 (a) Find the inverse Laplace transform by using convolution theorem.
$L^{-1}\left\{\frac{s^2+2s+3}{\left(s^2+2s+5\right)\left(s^2+2s+2\right)}\right\}$
6 M
6 (b) Find the directional derivative of ? = 4e2x-y+z at the point (1, 1, -1) in the direction towards the point (-3,5,6).
6 M
6 (c) Find the image of the circle x2+y2=1, under the transformation $w=\frac{5-4z}{4z-2}$
8 M

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