Applied Mathematics - 3 - May 2012

Electronics & Telecomm. (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.

This Qs paper appeared for Applied Mathematics - 3 of

Electronics Engineering . (Semester 3) 1 (a) If f(z)= (ax⁴ + bx²y² + cy⁴ + dx² - 2y²) + i(4x³y - exy³ + 4xy) is analytic, find the constants a,b,c,d,e(5 marks) 1 (b) Find the Fourier series expansion for f(x)= |sin x|, in (-?, ?)(5 marks) 1 (c) Find the Laplace transform of sin t? $$H\left(t-\frac{\pi{}}{2}\right)-H\left(t-\frac{3\pi{}}{2}\right)$$(5 marks) 1 (d) $$If\ \ \left\{f\left(x\right)\right\}=\left\{\begin{array}{l}4^k,\ \ \&for\ k<0\\3^k,\ \ \&for\ k\geq{}0\end{array}\right.\ \ find\ Z\left\{f\left(k\right)\right\}$$(5 marks) 2 (a) $$ if \ \int^{\infty}_{0}e^{-2t} \sin (t+\alpha) \cos (t-\alpha)dt=\dfrac {3}{8} \ then \ find\ \alpha $$(6 marks) 2 (b) Find the Fourier series expnasion for $$ f(x)= \sqrt{1-\cos x} in (0,2?) \ Hence \ deduce \ that \ \sum_{n=1}^{\infty}\dfrac{1}{4n^2 -1}=\dfrac{1}{2} $$(7 marks) 2 (c) Find the inverse of A if
$$\left[\begin{array}{ccc}1 & 0 & 0 \\2 & -1 & 0 \\-2 & 1 & 1\end{array}\right]\ A\ \left[\begin{array}{ccc}1 & -2 & 9 \\0 & 1 & -6 \\0 & 0 & 1\end{array}\right]=\ \left[\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]$$(7 marks) 3 (a) Find Laplace Transform of following
$$\left(i\right)\ e^{-4t}\ \int_0^1u\sin{3u\ du}$$
$$\left(ii\right)\ \frac{1}{t}(1-\cos{t)}$$(6 marks) 3 (b) Find non-singular matrices P & Q s.t. PAQ is in Normal form. Also find rank of A & A^-1
$$A=\left[\begin{array}{ccc}1 & 2 & 3 \\2 & 3 & 0 \\0 & 1 & 2\end{array}\right]$$(7 marks) 3 (c) Evaluate by Green's theorem $$\int_C\bar{F}\cdot{}d\bar{r}\ where\ \ \bar{F}=xy\left(xi-yi\right) \ and \ C \ is \ r=a(1+ \cos \theta) $$(7 marks) 4 (a) Obtain complex form of Fourier series for the functions f(x)= sin a x in (-?, ?)(6 marks) 4 (b) For what value of ?, the following system of equations possesses a non-trivial solution? Obtain the solution for real value of ?.
3x₁+x₂-? x₃=0, 4x₁2x₂-3x₃=0, 2? x₁+4x₂+? x₄=0(7 marks) 4 (c) Find inverse Laplace Transform of following
(i) 2 tanh^-1 s
$$\left(ii\right)\frac{s^2}{\left(s^2+1\right)\left(s^2+4\right)}$$(7 marks) 5 (a) Find the orthogonal trajectory of the family of curves 3x³y+2x²-y³-2y²= c (6 marks) 5 (b) Find the relation of linear dependence amongst the rows of the matrix
$$A=\left[\begin{array}{ccc}1 & 1 & -1 & 1 \\1 & -1 & 2 & -1 \\3 & 1 & 0 & 1\end{array}\right]$$(7 marks) 5 (c) Express the function $$f\left(x\right)=\left\{\begin{array}{l}-e^{kx},\ \ \&for\ x<0 \\e^{-kx},\ \ \&for\ x>0\end{array}\right.$$ as Fourier integral and prove that $$\int_0^{\infty{}}\frac{\omega{}\sin{\omega{}\ x}}{{\omega{}}^2+k^2}\ d\omega{}=\frac{\pi{}}{2}\ e^{-kx}\ \ if\ x>0,\ k>0$$(7 marks) 6 (a) Obtain half-range cosin series for f(x)=x in 0<x<l< a="">

</x<l<></span>(6 marks) 6 (b) Show that under the transformation $$ w= \dfrac {5-4Z}{4z-2}$$ the circle |z|=1 in the z-plane is transformed into a circle of unity in the w-plane. Also find the center of the circle(7 marks) 6 (c) A vector field is given by F=3x³yi + (x³-2yz²) j+ (3z²-2y²z) k is irrational. Also find ? such that F= ? ?. Also evaluate the line integral from (2,1,1), (2,0,1)(7 marks) 7 (a) Find inverse Z-transformation of $$ F \left(z\right)=\frac{z}{\left\{z-\left(\frac{1}{4}\right)\right]\ \left[z-\left(\frac{1}{5}\right)\right]},\frac{1}{5} <\left \vert{}z\right \vert{} <\frac{1}{4} $$ (6 marks) 7 (b) Find the analytic function f(z)=u+iv in terms of z if u-v=(x-y) (x² + 4xy + y²)(7 marks) 7 (c) Using laplace trasnform solve the following differential equation with given condition. (D²-3D+2) y=e^2t, y(0)=-3, y'(0)=5(7 marks)