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MU Electronics Engineering (Semester 3)
Applied Mathematics - 3
December 2011
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) Prove that f(z)=(x3-3xy2+2xy) + i(3x2y-x2+y2-y3) is analytic and find f'(z) & f(z) in terms of z
5 M
1 (b) Find the Fourier series expansion for f(x)=|x|, in (-π, π) Hence deduce that
$\frac{{\pi{}}^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}...$
5 M
1 (c) Find the inverse Laplace transform of $\frac{e^{-z^3}}{s^2-2s+2}$
5 M
1 (d) $If\ \left\{\ f\left(k\right)\right\}=\left\{\ 2^0,2^1,\ 2^3,...\right\}$ Find Z{ f(k) }
5 M

2 (a) Evaluate $\int_0^{\infty{}}e^{-2t}\sinh{t\frac{\sin t}{t}}\ dt$
6 M
2 (b) Find the Fourier series expansion for $f\left(x\right)={\left(\frac{\pi{}-x}{2}\right)}^2$ in the interval
0 ≤ x ≤ 2π & f(x+2π)=f(x) Also deduce that $\frac{{\pi{}}^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}...$
7 M
2 (c) Show that $\left[\begin{array}{cc}1 & -\tan{\frac{\theta{}}{2}} \\\tan{\frac{\theta{}}{2}} & 1\end{array}\right]\ \left[\begin{array}{cc}1 & \tan{\frac{\theta{}}{2}} \\-\tan{\frac{\theta{}}{2}} & 1\end{array}\right]=\left[\ \begin{array}{cc}\cos{\theta{}} & -\sin{\theta{}} \\\sin{\theta{}} & \cos{\theta{}}\end{array}\right]$
7 M

3 (a) Find Laplace Transform of following
$\left(i\right)\ \int_0^1\frac{1-e^{-au}}{u}\ du$
$\left(ii\right)\ \ {\left(t\sinh{2t}\right)}^2$
6 M
3 (b) Find non-singular matrices P & Q s.t. PAQ is in Normal form. Also find rank of A & A-1
$A=\left[\begin{array}{ccc}1 & 3 & 3 \\1 & 4 & 3 \\1 & 3 & 4\end{array}\right]$
7 M
3 (c) Evaluate by Green's theorem $\int_C^{\ }\ \left[\left(3x^2-8y^2\right)\ dx+\left(4y-6xy\right)\ dy\right]$ where C is the boundary of the region bounded by $y=\sqrt{x},\ \ y=x^2$
7 M

4 (a) Obtain complex form of Fourier series for the functions f(x)= eax in (0,a)
6 M
4 (b) For what value of λ the equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ2 have a solution and solve them completely in each case.
7 M
4 (c) Find inverse Laplace Transform of following
$\left(i\right)\ \log{\left(1+\frac{a^2}{s^2}\right)}$
$\left(ii\right)\ \frac{s}{{\left(s+1\right)}^2\left(s^2+1\right)}$
7 M

5 (a) Prove that u=e3 cos y+x3 - 3xy2 is harmonic
6 M
5 (b) Determine the linear dependence of vectors [2, -1, 3, 2], [1,3,4,2], & [3,-5,2,2] Find the relation between them if dependent
7 M
5 (c) Using Fourier Cosine integral prove that $e^{-x}\cos{x=\frac{2}{\pi{}}\\int_0^{\infty{}}\frac{\left({\omega{}}^2+2\right)}{\left({\omega{}}^4+4\right)}.\\cos{\omega{}\ x\ d\omega{}}}$
7 M

6 (a) Obtain half-range sine series for f(x)= x (2-x) in 0
6 M
6 (b) Find the bilinear transformation which maps the points 0, I, -2i of z-plane on to the points -4i, ∞, 0 respectively of w-plane. Also obtain fixed points of the transformation
7 M
6 (c) Verify Stoke's theorem for F=yzi + zxj+xy k and c is the boundary of the circel x2+y2+z2=1, z=0
7 M

7 (a) Find inverse Z-transform of $F(z)=\dfrac {z}{(z-1)(z-2)},|z|>2$
6 M
7 (b) Find the analytic function f(z)= u+iv in terms of z if u-v=ex (cos y - sin y)
7 M
7 (c) Using laplace trasnform solve the following differential equation with given condition. (D2-2D+1) x=et, with x=2, Dx=-1, at t=0
7 M

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