Applied Mathematics - 3 - Dec 2011

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) Find L.T. of $$f\left(t\right)=\ f\left(x\right)= \Bigg\{\begin{align*}{}1,\ \ \ \ \ 0<t<a \\-1,\ \ \ \ a<t<2a\end{align*} $$
And f(t) = f(t+2a)(5 marks) 1 (b) Find the Fourier series of f(x) = cos μx in (-π,π), where μ is not an integer.(5 marks) 1 (c) Find the value of P for which the following matrix A will be of rank one, rank two, rank three where
$$\left[A=\begin{array}{ccc}3 & p & p \\p & 3 & p \\p & p & 3\end{array}\right]$$(5 marks) 1 (d) Find Z-transform of {k² - 2k + 3}_{k ≥ 0}(5 marks) 2 (a) Solve by using L.T.
$$ \dfrac {dy}{dt}+2y+\int^{t}_{0}y \ dt=\sin t \ when \ y \ (0)=1$$(8 marks) 2 (b) Find Fourier series for f(x) = x + x² in (-π , π) Hence deduce that
$$ \dfrac{1}{1^2}+ \dfrac{1}{3^2}+ \dfrac{1}{5^2} +\frac{1}{7^2} +....= \dfrac{{\pi}^2}{8} $$(6 marks) 2 (c) Show that vectors X₁, X₂, X₃ are linearly independent and vector X₄ depends upon them where
$$X_1=\ \left(1,2,4\right)$$$$X_2=\left(2,-1,3\right)$$$$X_3=\left(0,1,2\right)\ and \\ X_4=\left(-3,7,2\right)$$(6 marks) 3 (a) Find the Fourier integral representation of the function $$f\left(x\right)=\left\{\begin{array}{l}1-x^2,\ \ \ & \ \ when\ \vert{}x\vert{}\ & \leq{}1 \\ x,\ \ \ & \ \ when\ \vert{}x\vert{}\ & \ >{}1\end{array}\right.$$
And hence evaluate $$ \int_0^{\infty{}}\left[\frac{x\cos x-\sin x}{x^3}\right]\cos{\frac{x}{2}}\ dx\ $$(8 marks) 3 (b) Find matrix A if adj $$\left[A=\begin{array}{ccc}-2 & 1 & 3 \\-2 & -3 & 11 \\2 & 1 & -5\end{array}\right]$$(6 marks) 3 (c) $$ Find L \ \left\{\frac{1-cost}{t^2}\right\} $$(6 marks) 4 (a) $${(i) Find L^{-1}\left\{{tan}^{-1}{\left(s+2\right)}^2\right\} } \\ {(ii) Find L\left\{t^2H\left(t-2\right)-cos h t\ \delta{}\left(t-4\right)\right\}} $$(6 marks) 4 (b) Find inverse Z-transform of $$\frac{z\left(z+1\right)}{\left(z-1\right)\left(z^2+z+1\right)}$$(6 marks) 4 (c) Find the nm singular matrices P and Q such that PAQ is in the normal form and hence find rank of A and rank of (PAQ) where
$$A=\left[\ \begin{array}{ccc}3 & 2 & -1 & 5 \\5 & 1 & 4 & -2 \\1 & -4 & 11 & 19\end{array}\right]$$(6 marks) 5 (a) Find half range cosine series for
$$ f\left(x\right)=\left \{\begin{array}{l}x,\ \ \ & 0 < x < \frac{\pi{}}{2} \\\pi{}-x,\ \ \ & \frac{\pi{}}{2}<x<\pi{}\end{array}\right.$$
hence find the sum $$ \sum^{\infty}_{n=1} \dfrac {1}{n^4} $$(8 marks) 5 (b) Discuss the consistancy of the following system of equation and if consistant solve them
3x+3y+2z=1
x+2y+=4
10y+3z= -2
2x-3y-z=5(6 marks) 5 (c) Evaluate by using L.T. $$\int_0^{\infty{}}{t^3e}^{-t}\ \ sin\ t\ dt$$(6 marks) 6 (a) (i) (i) Find complex form of Fourier series for f(x) = cosh3x + sinh3x in (-π,π)(4 marks) 6 (a) (ii) (ii) Show that the functions $${\left\{\sin{\left(2n-1\right)}\right\}}_{n=0}^{\infty{}}$$ are orthogonal on $$\left[0,\frac{\pi{}}{2}\right]$$hence construct an orthonormal set of functions from this.(4 marks) 6 (b) Apply Gauss- Seidal itterative method to solve the equations upto three itteratism
3x+20y-z=-18
2x-3y+20z=25
20x+y-2z=17
(6 marks) 6 (c) Find Z-transform of $$\{k^2\ a^{k-1}\ \cup{}\ \left(k-1\right)\}$$(6 marks) 7 (a) (i) By using cinvolution theorem finf
$$L^{-1}\left\{\frac{1}{{\left(s-4\right)}^4(s+3)}\right\}$$(5 marks) 7 (a) (ii) Find : - L(sin2t cost cosh2t)(3 marks) 7 (b) Find inverse Z-transform of $$\frac{2z^2-10z+13}{{\left(z-3\right)}^2\left(z-2\right)}$$
if R.O.C. is 2|z|<3.(6 marks) 7 (c) Find Fourier sin a integral of f(x) where
$$ f(x)=\left\{\begin{matrix} x & ; &0<x <1 \\ 2-x&; &1<x<2 \\ 0&; & x>2 \end{matrix}\right.$$(6 marks)