Applied Mathematics 3 - May 2013

Information Technology (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) $$Find\ L\left\{t\ e^{31}\sin t\right\}$$(5 marks) 1 (b) Show that every square matrix can be uniquely expressed as the sum of a Hermitian and skew-Hermitian matrix.(5 marks) 1 (c) Find Z-transform and region of convergence of f(k)=3k, k≥0. (5 marks) 1 (d) Find the Fourier expansion of f(x)= x² where -π ≤x ≤π. (5 marks) 2 (a) Prove that following matrix is orthogonal and hence find its inverse.
$$A=\frac{1}{9}\left[\begin{array}{ccc}-8 & 4 & 1 \\1 & 4 & -8 \\4 & 7 & 4\end{array}\right]$$(6 marks) 2 (b) $$Find\ L^{-1}\left\{\frac{s+2}{{\left(s^2+4s+5\right)}^2}\right\}$$(6 marks) 2 (c) Obtain the Fourier expansion of $$ f\left(x\right)={\left(\frac{\pi{}-x}{2}\right)}^2 $$ in the internal and 0≤x≤2π and f(x+2π)=f(x). Also deduce that,
$$ \left(i\right)\ \frac{{\pi{}}^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{} $$
$$ \left(ii\right)\frac{{\pi{}}^4}{90}=\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{} $$ (8 marks) 3 (a) Investing for what values of λ and μ the equations.
x+y+z=6
x+2y+3z=10
x+2y+λz=μ have,
(i) No Solution
(ii) a unique solution
(iii) Infinite no. of solutions.
(6 marks) 3 (b) Obtain complex form of Fourier series for f(x)=e^ax (-π,π) where is not integer. (6 marks) 3 (c) Solve (D² - D - 2) y=20 sin 2t with y(0)=1, y'(0)=2.(8 marks) 4 (a) Find Laplace transform of
$$ f\left(t\right)=a\sin{pt\ 0<t\leq{}\frac{\pi{}}{p}} \\ f\left(t\right)=0\frac{\pi{}}{P}< t\leq{}\frac{2\pi{}}{P} \\ and\ \ f\left(t\right)=\left(t+\frac{2\pi{}}{P}\right) $$ (6 marks) 4 (b) Find the inverse Z-transform for
$$f\left(z\right)=\frac{1}{\left(z-3\right)\left(z-2\right)}$$
for 2<|z|<3.(6 marks) 4 (c) Find inverse Laplace transform of
$$ \left(i\right)\ \frac{e^{4-3s}}{{\left(s+4\right)}^{\frac{5}{2}}}\\ (ii)\tan^{-1}\frac{2}{s} $$ (8 marks) 5 (a) Examine whether the following vectors are linearly independent or dependent [2,1,1], [1,3,1], [1,2,-1](6 marks) 5 (b) Using Convolution theorem prove that
$$ l^{-1}\left[\frac{1}{s}\ ln\ \left(\frac{s+1}{s+2}\right)\right]=\ \int_0^t\left(\frac{e^{-2u}-e^{-u}}{u}\right)du\ $$(6 marks) 5 (c) Using Fourier cosine Integral prove that
$$e^{-x}\cos{x=\frac{1}{\pi{}}\int_0^{\infty{}}\frac{w^2+2}{w^4+4}\ \cos{wx\ dw}}$$(8 marks) 6 (a) Find the Fourier Transform 0+f(x)=e^-1x1(6 marks) 6 (b) Find z[f(x)] where $$ f\left(k\right)=\cos{\left(\frac{k\pi{}}{u}+a\right)} $$ where k≥0. (6 marks) 6 (c) Find Fourier expansion of f(x)=2x - x² where 0 ≤ x ≤ 3 and period is 3. (8 marks) 7 (a) Reduce the following matrix to noraml form and find its rank.
$$A=\left[\begin{array}{ccc}1 & -1 & 3 & 6 \\1 & 3 & -3 & -4 \\5 & 3 & 3 & 11\end{array}\right]$$(6 marks) 7 (b) $$Evalute\ \int_0^{\infty{}}\frac{\cos{6t-\cos{4t}}}{t}dt$$(6 marks) 7 (c) Show that the set of function
$$\sin{\left(\frac{\pi{}x}{2L}\right)},\sin{\left(\frac{3\pi{}x}{2L}\right)},\sin{\left(\frac{5\pi{}x}{2L}\right)},\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}\bullet{}$$
is orthogonal over..(0,L) Hence construct corresponding orthonormal set.(8 marks)