MU Applied Mathematics - 3 - December 2015 Exam Question Paper

MU Electronics Engineering (Semester 3)
Applied Mathematics - 3
December 2015

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) Evaluate

\int_{0}^{\infty} e^{- l} (\frac{\cos 3 t - \cos 2 t}{t}) d t

$\int^\infty_0 e^{-l} \left ( \dfrac {\cos 3t - \cos 2t}{t} \right )dt$

5 M

1 (b) Obtain the Fourier Series expression for f(x)=2x-1 in (0,3)

5 M

1 (c) Find the value of 'p' such that the function

f (z) = \frac{1}{2} \log (x^{2} + y^{2}) + i \tan^{- 1} (\frac{p y}{x})

$f(z) = \dfrac {1}{2} \log (x^2 + y^2)+ i \tan^{-1} \left ( \dfrac {py}{x} \right )$ is analytic.

5 M

1 (d) If

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

$\overline F = (y \sin z-\sin x)\widehat{i} + (x \sin z + 2yz)\widehat{j}+ (xy \cos z + y^2)\widehat{k}.$ Show that

\bar{F}

$\overline F$ is irrotational. Also find its scalar potential.

5 M

2 (a) Solve the differential equation using Laplace Transform

\frac{d^{2} y}{d t^{2}} + 2 \frac{d y}{d t} + y = 3 t e^{- l},

$\dfrac {d^2y}{dt^2}+ 2 \dfrac {dy}{dt}+ y = 3te^{-l},$ given y(0)=4 and y'(0)=2.

6 M

2 (b) Prove that

J_{4} (x) (\frac{48}{x^{3}} - \frac{8}{x}) J_{1} (x) - (\frac{24}{x^{2}} - 1) J_{0} (x)

$J_4 (x) \left ( \dfrac {48}{x^3} - \dfrac {8}{x} \right ) J_1 (x) - \left ( \dfrac {24}{x^2}-1 \right )J_0 (x)$

6 M

2 (c) (i) In what direction is the directional derivative of φ x², y², z⁴ at (3, -1, 2) maximum. Find its magnitude.

4 M

2 (c) (ii) if

\bar{r} = x \hat{i} + y \hat{j} + z \hat{k}

$\overline r = x\widehat i + y \widehat j + z\widehat k$ Prove that ∇rⁿ=m^n-2r

8 M

3 (a) Obtain the Fourier Series expansion for the function

\begin{aligned} f (x) & = 1 + \frac{2 x}{π}, - π \leq x \leq 0 \\ = 1 - \frac{2 x}{π}, 0 \leq x \leq π \end{aligned}

$\begin {align*}f(x) &=1+\dfrac {2x}{\pi}, \ -\pi \le x\le 0 \\ &= 1- \dfrac {2x}{\pi}, \ 0\le x \le \pi \end{align*}$

6 M

3 (b) Find an analytic function f(z)=u+iv where.

u - v = \frac{x - y}{x^{2} + 4 x y + y^{2}}

$u-v = \dfrac {x-y}{x^2 + 4xy + y^2}$

6 M

3 (c) Find Laplace transform of

i) \cosh t \int_{0}^{1} e^{u} \sinh u i i) t \sqrt{1 + \sin t}

$i) \ \cosh t\int^{1}_0 e^{u} \sinh u \\ ii) \ t\sqrt{1+\sin t}$

8 M

4 (a) Obtain the complex form of Fourier series for f(x)=e^m in (-L, L)

6 M

4 (b) Prove that

\int x^{4} J_{1} (x) d x = x^{4} J_{1} (x) - 2 x^{3} J_{3} (x) + c

$\int x^4 J_1 (x)dx=x^4J_1(x)-2x^3J_3(x)+c$

6 M

4 (c) Find

i) L^{- 1} [\frac{2 s - 1}{s^{2} + 4 s + 29}] i i) L^{- 1} [\cot^{- 1} (\frac{s + 3}{2})]

$i) \ L^{-1} \left [ \dfrac {2s-1}{s^2 + 4s + 29} \right ] \\ ii) \ L^{-1} \left [ \cot^{-1} \left ( \dfrac {s+3}{2} \right ) \right ]$

8 M

5 (a) Find the Bi-linear Transformation which maps the points 1, i, -1 of z plane onto 0, 1, ∞ of w-plane.

6 M

5 (b) Using Convolution theorem find

L^{- 1} [\frac{s^{2}}{(s^{2} + 4)^{2}}]

$L^{-1} \left [ \dfrac {s^2}{(s^2 + 4)^2} \right ]$

6 M

5 (c) Verify Green's Theorem for

\int_{c} \bar{F} . \bar{d r}

$\int_c \overline F.\overline {dr}$ where

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

$\overline {F} = (x^2 - y^2)\widehat{i}+ (x+y)\widehat{j}$ and C is the triangle with vertices (0, 0), (1, 1) and (2, 1).

8 M

6 (a) Obtain half range sine series for

\begin{aligned} f (x) & = x, 0 \leq x \leq 2 \\ = 4 - x, 2 \leq x \leq 4 \end{aligned}

$\begin {align*} f(x) &= x, 0\le x \le 2 \\ &=4-x, 2\le x \le 4 \end{align*}$

6 M

6 (b) Prove that the transformation

w = \frac{1}{z + l}

$w = \dfrac {1}{z+l}$ transforms the real axis of the z-plane into a circle in the w-plane.

6 M

6 (c) (i) Use Stoke's Theorem to evaluate

\int_{c} \bar{F} . \bar{d r}

$\int_c \overline F. \overline {dr}$ where

\bar{F} = (x^{3} - y^{2}) \hat{i} + 2 x y \hat{j}

$\overline {F}=(x^3 - y^2) \widehat{i}+2xy\widehat{j}$ and C is the rectangle in the plane z=0, bounded by x=0, y=0, x=a and y=b.

4 M

6 (c) (ii) Use Gauss Divergence Theorem to evaluate.

\iint_{s} \bar{F} . \hat{n} d s where \bar{F} = 4 x \hat{i} + 3 y \hat{j} - 2 z \hat{k}

$\iint_s \overline F. \widehat{n}ds \text{ where }\overline F= 4x\widehat{i} + 3y\widehat{j} - 2z\widehat{k}$ and S is the surface bounded by x=0, y=0 and 2x+2y+z=4.

4 M

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