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

This Qs paper appeared for Applied Mathematics - 3 of Electronics Engineering . (Semester 3)

1 (a) Prove that f(z)=(x³-3xy²+2xy) + i(3x²y-x²+y²-y³) is analytic and find f'(z) & f(z) in terms of z

5 M

1 (b) Find the Fourier series expansion for f(x)=|x|, in (-?, ?) Hence deduce that

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

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

5 M

1 (c) Find the inverse Laplace transform of

\frac{e^{- z^{3}}}{s^{2} - 2 s + 2}

$\frac{e^{-z^3}}{s^2-2s+2}$

5 M

1 (d)

I f {f (k)} = {2^{0}, 2^{1}, 2^{3}, . . .}

$If\ \left\{\ f\left(k\right)\right\}=\left\{\ 2^0,2^1,\ 2^3,...\right\}$ Find Z{ f(k) }

5 M

2 (a) Evaluate

\int_{0}^{\infty} e^{- 2 t} \sinh t \frac{\sin t}{t} d t

$\int_0^{\infty{}}e^{-2t}\sinh{t\frac{\sin t}{t}}\ dt$

6 M

2 (b) Find the Fourier series expansion for

f (x) = {(\frac{π - x}{2})}^{2}

$f\left(x\right)={\left(\frac{\pi{}-x}{2}\right)}^2$ in the interval
0 ? x ? 2? & f(x+2?)=f(x) Also deduce that

\frac{π^{2}}{6} = \frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} . . .

$\frac{{\pi{}}^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}...$

7 M

2 (c) Show that

[\begin{array}{cc} 1 & - \tan \frac{θ}{2} \\ \tan \frac{θ}{2} & 1 \end{array}] [\begin{array}{cc} 1 & \tan \frac{θ}{2} \\ - \tan \frac{θ}{2} & 1 \end{array}] = [\begin{array}{cc} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{array}]

$\left[\begin{array}{cc}1 & -\tan{\frac{\theta{}}{2}} \\\tan{\frac{\theta{}}{2}} & 1\end{array}\right]\ \left[\begin{array}{cc}1 & \tan{\frac{\theta{}}{2}} \\-\tan{\frac{\theta{}}{2}} & 1\end{array}\right]=\left[\ \begin{array}{cc}\cos{\theta{}} & -\sin{\theta{}} \\\sin{\theta{}} & \cos{\theta{}}\end{array}\right]$

7 M

3 (a) Find Laplace Transform of following

(i) \int_{0}^{1} \frac{1 - e^{- a u}}{u} d u

$\left(i\right)\ \int_0^1\frac{1-e^{-au}}{u}\ du$

(i i) {(t \sinh 2 t)}^{2}

$\left(ii\right)\ \ {\left(t\sinh{2t}\right)}^2$

6 M

3 (b) Find non-singular matrices P & Q s.t. PAQ is in Normal form. Also find rank of A & A^-1

A = [\begin{array}{ccc} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{array}]

$A=\left[\begin{array}{ccc}1 & 3 & 3 \\1 & 4 & 3 \\1 & 3 & 4\end{array}\right]$

7 M

3 (c) Evaluate by Green's theorem

\int_{C}^{} [(3 x^{2} - 8 y^{2}) d x + (4 y - 6 x y) d y]

$\int_C^{\ }\ \left[\left(3x^2-8y^2\right)\ dx+\left(4y-6xy\right)\ dy\right]$ where C is the boundary of the region bounded by

y = \sqrt{x}, y = x^{2}

$y=\sqrt{x},\ \ y=x^2$

7 M

4 (a) Obtain complex form of Fourier series for the functions f(x)= e^ax in (0,a)

6 M

4 (b) For what value of ? the equations x+y+z=1, x+2y+4z=?, x+4y+10z=?² have a solution and solve them completely in each case.

7 M

4 (c) Find inverse Laplace Transform of following

(i) \log (1 + \frac{a^{2}}{s^{2}})

$\left(i\right)\ \log{\left(1+\frac{a^2}{s^2}\right)}$

(i i) \frac{s}{{(s + 1)}^{2} (s^{2} + 1)}

$\left(ii\right)\ \frac{s}{{\left(s+1\right)}^2\left(s^2+1\right)}$

7 M

5 (a) Prove that u=e³ cos y+x³ - 3xy² is harmonic

6 M

5 (b) Determine the linear dependence of vectors [2, -1, 3, 2], [1,3,4,2], & [3,-5,2,2] Find the relation between them if dependent

7 M

5 (c) Using Fourier Cosine integral prove that

e^{- x} \cos x = \frac{2}{π} i n t_{0}^{\infty} \frac{(ω^{2} + 2)}{(ω^{4} + 4)} . c o s ω x d ω

$e^{-x}\cos{x=\frac{2}{\pi{}}\\int_0^{\infty{}}\frac{\left({\omega{}}^2+2\right)}{\left({\omega{}}^4+4\right)}.\\cos{\omega{}\ x\ d\omega{}}}$

7 M

6 (a) Obtain half-range sine series for f(x)= x (2-x) in 0

6 M

6 (b) Find the bilinear transformation which maps the points 0, I, -2i of z-plane on to the points -4i, ?, 0 respectively of w-plane. Also obtain fixed points of the transformation

7 M

6 (c) Verify Stoke's theorem for F=yzi + zxj+xy k and c is the boundary of the circel x²+y²+z²=1, z=0

7 M

7 (a) Find inverse Z-transform of

F (z) = \frac{z}{(z - 1) (z - 2)}, | z | > 2

$F(z)=\dfrac {z}{(z-1)(z-2)},|z|>2$

6 M

7 (b) Find the analytic function f(z)= u+iv in terms of z if u-v=e^x (cos y - sin y)

7 M

7 (c) Using laplace trasnform solve the following differential equation with given condition. (D²-2D+1) x=e^t, with x=2, Dx=-1, at t=0

7 M

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