Applied Mathematics - 3 - Dec 2011

Electronics Engineering (Semester 3)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 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 marks) 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 marks) 1 (c) Find the inverse Laplace transform of $$\frac{e^{-z^3}}{s^2-2s+2}$$(5 marks) 1 (d) $$If\ \left\{\ f\left(k\right)\right\}=\left\{\ 2^0,2^1,\ 2^3,...\right\}$$ Find Z{ f(k) }(5 marks) 2 (a) Evaluate $$\int_0^{\infty{}}e^{-2t}\sinh{t\frac{\sin t}{t}}\ dt$$(6 marks) 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 marks) 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 marks) 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 marks) 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 marks) 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 marks) 4 (a) Obtain complex form of Fourier series for the functions f(x)= e^ax in (0,a)(6 marks) 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 marks) 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 marks) 5 (a) Prove that u=e³ cos y+x³ - 3xy² is harmonic(6 marks) 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 marks) 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 marks) 6 (a) Obtain half-range sine series for f(x)= x (2-x) in 0<x<2< a="">

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