MU Applied Mathematics - 3 - December 2015 Exam Question Paper

MU Electronics and Telecom 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)

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

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

$f(x)=\dfrac {1}{2} \log (x^2 + y^2)+t\tan^{-1}\left ( \dfrac {py}{x} \right )$

5 M

1 (d)

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

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

5 M

2 (a) Solve the differential equations using Laplace Transform

$\dfrac {d^2y}{dt^2} + 2 \dfrac {dy}{dt}+ y = 3te^{-t}$ given y(0)=4 and y'(0)=2.

6 M

2 (b) Prove that

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

6 M

2 (c) (i) In what direction is the directional derivative of ϕ=x²y²z⁴ at (3, -1, 2) maximum. Find its magnitude.

4 M

2 (c) (ii)

$\text{If } \overline{r}=x\widehat{i}+y\widehat{j}+z\widehat{k} \ \text{prove that, } \nabla r^n=nr^{n-2}\overline{r}$

4 M

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

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

6 M

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

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

6 M

3 (c) Find Laplace transform of

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

8 M

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

6 M

4 (b) Prove that

$\int x^4 J_1 (x)dx = x^4 J_2(x)-2x^3J_3 (x) +c$

6 M

4 (c) Find

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

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

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

6 M

5 (c) Verify Green's Theorem for

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

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

8 M

6 (a) Obtain half range sine series for

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

6 M

6 (b) Prove that the transformation

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

6 M

6 (c) (i) Use Stroke's theorem to evaluate

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

$\overline {F} = (x^2 - y^2)\widehat{i} + 2xy \widehat{j}$ 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

$\iint_s \overline {F} \cdot \widehat {n}ds$ where

$\overline {F} 4x\widehat{i} + 3y\widehat{j},$ 2zk and S is the surface bounded by x=0, y=0, z=0 and 2x+2y+z=4.

4 M

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