MU Applied Mathematics - 3 - May 2013 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 3)
Applied Mathematics - 3
May 2013

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) Show that

f (z) = \frac{\bar{z}}{{| z |}^{2}}, | z | ? 0

$f(z)=\frac{\bar{z}}{{\left\vert{}z\right\vert{}}^2} \ \ \ \ \ \ , |z|?0$ is analytic function.Hence find f'(z)

5 M

1 (b) Find the Fourier Expansion of f(x)=sinx in (-?,?).

5 M

1 (c) Find the Laplace Transform of t?

\sqrt{1 + s i n t}

$\sqrt{1+sint}$

5 M

1 (d) Find z transformation of { a^k sin ak}, k ? 0. where a is constant

5 M

2 (a) Using Laplace Transform evaluate

\int_{0}^{\infty} e^{- t} \frac{\sin 3 t}{t} d t .

$\int_0^{\infty{}}e^{-t}\frac{\sin{3t}}{t}dt.\ \ \ \ \ \ \ \ \ \ \ \ \ \$

6 M

2 (b) Find the Fourier expansion of f(x)=cospx where p is not an integer in (0,2?)

7 M

2 (c) Find the matrix A, if adj A =

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

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

7 M

3 (a) Find inverse Laplace transform of
(I) log (s-2/s-3)
(II) s+1/s² - 4

6 M

3 (b) Find non Singular matrices P and Q such that PAQ is in normal form. Also find rank of a matrix A where
A =

\begin{array}{cccc} 2 & - 4 & 3 & 1 & 0 \\ 1 & - 2 & 1 & - 4 & 2 \\ 0 & 1 & - 1 & 3 & 1 \\ 4 & - 7 & 4 & - 4 & 5 \end{array}

$\begin{array}{cccc}2 & -4 & 3 & 1 & 0 \\1 & -2 & 1 & -4 & 2 \\0 & 1 & -1 & 3 & 1 \\4 & -7 & 4 & -4 & 5\end{array}$

7 M

3 (c) Verify Green's Theorem in the plane for ?_c (xy+y²) dx+x² by where c is a closed
curve of a region bounded by y=x and y²=x

7 M

4 (a) Obtain Complex form of Fourier series for f(x)=e^-ax in (-2,2)where a is not an integer.

6 M

4 (b) if A=

[\begin{array}{ccc} 1 & 1 & 1 \\ 2 & 5 & 7 \\ 2 & 1 & - 1 \end{array}]

$\left[\begin{array}{ccc}1 & 1 & 1 \\2 & 5 & 7 \\2 & 1 & -1\end{array}\right]$ compute A ^l and hence, Solve the system of equation
x + y + z = 9, 2x + 5y + 7z =52, 2x + y - z = 0.

7 M

4 (c) Find the Laplace transform of
f(t) = 1, 0 ? t ?a
f(t) =1, a < t ? 2a &
f(t + 2a)= f(t)

7 M

5 (a) Find the analytic function f(z)=u+iv if u=(r+ a²/r) cos ?.

6 M

5 (b) Show that the equations.
ax + by +cz =0
bx + cy + az =0
cx + ay + bz = 0
has non trivial solution if a+b+c=0 or if a=b=c. Find the non trivial solution when the condition is satisfied.

7 M

5 (c) Find Fourier integral representing f(x) =

{\begin{array}{l} 1 - x^{2} | x | \leq 1 \\ 0 | x | > 1 \end{array} .

$\left\{\begin{array}{l}1-x^2\ \ \ \ \ \ \ \ \left\vert{}x\right\vert{}\leq{}1 \\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\vert{}x\right\vert{}>1\end{array}.\right.$

7 M

6 (a) Find the half range cosine series for f(x)=2x-x² in (0,2)

6 M

6 (b) Find the Bilinear Transformation which maps the points 2,i,-2 on the points 1,i,-1

7 M

6 (c) Using Lapace transform solve the differential equation

\frac{d^{2} y}{{d t}^{2}} - 2 \frac{d y}{d t} - 8 y = 4, y (0) = 0, y^{^{'}} (0) = 1.

$\frac{d^2y}{{dt}^2}-2\frac{dy}{dt}-8y=4\ ,\ y\left(0\right)=0,\ y^{'}\left(0\right)=1.$

7 M

7 (a) Find inverse z-transform of F(z)= 1/(z-2)(z-3) if ROC is 2 < |z| < 3.

6 M

7 (b) Verify Stoke's theorem for

\bar{F} = x^{2} \bar{i} + x y \bar{j}

$\bar{F}=x^2\bar{i}+xy\bar{j}$ & C is the boundary of the rectangle x=0, x=2, y=3.

7 M

7 (c) Using Gauss Divergence Theorem evaluate

\iint_{S} \vec{F} ∙ \hat{n} d s w h e r e \vec{F} = 4 x \bar{i} + 3 y \bar{j} - 4 z^{2} \vec{k}

$\iint_S\vec{F}\bullet{}\hat{n}ds\ where\ \vec{F}=4x\bar{i}+3y\bar{j}-4z^2\vec{k}$
and S is the closed surface bounded the planes x=0,y=0,z=0 and 2x+2y+z=4

7 M

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