MU Applied Mathematics - 3 - May 2012 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 3)
Applied Mathematics - 3
May 2012

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) If f(z)= (ax⁴ + bx²y² + cy⁴ + dx² - 2y²) + i(4x³y - exy³ + 4xy) is analytic, find the constants a,b,c,d,e

5 M

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

5 M

1 (c) Find the Laplace transform of sin t?

H (t - \frac{π}{2}) - H (t - \frac{3 π}{2})

$H\left(t-\frac{\pi{}}{2}\right)-H\left(t-\frac{3\pi{}}{2}\right)$

5 M

1 (d)

I f {f (x)} = {\begin{cases} 4^{k}, & f o r k < 0 \\ 3^{k}, & f o r k \geq 0 \end{cases} f i n d Z {f (k)}

$If\ \ \left\{f\left(x\right)\right\}=\left\{\begin{array}{l}4^k,\ \ \&for\ k<0\\3^k,\ \ \&for\ k\geq{}0\end{array}\right.\ \ find\ Z\left\{f\left(k\right)\right\}$

5 M

2 (a)

i f \int_{0}^{\infty} e^{- 2 t} \sin (t + α) \cos (t - α) d t = \frac{3}{8} t h e n f i n d α

$if \ \int^{\infty}_{0}e^{-2t} \sin (t+\alpha) \cos (t-\alpha)dt=\dfrac {3}{8} \ then \ find\ \alpha$

6 M

2 (b) Find the Fourier series expnasion for

f (x) = \sqrt{1 - \cos x} i n (0, 2 ?) H e n c e d e d u c e t h a t \sum_{n = 1}^{\infty} \frac{1}{4 n^{2} - 1} = \frac{1}{2}

$f(x)= \sqrt{1-\cos x} in (0,2?) \ Hence \ deduce \ that \ \sum_{n=1}^{\infty}\dfrac{1}{4n^2 -1}=\dfrac{1}{2}$

7 M

2 (c) Find the inverse of A if

[\begin{array}{ccc} 1 & 0 & 0 \\ 2 & - 1 & 0 \\ - 2 & 1 & 1 \end{array}] A [\begin{array}{ccc} 1 & - 2 & 9 \\ 0 & 1 & - 6 \\ 0 & 0 & 1 \end{array}] = [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

$\left[\begin{array}{ccc}1 & 0 & 0 \\2 & -1 & 0 \\-2 & 1 & 1\end{array}\right]\ A\ \left[\begin{array}{ccc}1 & -2 & 9 \\0 & 1 & -6 \\0 & 0 & 1\end{array}\right]=\ \left[\begin{array}{ccc}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]$

7 M

3 (a) Find Laplace Transform of following

(i) e^{- 4 t} \int_{0}^{1} u \sin 3 u d u

$\left(i\right)\ e^{-4t}\ \int_0^1u\sin{3u\ du}$

(i i) \frac{1}{t} (1 - \cos t)

$\left(ii\right)\ \frac{1}{t}(1-\cos{t)}$

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 & 2 & 3 \\ 2 & 3 & 0 \\ 0 & 1 & 2 \end{array}]

$A=\left[\begin{array}{ccc}1 & 2 & 3 \\2 & 3 & 0 \\0 & 1 & 2\end{array}\right]$

7 M

3 (c) Evaluate by Green's theorem

\int_{C} \bar{F} \cdot d \bar{r} w h e r e \bar{F} = x y (x i - y i) a n d C i s r = a (1 + \cos θ)

$\int_C\bar{F}\cdot{}d\bar{r}\ where\ \ \bar{F}=xy\left(xi-yi\right) \ and \ C \ is \ r=a(1+ \cos \theta)$

7 M

4 (a) Obtain complex form of Fourier series for the functions f(x)= sin a x in (-?, ?)

6 M

4 (b) For what value of ?, the following system of equations possesses a non-trivial solution? Obtain the solution for real value of ?.
3x₁+x₂-? x₃=0, 4x₁2x₂-3x₃=0, 2? x₁+4x₂+? x₄=0

7 M

4 (c) Find inverse Laplace Transform of following
(i) 2 tanh^-1 s

(i i) \frac{s^{2}}{(s^{2} + 1) (s^{2} + 4)}

$\left(ii\right)\frac{s^2}{\left(s^2+1\right)\left(s^2+4\right)}$

7 M

5 (a) Find the orthogonal trajectory of the family of curves 3x³y+2x²-y³-2y²= c

6 M

5 (b) Find the relation of linear dependence amongst the rows of the matrix

A = [\begin{array}{ccc} 1 & 1 & - 1 & 1 \\ 1 & - 1 & 2 & - 1 \\ 3 & 1 & 0 & 1 \end{array}]

$A=\left[\begin{array}{ccc}1 & 1 & -1 & 1 \\1 & -1 & 2 & -1 \\3 & 1 & 0 & 1\end{array}\right]$

7 M

5 (c) Express the function

f (x) = {\begin{cases} - e^{k x}, & f o r x < 0 \\ e^{- k x}, & f o r x > 0 \end{cases}

$f\left(x\right)=\left\{\begin{array}{l}-e^{kx},\ \ \&for\ x<0 \\e^{-kx},\ \ \&for\ x>0\end{array}\right.$ as Fourier integral and prove that

\int_{0}^{\infty} \frac{ω \sin ω x}{ω^{2} + k^{2}} d ω = \frac{π}{2} e^{- k x} i f x > 0, k > 0

$\int_0^{\infty{}}\frac{\omega{}\sin{\omega{}\ x}}{{\omega{}}^2+k^2}\ d\omega{}=\frac{\pi{}}{2}\ e^{-kx}\ \ if\ x>0,\ k>0$

7 M

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

6 M

6 (b) Show that under the transformation

w = \frac{5 - 4 Z}{4 z - 2}

$w= \dfrac {5-4Z}{4z-2}$ the circle |z|=1 in the z-plane is transformed into a circle of unity in the w-plane. Also find the center of the circle

7 M

6 (c) A vector field is given by F=3x³yi + (x³-2yz²) j+ (3z²-2y²z) k is irrational. Also find ? such that F= ? ?. Also evaluate the line integral from (2,1,1), (2,0,1)

7 M

7 (a) Find inverse Z-transformation of

F (z) = \frac{z}{{z - (\frac{1}{4})] [z - (\frac{1}{5})]}, \frac{1}{5} < | z | < \frac{1}{4}

$F \left(z\right)=\frac{z}{\left\{z-\left(\frac{1}{4}\right)\right]\ \left[z-\left(\frac{1}{5}\right)\right]},\frac{1}{5} <\left \vert{}z\right \vert{} <\frac{1}{4}$

6 M

7 (b) Find the analytic function f(z)=u+iv in terms of z if u-v=(x-y) (x² + 4xy + y²)

7 M

7 (c) Using laplace trasnform solve the following differential equation with given condition. (D²-3D+2) y=e^2t, y(0)=-3, y'(0)=5

7 M

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