Applied Mathematics - 3 - Dec 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) $$A=\frac{1}{7}\ \left[\begin{array}{ccc}2 & 6 & 3 \\6 & -3 & 2 \\3 & 2 & -6\end{array}\right]$$
Show that-
(i) |A| = 1
(ii) .adj A=A'(5 marks) 1 (b) Show that the Fourier cosine transform of-
$$f\left(x\right)=\ \frac{1}{\sqrt{x}}\ is\ \frac{1}{\sqrt{s}}$$(5 marks) 1 (c) Show that
$$ L^{-1}\ {tan}^{-1}\left(\frac{a}{s}\right)=\ \frac{sin?(at)}{t}\ $$(5 marks) 1 (d) Show that
$$ \int_0^{\infty}e^{-t} \sin \left( \frac{t}{2}\right) \sin h{\ \left(\frac{\sqrt{3}t}{2}\right)}\ dt=\ \frac{\sqrt{3}}{2} $$(5 marks) 2 (a) If z is any zero complex number show that -
[A=frac{1}{sqrt{2} vert{}zvert{}}left[?egin{array}{cc}z (6 marks) 2 (b) Find the Fourier series expansion of f(x)=x² in [0,2π](6 marks) 2 (c) Show that
$$\int_0^{\infty{}}\frac{\sin{2t+\sin{3t}}}{te^t}dt=\ \frac{3\pi{}}{4}$$(8 marks) 3 (a) Find the half range sine series for f(x) = ?x - x² in [ 0, ? ]. Hence find $$\sum_{n=1}^{\infty{}}\frac{1}{{(\ 2n-1)}^6}$$(6 marks) 3 (b) Show that
$$L(cos\ at\ cosh\ at)=\frac{s^3}{s^4+\ 4a^4}$$(6 marks) 3 (c) if a,b,c are distinct real numbers such that a+b+c ≠ 0. Show that the vectors (a,b,c), (b,c,a) and (c,a,b) are linearly independent. (8 marks) 4 (a) Find Fourier series expansion of -
$$ f(x)=\left\{\begin{matrix} x &-π<x<0 \\ 0 &0<x<\dfrac {π}{2} \\ x-\dfrac {π}{2}& \dfrac {π}{2}<x,<π \end{matrix}\right. $$(6 marks) 4 (b) Find the all possible ranks of the matrix
$$A=\left[\ \begin{array}{ccc}k & -3 & -3 \\-3 & k & -3 \\-3 & -3 & k\end{array}\right]$$
where k is any real number (6 marks) 4 (c) Find
$$ (i) \ L^{-1}\ \dfrac{se^{-πs}}{s^2+3s+2}; \ (ii) \ L [t. H(t - 4)+t^2 \ δ(t-4)] $$(8 marks) 5 (a) Solve the system of equation using the Gauss- Seidel method :
3x+4y+6z=6
2x+5y+4z=11
2x+y+z+=5(6 marks) 5 (b) Find $${\ L}^{-1}\frac{s}{{\left(s^2+\ 2\right)}^2}$$using convolution theorem(6 marks) 5 (c) Show that
$$z[\cos\ {(\alpha \ k)]=\ \frac{z^2-z\cos \alpha}{z^2-2(\cos{\alpha)z+1}}}$$(8 marks) 6 (a) Find Lf(t) where f(t) = cos t, 0 < t < π
sin t, sin t ≥ π
Using the Heaviside unit step function.(6 marks) 6 (b) Find $$Z^{-1}\frac{1}{(z-2)(z-3)}$$for 2<|z|<3 using residues. (6 marks) 6 (c) Express the matrix
$$A=\left[\begin{array}{ccc}\sin{\alpha{}} & 0 & \sin{\alpha{}} \\\left(sin\ \alpha{}\right)\ (sin\beta{}) & cos\beta{} &-\left(sin\beta{}\right)\left(cos\alpha{}\right) \\-(cos\beta{})(sin\alpha{}) & sin\beta{} & (cos\alpha{})(cos\beta{})\end{array}\right]$$
as a product of two orthogonal matrices. (8 marks) 7 (a) Solve the $$DE\ \ y+\int_0^ty\ dt=1-\ e^{-t}$$using Laplace transforms.(6 marks) 7 (b) if 0 ≤ x ≤ π show that-
$$ x^2= \ \frac{2}{π}\left[ \left(\frac{\pi^2}{1}-\frac{4}{1^3}\right)\sin x- \ \left(\frac{\pi^2}{2}\right)\ \sin 2x\\ \ \ \ \ \ \ +\ \left(\frac{\pi^2}{3}-\frac{4}{3^3}\right)\ \sin 3x+.....\right]$$(6 marks) 7 (c) Show that {cos nx}_n≥1 is an orthogonal family of functions in [ -π , π ]. Also find the corresponding orthonormal set.(8 marks)