MORE IN Applied Mathematics 3
MU Computer Engineering (Semester 3)
Applied Mathematics 3
May 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 $L^{-1}\left[\frac{e^{-\pi{}s}}{s^2+2s+2}\right]$
5 M
1 (b) State true or false with proper justification "There does not exist ar. Analytic function whose real part is x3 - 3x3y-y3.
5 M
1 (c) prove that $f_1\left(x\right)=1,\ f_2\left(x\right)=x,\ f_3\left(x\right)=\frac{\left(3x^2-1\right)}{2}$ are orthogonal over (-1,1)
5 M
1 (d) Using Green's theorem in the plane, evaluate $\int_c^{\ }\left(x^2-y\right)dx+\left(2y^2+x\right)\$ by around the boundry of the region defined by y=x2 and y=4.
5 M

2 (a) Find the fourier cosine integral representation of the function f(x)=e-ax, x>0 and hence show that
$\int_0^{\infty{}}\frac{\cos{ws}}{1+w^2}dw=\frac{\pi{}}{2}e^{-x},\ x\geq{}0$
6 M
2 (b) Verify laplace equations for $U=\left(r+\frac{a^2}{r}\right)\cos{\theta{}}$ Also find V and f(z).
6 M
2 (c) Solve the following equation by using laplace transform $\frac{dy}{dt}+2y+\int_0^tydt=\sin t$ given that y(0)=1
8 M

3 (a) Expland $f\left(x\right)=\left\{\begin{array}{l}\pi{}x,\ 0<x<1 \\ 0,\ 1<x<2\end{array}\right.$ with period 2 into a fourier series.
6 M
3 (b) A vector field is given by $\bar{F}=\left(x^2+xy^2\right)i+\left(y^2+x^2y\right)j\ show\ that\ \bar{F}$ is irrotational and find its scalar potential.
6 M
3 (c) Find the inverse z-transform of- $f\left(Z\right)=\frac{z+2}{z^2-2z+1},\left\vert{}z\right\vert{}>1$
8 M

4 (a) Find the constants 'a' and 'b' so that the surface ax2-byz=(a+2) x will be orthogonal to the surface 4x2y+z3=4 at (1, -1, 2)
6 M
4 (b) $Given\ \ L\left(erf\sqrt{t}\right)=\frac{1}{S\sqrt{S+1}},\ evaluate\ \int_0^{\infty{}}te^{-t}erf (\sqrt{t})\ dt$
6 M
4 (c) Obtain the expansion of f(x)=x(π - x), 0 $\left(ii\right)\ \ \sum_1^{\infty{}}\frac{1}{n^4}=\frac{{\pi{}}^4}{90}$
8 M

5 (a) If the imaginary part of the analytic function $W=f\left(z\right)is\ V=x^2-y^2+\frac{x}{x^2+y^2}$ find the real part U.
6 M
5 (b) If f(k)=4k U(K) and g(k)=5k U(K), then find the z-transform of f(k)· g(k)
6 M
5 (c) Use Gauss's Divergence theorem to evaluate $\iint_s^{\ }\ \bar{N}\cdot{}\bar{F\ }ds\ \ \ where\ \bar{F}=4xi+3yj-2zk$ and S is the surface bounded by x=0, y=0, z=0 and 2x+2y+z=4
8 M

6 (a) Obtain complex form of Fourier series for f(x)= c0s h 3x sin h 3x in (-3,3).
6 M
6 (b) Find the inverse Laplace transform of $\frac{{\left(S-1\right)}^2}{{\left(s^2-2s+5\right)}^2}$
6 M
6 (c) Find the bilinear transformation under which 1, I, -1 from the z-plane are mapped onto 0,1,&infin of w-plane. Also show that under this transformation the unit circle in the w-plane is mapped onto a straight line in the z-plane. Write the name of this line.
8 M

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