MU Applied Mathematics - 3 - May 2012 Exam Question Paper

MU Electronics 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

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 (-π, π)

5 M

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

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

5 M

1 (d)

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

5 M

2 (a)

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

6 M

2 (b) Find the Fourier series expnasion for

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

7 M

2 (c) Find the inverse of A if

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

7 M

3 (a) Find Laplace Transform of following

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

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

6 M

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

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

7 M

3 (c) Evaluate by Green's theorem

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

7 M

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

6 M

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

7 M

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

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

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

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

7 M

5 (c) Express the function

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

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

7 M

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

6 M

6 (b) Show that under the transformation

$w= \dfrac {5-4Z}{4z-2}$ 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

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

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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