MU Applied Mathematics - 3 - December 2011 Exam Question Paper

MU Electronics and Telecom 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

1 (a) Find L.T. of

f (t) = f (x) = {\begin{aligned} 1, 0 < t < a \\ - 1, a < t < 2 a \end{aligned}

$f\left(t\right)=\ f\left(x\right)= \Bigg\{\begin{align*}{}1,\ \ \ \ \ 0<t<a \\-1,\ \ \ \ a<t<2a\end{align*}$
And f(t) = f(t+2a)

5 M

1 (b) Find the Fourier series of f(x) = cos μx in (-π,π), where μ 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

[A = \begin{array}{ccc} 3 & p & p \\ p & 3 & p \\ p & p & 3 \end{array}]

$\left[A=\begin{array}{ccc}3 & p & p \\p & 3 & p \\p & p & 3\end{array}\right]$

5 M

1 (d) Find Z-transform of {k² - 2k + 3}_{k ≥ 0}

5 M

2 (a) Solve by using L.T.

\frac{d y}{d t} + 2 y + \int_{0}^{t} y d t = \sin t w h e n y (0) = 1

$\dfrac {dy}{dt}+2y+\int^{t}_{0}y \ dt=\sin t \ when \ y \ (0)=1$

8 M

2 (b) Find Fourier series for f(x) = x + x² in (-π , π) Hence deduce that

\frac{1}{1^{2}} + \frac{1}{3^{2}} + \frac{1}{5^{2}} + \frac{1}{7^{2}} + . . . . = \frac{π^{2}}{8}

$\dfrac{1}{1^2}+ \dfrac{1}{3^2}+ \dfrac{1}{5^2} +\frac{1}{7^2} +....= \dfrac{{\pi}^2}{8}$

6 M

2 (c) Show that vectors X₁, X₂, X₃ are linearly independent and vector X₄ depends upon them where

X_{1} = (1, 2, 4)

$X_1=\ \left(1,2,4\right)$

X_{2} = (2, - 1, 3)

$X_2=\left(2,-1,3\right)$

X_{3} = (0, 1, 2) a n d X_{4} = (- 3, 7, 2)

$X_3=\left(0,1,2\right)\ and \\ X_4=\left(-3,7,2\right)$

6 M

3 (a) Find the Fourier integral representation of the function

f (x) = {\begin{cases} 1 - x^{2}, & w h e n | x | & \leq 1 \\ x, & w h e n | x | & > 1 \end{cases}

$f\left(x\right)=\left\{\begin{array}{l}1-x^2,\ \ \ & \ \ when\ \vert{}x\vert{}\ & \leq{}1 \\ x,\ \ \ & \ \ when\ \vert{}x\vert{}\ & \ >{}1\end{array}\right.$
And hence evaluate

\int_{0}^{\infty} [\frac{x \cos x - \sin x}{x^{3}}] \cos \frac{x}{2} d x

$\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

[A = \begin{array}{ccc} - 2 & 1 & 3 \\ - 2 & - 3 & 11 \\ 2 & 1 & - 5 \end{array}]

$\left[A=\begin{array}{ccc}-2 & 1 & 3 \\-2 & -3 & 11 \\2 & 1 & -5\end{array}\right]$

6 M

3 (c)

F i n d L {\frac{1 - c o s t}{t^{2}}}

$Find L \ \left\{\frac{1-cost}{t^2}\right\}$

6 M

4 (a)

(i) F i n d L^{- 1} {{t a n}^{- 1} {(s + 2)}^{2}} (i i) F i n d L {t^{2} H (t - 2) - c o s h t δ (t - 4)}

${(i) Find L^{-1}\left\{{tan}^{-1}{\left(s+2\right)}^2\right\} } \\ {(ii) Find L\left\{t^2H\left(t-2\right)-cos h t\ \delta{}\left(t-4\right)\right\}}$

6 M

4 (b) Find inverse Z-transform of

\frac{z (z + 1)}{(z - 1) (z^{2} + z + 1)}

$\frac{z\left(z+1\right)}{\left(z-1\right)\left(z^2+z+1\right)}$

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 = [\begin{array}{ccc} 3 & 2 & - 1 & 5 \\ 5 & 1 & 4 & - 2 \\ 1 & - 4 & 11 & 19 \end{array}]

$A=\left[\ \begin{array}{ccc}3 & 2 & -1 & 5 \\5 & 1 & 4 & -2 \\1 & -4 & 11 & 19\end{array}\right]$

6 M

5 (a) Find half range cosine series for

f (x) = {\begin{cases} x, & 0 < x < \frac{π}{2} \\ π - x, & \frac{π}{2} < x < π \end{cases}

$f\left(x\right)=\left \{\begin{array}{l}x,\ \ \ & 0 < x < \frac{\pi{}}{2} \\\pi{}-x,\ \ \ & \frac{\pi{}}{2}<x<\pi{}\end{array}\right.$
hence find the sum

\sum_{n = 1}^{\infty} \frac{1}{n^{4}}

$\sum^{\infty}_{n=1} \dfrac {1}{n^4}$

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^{3} e}^{- t} s i n t d 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 (-π,π)

4 M

6 (a) (ii) (ii) Show that the functions

{\sin (2 n - 1)}_{n = 0}^{\infty}

${\left\{\sin{\left(2n-1\right)}\right\}}_{n=0}^{\infty{}}$ are orthogonal on

[0, \frac{π}{2}]

$\left[0,\frac{\pi{}}{2}\right]$ 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 (k - 1)}

$\{k^2\ a^{k-1}\ \cup{}\ \left(k-1\right)\}$

6 M

7 (a) (i) By using cinvolution theorem finf

L^{- 1} {\frac{1}{{(s - 4)}^{4} (s + 3)}}

$L^{-1}\left\{\frac{1}{{\left(s-4\right)}^4(s+3)}\right\}$

5 M

7 (a) (ii) Find : - L(sin2t cost cosh2t)

3 M

7 (b) Find inverse Z-transform of

\frac{2 z^{2} - 10 z + 13}{{(z - 3)}^{2} (z - 2)}

$\frac{2z^2-10z+13}{{\left(z-3\right)}^2\left(z-2\right)}$
if R.O.C. is 2|z|<3.

6 M

7 (c) Find Fourier sin a integral of f(x) where

f (x) = {\begin{matrix} x & ; & 0 < x < 1 \\ 2 - x & ; & 1 < x < 2 \\ 0 & ; & x > 2 \end{matrix}

$f(x)=\left\{\begin{matrix} x & ; &0<x <1 \\ 2-x&; &1<x<2 \\ 0&; & x>2 \end{matrix}\right.$

6 M

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