MORE IN Applied Mathematics - 3
MU Electronics Engineering (Semester 3)
Applied Mathematics - 3
May 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) Show that $f(z)=\frac{\bar{z}}{{\left\vert{}z\right\vert{}}^2} \ \ \ \ \ \ , |z|≠0$ is analytic function.Hence find f'(z)
5 M
1 (b) Find the Fourier Expansion of f(x)=sinx in (-π,π).
5 M
1 (c) Find the Laplace Transform of $t\sqrt{1+sint}$
5 M
1 (d) Find z transformation of { ak sin ak}, k ≥ 0. where a is constant
5 M

2 (a) Using Laplace Transform evaluate $\int_0^{\infty{}}e^{-t}\frac{\sin{3t}}{t}dt.\ \ \ \ \ \ \ \ \ \ \ \ \ \$
6 M
2 (b) Find the Fourier expansion of f(x)=cospx where p is not an integer in (0,2π)
7 M
2 (c) Find the matrix A, if adj A = $\ \left[\begin{array}{ccc}-2 & 1 & 3 \\-2 & -3 & 11 \\2 & 1 & -5\end{array}\right]$
7 M

3 (a) Find inverse Laplace transform of
(I) log (s-2/s-3)
(II) s+1/s2 - 4
6 M
3 (b) Find non Singular matrices P and Q such that PAQ is in normal form. Also find rank of a matrix A where
A =$\begin{array}{cccc}2 & -4 & 3 & 1 & 0 \\1 & -2 & 1 & -4 & 2 \\0 & 1 & -1 & 3 & 1 \\4 & -7 & 4 & -4 & 5\end{array}$
7 M
3 (c) Verify Green's Theorem in the plane for ∮c (xy+y2) dx+x2 by where c is a closed
curve of a region bounded by y=x and y2=x
7 M

4 (a) Obtain Complex form of Fourier series for f(x)=e-ax in (-2,2)where a is not an integer.
6 M
4 (b) if A= $\left[\begin{array}{ccc}1 & 1 & 1 \\2 & 5 & 7 \\2 & 1 & -1\end{array}\right]$ compute A l and hence, Solve the system of equation
x + y + z = 9, 2x + 5y + 7z =52, 2x + y - z = 0.
7 M
4 (c) Find the Laplace transform of
f(t) = 1, 0 ≤ t ≤a
f(t) =1, a < t ≤ 2a &
f(t + 2a)= f(t)
7 M

5 (a) Find the analytic function f(z)=u+iv if u=(r+ a2/r) cos θ.
6 M
5 (b) Show that the equations.
ax + by +cz =0
bx + cy + az =0
cx + ay + bz = 0
has non trivial solution if a+b+c=0 or if a=b=c. Find the non trivial solution when the condition is satisfied.
7 M
5 (c) Find Fourier integral representing f(x) = $\left\{\begin{array}{l}1-x^2\ \ \ \ \ \ \ \ \left\vert{}x\right\vert{}\leq{}1 \\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\vert{}x\right\vert{}>1\end{array}.\right.$
7 M

6 (a) Find the half range cosine series for f(x)=2x-x2 in (0,2)
6 M
6 (b) Find the Bilinear Transformation which maps the points 2,i,-2 on the points 1,i,-1
7 M
6 (c) Using Lapace transform solve the differential equation
$\frac{d^2y}{{dt}^2}-2\frac{dy}{dt}-8y=4\ ,\ y\left(0\right)=0,\ y^{'}\left(0\right)=1.$
7 M

7 (a) Find inverse z-transform of F(z)= 1/(z-2)(z-3) if ROC is 2 < |z| < 3.
6 M
7 (b) Verify Stoke's theorem for $\bar{F}=x^2\bar{i}+xy\bar{j}$ & C is the boundary of the rectangle x=0, x=2, y=3.
7 M
7 (c) Using Gauss Divergence Theorem evaluate $\iint_S\vec{F}\bullet{}\hat{n}ds\ where\ \vec{F}=4x\bar{i}+3y\bar{j}-4z^2\vec{k}$
and S is the closed surface bounded the planes x=0,y=0,z=0 and 2x+2y+z=4
7 M

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