MORE IN Applied Mathematics - 3
MU Electronics and Telecom Engineering (Semester 3)
Applied Mathematics - 3
May 2016
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) Evaluate $$\int ^{\infty}_0e^{-2t}\left ( \dfrac{\sinh t \sin t}{t} \right )dt$$
5 M
1(b) Obtain the Fourier Series expression for
f(x) = 9-x2 in (-3,3)
5 M
1(c) Find the value of 'p' sucj that the function f(z) expressed in polar co-ordinates as
f(z) = r3 cos p&theta
5 M
1(d) If $$\bar{F}=(y^2-z^2+3yz-2x)\hat{i}+(3xz+2xy)\hat{j}+(3xy-2xz+2z)\hat{k}.$$
Show that $$\bar{F}$$ is irrotational and solenoidal.
5 M

2(a) Solve the differential equation using Laplace Transform $$\dfrac{d^2y}{dt^2}+4\frac{dy}{dt}+8y=1$$, given y(0)=0 and y'(0)=1
6 M
2(b) Prove that $J_4(x)=\left ( \dfrac{48}{x^3}-\dfrac{8}{x} \right )J_1(x)-\left ( \dfrac{24}{x^2}-1 \right )J_0(x)$
6 M
2(c)(i) Find the directional derivative of ϕ = 4xz3 - 3x2y2z at (2,-1,2) in the direction of $$2\hat{i}+3\hat{j}+6\hat{k}.$$
8 M
2(c)(ii) if $$\bar{r}=x\hat{i}+y\hat{j}+z\hat{k}$$
prove that $$\nabla \log r=\dfrac{\bar{r}}{r^2}$$
8 M

3(a) Show that { cosx, cos2x, cos3x.....} is a set of orthogonal fundtions over (-π, π), Hence construct an orthogonal set.
6 M
3(b) Find an analytic function f(z)=u+iy where. $u=\dfrac{x}{2}\log(x^2+y^2)-y\tan^{-1}\left ( \dfrac{y}{x} \right )+\sin x \cosh y$
6 M
Find the Laplace transform of
3(c)(i) $$\int ^1_0 ue^{-3u}\cos^2u du$$
4 M
3(c)(ii) $$t\sqrt{1+\sin t}$$
4 M

4(a) Find the Fourier Series for $$f(x)=\dfrac{3x^2-6\pi x+2\pi^2}{12}$$ in (0,2π)
Hence deduce that $$\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}\cdots=\dfrac{\pi^2}{6}$$
6 M
4(b) Prove that $\int ^b_0 xJ_0(ax)dx=\dfrac{b}{a}J_1(ab)$
6 M
Find
4(c)(i) $$L^{-1}\left [ \log\left ( \dfrac{s^2+1}{s(s+1)} \right ) \right ]$$
4 M
4(c)(ii) $$L^{-1}\left [ \log\left ( \dfrac{s+2}{s^2-2s+17} \right ) \right ]$$
4 M

5(a) Obtain the half range cosine series for \begin {align*} f(x)&=x, 0 6 M 5(b) Find the Bi-linear Transformation which maps the points 1,i,-1 of z plane onto i,0,-i of w-plane 6 M 5(c) Verify Green's Theorem for $$\int _c \overline{F} .\overline{br}$$ where $$\bar{F}=(x^2-xy)\hat{i}+(x^2-y^2)\hat{j}$$ and C is the curve bounded by x2 = 2y and x=y 8 M 6(a) Show that the transformation $$w=\dfrac{i-iz}{1+z}$$ maps the unit circle |z|=1 into real axis of w plane. 6 M 6(b) Using Convolution theorem find \[L^{-1}\left [ \dfrac{s}{(s^2+1)(s^2+4)} \right ]
6 M
6(c)(i) Use Gauss Divergence Theorem to evaluate $$\iint _s \bar{F}.\hat{n}ds\ \text{where}\ \bar{F}=x\hat{i}+y\hat{j}+z\hat{k}$$ and S is the sphere x2 + y2 + z2 = 9 and $$\hat{n}$$ is the outward normal to S
8 M
6(c)(ii) Use Stoke's Theorem to evaluate $$\int _c\overline{F}.\overline{dr}\ \text{where}\ \bar{F}=x^2\hat{i}-xy\hat{j}$$ and C is the square in the plane z=0 and bounded by x=0, y=0, x=a and y=a.
8 M

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