MORE IN Applied Mathematics - 3
MU Electronics and Telecom 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) Evaluate $\int_0^{\infty{}}\frac{\left(\cos{6t-cos4t}\right)}{t}\ dt$
5 M
1 (b) Obtain complex form of fourier series for f(x)= eax in (-1, 1)
5 M
1 (c) Find the work done in moving a particle in a force field given by $\bar{F}=3xy\ \hat{i}-5z\hat{j}+10x\hat{k}$ along the curve x=t2+1, y=2t2, z=t3 from t=1 to t=2
5 M
1 (d) Find the orthogonal trajectory of the curves 3x2y+2x3-y3-2y2 = ?, where &lpha; is a constant
5 M

2 (a) Evaluate $\frac{d^2y}{dt^2}+2\frac{dy}{dt}-3y=sint,$ y(0)=0, y'(0)=0, by Laplace transform
6 M
2 (b) Show that $J_{\frac{5}{2}}=\ \sqrt{\frac{2}{\pi{}x}} \left[\frac{3-x^2}{x^2}\sin{x-\frac{3}{x}\cos{x\ }}\right]$
6 M
2 (c) (i) Find the constant a,b,c so that $\bar{F}=\left(x+2y+az\right)\hat{i}+\left(bx-3y-z\right)\hat{j}+(4x+\left(y+2z\right)\hat{k}$
4 M
2 (c) (ii) Prove that the angle between two surface x2+y2+z2=9 and x2+y2-z=3 at the point (2,-1,2) is ${\cos}^{-1}{\left(\frac{8}{3\sqrt{21}}\right)}$
4 M

3 (a) Obtain the fourier series of f(x) given by
$f\left(x\right)=\left\{\begin{array}{l}0,\ \ \&-\pi{}\leq{}x\leq{}0 \\x^2,\ \ \&0\leq{}x\leq{}\pi{}\end{array}\right.$
6 M
3 (b) Find the analytic function f(z)= u+iv where u=r2 cos2θ-r cosθ+2
6 M
3 (c) Find Laplace transform of
(i) te-3t cos2t.cos3t
(ii) $\frac{d}{dt}\left[\frac{\sin{3t}}{t}\right]$
8 M

4 (a) Evaluate ∫ J3(x) dx and Express the result in terms of J0 and J1
6 M
4 (b) Find half range sine series for f(x)= πx-x2 in (0, π) Hence deduce that $\frac{{\pi{}}^3}{32}=\frac{1}{12}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\pi$
6 M
4 (c) Find inverse Laplace transform of :-
$\left(i\right)\frac{1}{s}{\tan h}^{-1}{\left(s\right)}$
$\left(ii\right)\ \frac{se^{-2s}}{\left(s^2+2s+2\right)}$
8 M

5 (a) Under the transformation w+2i=z 1/z, show that the map of the circle |z|=2 is an ellipse in w-plane
6 M
5 (b) Find half range cosine series of f(x)= sinx in 0 ≤ x ≤ π Hence deduce that
$\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+?=\frac{1}{2}$
6 M
5 (c) Verify Green's theorem, for $\oint_C\left(3x^2-8y^2\right)dx+\left(4y-6xy\right)$ by where c is boundary of the region defined by x=0, y=0, and x+y=1
8 M

6 (a) Using convolution theorem; evaluate
$L^{-1}\left\{\frac{1}{\left(S-1\right)\left(s^24\right)}\right\}$
6 M
6 (b) Find the bilinear transformation which maps the points z=1, I, -1 onto w=0, 1, ?
6 M
6 (c) By using the appropriate theorem, evaluate the following :-
$\left(i\right)\ \int\bar{F}c \dot{}d\bar{r}\ where\\bar{F}=\left(2x-y\right)\hat{i}-\left(yz^2\right)\hat{j}-\left(y^2z\right)\hat{k}$
and c is the boundary of the upper half of the sphere x2+y2+z2=4
$\left(ii\right)\ \iint_s\ (9x\hat{i}+6y\hat{j}-10z\hat{k})\ c \dot{}d\bar{s}$
where s is the surface of sphere with radius 2 uints.
8 M

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