MU Applied Mathematics - 3 - December 2013 Exam Question Paper

MU Electronics and Telecom Engineering (Semester 3)
Applied Mathematics - 3
December 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) Prove that real and imaginary parts of an analytic function F(z)=u+iv are harmonic function.

5 M

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

5 M

1 (c) Find the Laplace Transform of

\int_{0}^{t} {u e}^{- 3 u} \sin 4 u d u

$\int_0^t\ {ue}^{-3u}\sin{4udu}$

5 M

1 (d)

I f \vec{F} = x y e^{2 z} i + x y^{2} c o s z j + x^{2} c o s x y k . T h e n f i n d d i v \bar{F} a n d \curl \bar{F .}

$If\ \ \vec{F}=xye^{2z}i+xy^2coszj\ \ \ +x^2cosxyk.Then\ find\ div\ \bar{F\ }and\curl\ \bar{F.}$

5 M

2 (a) Using laplace transofrm solve-
(D² + 3D + 2) y= e^-2t. Sin t where y(0) = 0 and y' (0) =0.

6 M

2 (b) Find the directional derivative of d=x² y cos?z at (1,2, ?/2)in the direction of t=2i + 3j + 2k.

6 M

2 (c) Find the Fourier expansion of

f (x) = \sqrt{1 - c o s x} i n (0, 2 π) . H e n c e p r o v e f r a c 12 = \sum_{n = 1}^{\infty} \frac{1}{4 n^{2} - 1} .

$f(x)=\sqrt{1-cosx}\ in\left(0,2\pi{}\right).Hence\ prove\\frac{1}{2}=\sum_{n=1}^{\infty{}}\frac{1}{4n^2-1}.$

8 M

3 (a) Prove that

J_{\frac{3}{2}} = \sqrt{\frac{2}{π x}} . (\frac{\sinx}{x} - \cosx)

$J_{\frac{3}{2}}=\sqrt{\frac{2}{\pi{}x}}.\left(\frac{\sinx}{x}-\cosx\right)$

6 M

3 (b) Evaluate by Green's theorem

\oint_{c} (x^{2} y d x + y^{3} d y)

$\oint_c\left(x^2ydx+y^3dy\right)$ where c is closed path formed by y=x,y=x²

6 M

3 (c) i) Find the Laplace Transform of

\frac{\cos b t - \cos a t}{t}

$\frac{\cos{bt-\cos{at}}}{t}$
ii) Find the Laplace Transform of

\frac{d}{d t} [\frac{s i n t}{t}] .

$\frac{d}{dt}\left[\frac{sint}{t}\right].$

8 M

4 (a) Show that the set of functions {sin??x,sin??3x?} OR {sin??(2n+1)x:n=0,1,2,3} is orthogonal over [0,??2],Hence construct orthonormal set of functions.

6 M

4 (b) Find the imaginary part whose real part is u= x³ - 3xy² + 3x² + 1

6 M

4 (c) Find Inverse Laplace Transform of?

i) l o g (\frac{s^{2} + 4}{s^{2} + 9})

$i)log\left(\frac{s^2+4}{s^2+9}\right)$

i i) \frac{s}{(s^{2} + 4) (s^{2} + 9)}

$ii)\frac{s}{\left(s^2+4\right)\left(s^2+9\right)}$

8 M

5 (a) Obtain half range sine series for f(x)=x² in 0

6 M

5 (b) A Vector field F is given by

\bar{F} = (x^{2} - y z) \hat{i} + (y^{2} - z x) \hat{j} + (z^{2} - x y) \hat{k}

$\bar{F}=\left(x^2-yz\right)\hat{i}+\left(y^2-zx\right)\hat{j}+\left(z^2-xy\right)\hat{k}$ is irroational and hence find scalar point function ? such that F = ? ?

6 M

5 (c) Show that the function V=e^x (xsiny+ycosy) satisfies Laplace equation and find its corresponding analytic function

8 M

6 (a) By using stoke's theorem ,evaluate

\oint_{c} [(x^{2} + y^{2}) \hat{i} + (x^{2} - y^{2}) \hat{j}] ∙ d \bar{r}

$\oint_c\left[\left(x^2+y^2\right)\hat{i}+\left(x^2-y^2\right)\hat{j}\right]\bullet{}d\bar{r}$
where c is the boundary of a region enclosed by circles x² + y² =4, x^{2 + y² = 16.}

6 M

6 (b) Show that under the transformation w= 5-4z/4z-2 the circle |z|=1 in the z plane is transformed into a circle of unity in w-plane.

6 M

6 (c) Prove that

\intJ_{3} (x) d x = - \frac{2 J_{1} (x)}{x} - J_{2} (x)

$\intJ_3\left(x\right)dx=\ -\frac{2J_1(x)}{x}-J_2(x)$

8 M

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