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. 1 (a) Prove that
$$\int_0^{\infty{}}e^{-t}\dfrac{{sin}^2t}{t}dt= \dfrac{1}{4} log 5$$(5 marks) 1 (b) Is the matrix orthogonal ? If not then can it be converted to an orthogonal matrix :-
$$A=\left[\begin{array}{ccc}-8 & 1 & 4 \\4 & 4 & 7 \\1 & -8 & 4\end{array}\right]$$(5 marks) 1 (c) Obtain complex form of Fourier series for f(x) = e^ax in (-l ,l).(5 marks) 1 (d) Find the Z-transform of f(k) =a^k, k≥0.(5 marks) 2 (a) Find the Fourier sine transform of f(x) if $$ \begin {align*} f(x)&=\sin kx, &0 \le x <a \\ &=0, &x>a \end{align*} $$(6 marks) 2 (b) Find the Matrix A if
$$\left[\begin{array}{cc}2 & 1 \\3 & 2\end{array}\right]\ A\ \left[\begin{array}{cc}-3 & 2 \\5 & -3\end{array}\right]=\ \left[\begin{array}{cc}-2 & 4 \\3 & -1\end{array}\right]$$(6 marks) 2 (c) (D²- 3D+2) y=4 e²¹, with y(0) = -3, y'(0)=5 solve using Laplace transform. (8 marks) 3 (a) Reduce the matrix to normal form and find its rank :-
$$\left[\begin{array}{cccc}2 & 3 & -1 & -1 \\1 & -1 & -2 & -4 \\3 & 1 & 3 & -2 \\6 & 3 & 0 & -7\end{array}\right]$$(6 marks) 3 (b) Find the inverse Laplace transform of ?
$$ \left(i\right)\ \frac{e^{-2s}}{s^2+8s+25} \ltbr\gt \\ \left(ii\right)\ \frac{e^{-3s}}{{\left(s+4\right)}^3}$$(6 marks) 3 (c) $$ \left.\begin{matrix} f(x)&= \pi x 0\leq x \leq 1 \\ f(x)&=\pi (2-x)1 \leq x \leq 2 \end{matrix}\right\} with \ period \ 2 $$
Find the Fourier series expansion (8 marks) 4 (a) Show that the set of functions $$ {\ \left(\frac{\pi{}x}{2l}\right), sin\left(\frac{3\pi{}x}{2l}\right),\ \sin\left(\frac{5\pi{}x}{2l}\right),.....}$$is orthogonal over (0,l).(6 marks) 4 (b) If f(k)= 4^kU(K), g(k)= 5^kU(k), then find the z-transform of f(k) x g(k).(6 marks) 4 (c) Solve the following equations by Gauss-Seidel Method.
28x+4y-z=32
2x+17y+4z=35
x+3y+10z=24.(8 marks) 5 (a) Obtain Fourier series for
$$ {\ f(x) = x + \frac{\pi{}}{2}, -\pi{} < x < 0} \\ {= \frac{\pi{}}{2}-x\ 0 < x < \pi{}} $$
Hence deduce that, $$ {\ \frac{\pi^2}{8} = \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + ....} $$(6 marks) 5 (b) State Convolution theorem and hence find inverse Laplace transform of the function using the same :-
$$f(s)\ =\frac{{\left(s+3\right)}^2}{{\left(s^2+6s+5\right)}^2}$$(6 marks) 5 (c) For what value of λ the equations 3x-2y+ λ z=1, 2x+y+z=2, x+2y- λz= -1 will have no unique solution ? Will the equations have any solution for this value of λ ? (8 marks) 6 (a) Show that every square matrix A can be uniquely expressed as P+iQ when P and Q are Hermitian matrices (6 marks) 6 (b) If L[f(t)] = f(s), then prove that L[ tⁿ f(t)] = (-1)ⁿ dⁿ/dsⁿ f(s), Hence find the Laplace transform of f(t) = t cos²t(6 marks) 6 (c) Obtain the half rang sine series for f(x) when
$$ {\ f(x) = x 0 < x < \frac{\pi{}}{2}}\\{= \pi{} - x \frac{\pi{}}{2}< x < \pi{}}\ltbr\gt \\ Hence \ find \ the \ sum \ of \ \sum_{2n-1}^{\infty{}}\ \frac{1}{n^4}$$(8 marks) 7 (a) Find the Fourier transform of-
f(x) = (1-x²), |x|<|
= 0 , |x|>|,
then $$f\left(s\right)=\ -2\sqrt{\frac{2}{\pi{}}}\left[\frac{scoss-sins}{s^3}\right]$$(6 marks) 7 (b) Find the inverse z transform of $$ F\left(z\right)=\ \frac{z}{\left(z-1\right)\left(z-2\right)},\ \left\vert{}z\right\vert{}>2$$(6 marks) 7 (c) Find the non-singular matrices P and Q such that -
$$A=\ \left[\begin{array}{ccc}1 & 2 & 3 & 2 \\2 & 3 & 5 & 1 \\1 & 3 & 4 & 5\end{array}\right]$$
is reduced to normal form. Also find its rank. (8 marks)