MU Applied Mathematics - 3 - May 2016 Exam Question Paper

MU Electronics Engineering (Semester 3)
Applied Mathematics - 3
May 2016

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

$\int ^{\infty}_0e^{-2t}\left ( \dfrac{\sinh t \sin t}{t} \right )dt$

5 M

1(b) Obtain the Fourier Series expression for
f(x) = 9-x² in (-3,3)

5 M

1(c) Find the value of 'p' sucj that the function f(z) expressed in polar co-ordinates as
f(z) = r³ cos p&theta

5 M

1(d) If

$\bar{F}=(y^2-z^2+3yz-2x)\hat{i}+(3xz+2xy)\hat{j}+(3xy-2xz+2z)\hat{k}.$
Show that

$\bar{F}$ is irrotational and solenoidal.

5 M

2(a) Solve the differential equation using Laplace Transform

$\dfrac{d^2y}{dt^2}+4\frac{dy}{dt}+8y=1$ , given y(0)=0 and y'(0)=1

6 M

2(b) Prove that

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

6 M

2(c)(i) Find the directional derivative of ϕ = 4xz³ - 3x²y²z at (2,-1,2) in the direction of

$2\hat{i}+3\hat{j}+6\hat{k}.$

8 M

2(c)(ii) if

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

$\nabla \log r=\dfrac{\bar{r}}{r^2}$

8 M

3(a) Show that { cosx, cos2x, cos3x.....} is a set of orthogonal fundtions over (-π, π), Hence construct an orthogonal set.

6 M

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

$u=\dfrac{x}{2}\log(x^2+y^2)-y\tan^{-1}\left ( \dfrac{y}{x} \right )+\sin x \cosh y$

6 M

Find the Laplace transform of

3(c)(i)

$\int ^1_0 ue^{-3u}\cos^2u du$

4 M

3(c)(ii)

$t\sqrt{1+\sin t}$

4 M

4(a) Find the Fourier Series for

$f(x)=\dfrac{3x^2-6\pi x+2\pi^2}{12}$ in (0,2π)
Hence deduce that

$\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}\cdots=\dfrac{\pi^2}{6}$

6 M

4(b) Prove that

$\int ^b_0 xJ_0(ax)dx=\dfrac{b}{a}J_1(ab)$

6 M

Find

4(c)(i)

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

4 M

4(c)(ii)

$L^{-1}\left [ \log\left ( \dfrac{s+2}{s^2-2s+17} \right ) \right ]$

4 M

5(a) Obtain the half range cosine series for \[\begin {align*} f(x)&=x, 0

6 M

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

6 M

5(c) Verify Green's Theorem for

$\int _c \overline{F} .\overline{br}$ where

$\bar{F}=(x^2-xy)\hat{i}+(x^2-y^2)\hat{j}$ and C is the curve bounded by x² = 2y and x=y

8 M

6(a) Show that the transformation

$w=\dfrac{i-iz}{1+z}$ maps the unit circle |z|=1 into real axis of w plane.

6 M

6(b) Using Convolution theorem find

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

6 M

6(c)(i) Use Gauss Divergence Theorem to evaluate

$\iint _s \bar{F}.\hat{n}ds\ \text{where}\ \bar{F}=x\hat{i}+y\hat{j}+z\hat{k}$ and S is the sphere x² + y² + z² = 9 and

$\hat{n}$ is the outward normal to S

8 M

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

$\int _c\overline{F}.\overline{dr}\ \text{where}\ \bar{F}=x^2\hat{i}-xy\hat{j}$ and C is the square in the plane z=0 and bounded by x=0, y=0, x=a and y=a.

8 M

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