MORE IN Applied Mathematics - 3
MU Electronics Engineering (Semester 3)
Applied Mathematics - 3
December 2011
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

This Qs paper appeared for Applied Mathematics - 3 of Electronics & Telecomm. (Semester 3)
1 (a) Find L.T. of f\left(t\right)=\ f\left(x\right)= \Bigg\{\begin{align*}{}1,\ \ \ \ \ 0 And f(t) = f(t+2a) 5 M 1 (b) Find the Fourier series of f(x) = cos ?x in (-?,?), where ? is not an integer. 5 M 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 M
1 (d) Find Z-transform of {k2 - 2k + 3}k ? 0
5 M

2 (a) Solve by using L.T.
$\dfrac {dy}{dt}+2y+\int^{t}_{0}y \ dt=\sin t \ when \ y \ (0)=1$
8 M
2 (b) Find Fourier series for f(x) = x + x2 in (-? , ?) Hence deduce that
$\dfrac{1}{1^2}+ \dfrac{1}{3^2}+ \dfrac{1}{5^2} +\frac{1}{7^2} +....= \dfrac{{\pi}^2}{8}$
6 M
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 M

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 M
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 M
3 (c) $Find L \ \left\{\frac{1-cost}{t^2}\right\}$
6 M

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 M
4 (b) Find inverse Z-transform of $\frac{z\left(z+1\right)}{\left(z-1\right)\left(z^2+z+1\right)}$
6 M
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 M

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} hence find the sum \[ \sum^{\infty}_{n=1} \dfrac {1}{n^4}$
8 M
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 M
5 (c) Evaluate by using L.T. $\int_0^{\infty{}}{t^3e}^{-t}\ \ sin\ t\ dt$
6 M

6 (a) (i) (i) Find complex form of Fourier series for f(x) = cosh3x + sinh3x in (-?,?)
4 M
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 M
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 M
6 (c) Find Z-transform of $\{k^2\ a^{k-1}\ \cup{}\ \left(k-1\right)\}$
6 M

7 (a) (i) By using cinvolution theorem finf
$L^{-1}\left\{\frac{1}{{\left(s-4\right)}^4(s+3)}\right\}$
5 M
7 (a) (ii) Find : - L(sin2t cost cosh2t)
3 M
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 M
7 (c) Find Fourier sin a integral of f(x) where
$f(x)=\left\{\begin{matrix}x & ; &02\end{matrix}\right.$
6 M

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