MU Applied Mathematics - 4 - December 2015 Exam Question Paper

MU Chemical Engineering (Semester 4)
Applied Mathematics - 4
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 by Stokes Theorem ∮_c (e^x dx+2ydy-dx) where C is the curve bounded by x ² +y² =4, z=2.

5 M

1(b) Show that the set of functions {sin(2n+1)x}, n=0, 1, 2, ..... is orthogonal in the Interval

[0, \frac{π}{2}]

$\left [ 0,\dfrac{\pi}{2} \right ]$ . Hence construct corresponding orthogonal set of functions.

5 M

1(c) Find the values sine and cosine transforms of f(x)= x^m-1

5 M

1(d) For what values of x and y the given partial differential equation is hyperbolic, parabolic or elliptic

(y + 1) \frac{\partial^{2} u}{\partial x^{2}} + 2 x \frac{\partial^{2} u}{\partial x \partial y} + \frac{\partial^{2}}{\partial y^{2}} = x + y

$(y+1)\dfrac{\partial^2 u}{\partial x^2}+2x\dfrac{\partial^2 u}{\partial x \partial y}+\dfrac{\partial^2 }{\partial y^2}=x+y$

5 M

2(a) Find the Fourier series of f(x)=x|x| in the Interval (-1, 1)

6 M

2(b) Find the Fourier transform of f(x) =1, |x| =0, |x|>a
Hence find the value of

\int_{0}^{\infty} \frac{\sin x}{x} d x

$\int ^{\infty}_0 \dfrac{\sin x}{x}dx$

6 M

2(c) Verify Green's Theorem for ∮_c (y-sinx)dx+cosxdy where C is the plane triangle bounded by the lines y=0,

x = \frac{π}{2}

$x=\dfrac{\pi}{2}$ ,

y = \frac{2 x}{π}

$y=\dfrac{2x}{\pi}$

8 M

3(a) If

\vec{F} = 2 x y z t + (x^{2} z + 2 y)) \hat{j} + (x^{2} y) \hat{k}

$\overrightarrow {F}=2xyzt+(x^2z+2y))\hat{j}+(x^2y)\hat{k}$ then
I) Prove that

\vec{F}

$\overrightarrow {F}$ is irrotational
II) Find its scalar potential φ
III) Find the work done in moving a particle under this force field from (0, 1, 1) to (1, 2, 0).

6 M

3(b) The vibrations of an elastic string is governed by the partial differential equation

\frac{\partial^{2} u}{\partial t^{2}} = C^{2} \frac{\partial^{2} u}{\partial x^{2}} . A

$\dfrac{\partial^2 u}{\partial t^2}=C^2\dfrac{\partial^2 u}{\partial x^2}.A$ string is stretched and fastened to two points l apart. Motion is started by displacing the string in the form

y = a s i n (\frac{π x}{l})

$y=asin\left ( \dfrac{\pi x}{l} \right )$ from which it is released at time t=0. Show that the displacement of any point at a distance x from one end at time is given by

y (x, t) = a \sin \frac{π x}{l} \cos \frac{π c t}{l}

$y(x,t)=a\sin\frac{\pi x}{l}\cos \frac{\pi ct}{l}$ .

6 M

3(c) Find the Fourier series of
f(x)=πx, 0≤x<1;
=0, x=1
=π(x-2), 1 Hence deduce that

\frac{π}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots

$\dfrac{\pi}{4}=1-\dfrac{1}{3}+\dfrac{1}{5}-\dfrac{1}{7}+\cdots$

8 M

4(a) Find the complex form of Fourier series of f(x)=sinax in the interval (-π, x) where a is not an integer.

6 M

4(b) Evaluate

\int \int_{S} x^{3} d y d z + x^{2} y d z d x + x^{2} z d x d y

$\int \int _S x^3dydz+x^2ydzdx+x^2zdxdy$ where S is the closed surface consisting of the circular cylinder x² +y² =a², z=0 and z=b.

6 M

4(c) The equation of one dimensional heat flow is given by

\frac{\partial u}{\partial t} = C^{2} \frac{\partial^{2} u}{\partial x^{2}}

$\dfrac{\partial u}{\partial t}=C^2 \dfrac{\partial^2 u}{\partial x^2}$ .
A bar of 10cm long with Insulated sides has its ends A and B maintained at temperature 50°C and 100°C respectively, until steady-state conditions prevall. The temperature A is suddenly to raised 90°C and at the same time at B is lowered to 60°C. Find the temperature distribution in the bar at time t.

8 M

5(a) Evaluate by Green's theorem ∮_c (y³ -xy)dx+(xy+3xy²)dy where C is the bounded by the square with vertices (0,0),

(\frac{π}{2}, 0), (\frac{π}{2}, \frac{π}{2}), (0, \frac{π}{2})

$\left ( \dfrac{\pi}{2},0 \right ),\left ( \dfrac{\pi}{2},\dfrac{\pi}{2} \right ),\left ( 0,\dfrac{\pi}{2} \right )$ .

6 M

5(b) Find the Fourier sine integral of the function
f(x)=x, 0 =2-x, 1 =0, x>2

6 M

5(c) Solve

\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0

$\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=0$ for 02x.

8 M

6(a) Using Stokes' theorem find the work done in moving a particle once around the perimeter of the triangle with vertices at (2, 0, 0), (0, 3, 0) and (0, 0, 0) under the force field

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

$\overrightarrow {F}=(x+y)\hat{i}+(2s-z)\hat{j}+(y+z)\hat{k}$ .

6 M

6(b) Find the half range sine series of
f(x)=x, 0≤ x ≤ 2;
=4-x, 2≤ x≤ 4

6 M

6(c) Find the Fourier cosine transform of

f (x) = \frac{1}{1 + x^{2}}

$f(x)=\dfrac{1}{1+x^2}$ . Hence derive the Fourier sine transform of

f (x) = \frac{x}{1 + x^{2}}

$f(x)=\dfrac{x}{1+x^2}$

8 M

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