Question Paper: Applied Mathematics - 4 Question Paper - December 2015 - Chemical Engineering (Semester 4) - Mumbai University (MU)
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## Applied Mathematics - 4 - December 2015

### MU Chemical Engineering (Semester 4)

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 (ex dx+2ydy-dx) where C is the curve bounded by x 2 +y2 =4, z=2. 5 marks

1(b) Show that the set of functions {sin(2n+1)x}, n=0, 1, 2, ..... is orthogonal in the Interval [0,π2][0,π2] \left [ 0,\dfrac{\pi}{2} \right ] . Hence construct corresponding orthogonal set of functions. 5 marks

1(c) Find the values sine and cosine transforms of f(x)= xm-1 5 marks

1(d) For what values of x and y the given partial differential equation is hyperbolic, parabolic or elliptic (y+1)∂2u∂x2+2x∂2u∂x∂y+∂2∂y2=x+y(y+1)∂2u∂x2+2x∂2u∂x∂y+∂2∂y2=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 marks

2(a) Find the Fourier series of f(x)=x|x| in the Interval (-1, 1) 5 marks

2(b) Find the Fourier transform of f(x) =1,    |x|  =0,    |x|>a
Hence find the value of ∫∞0sinxxdx∫0∞sin⁡xxdx \int ^{\infty}_0 \dfrac{\sin x}{x}dx
5 marks

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=π2x=π2 x=\dfrac{\pi}{2} , y=2xπy=2xπ y=\dfrac{2x}{\pi} 6 marks

3(a) If →F=2xyzt+(x2z+2y))^j+(x2y)^kF→=2xyzt+(x2z+2y))j^+(x2y)k^ \overrightarrow {F}=2xyzt+(x^2z+2y))\hat{j}+(x^2y)\hat{k} then
I) Prove that →FF→ \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 marks

3(b) The vibrations of an elastic string is governed by the partial differential equation ∂2u∂t2=C2∂2u∂x2.A∂2u∂t2=C2∂2u∂x2.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=asin(πxl)y=asin(πxl) 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)=asinπxlcosπctly(x,t)=asin⁡πxlcos⁡πctl y(x,t)=a\sin\frac{\pi x}{l}\cos \frac{\pi ct}{l} . 6 marks

3(c) Find the Fourier series of
f(x)=πx, 0≤x<1;
=0, x=1
=π(x-2), 1<x≤2 <br=""> Hence deduce that π4=1−13+15−17+⋯π4=1−13+15−17+⋯ \dfrac{\pi}{4}=1-\dfrac{1}{3}+\dfrac{1}{5}-\dfrac{1}{7}+\cdots </x≤2>
6 marks

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 marks

4(b) Evaluate ∫∫Sx3dydz+x2ydzdx+x2zdxdy∫∫Sx3dydz+x2ydzdx+x2zdxdy \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 marks

4(c) The equation of one dimensional heat flow is given by ∂u∂t=C2∂2u∂x2∂u∂t=C2∂2u∂x2 \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.
6 marks

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), (π2,0),(π2,π2),(0,π2)(π2,0),(π2,π2),(0,π2) \left ( \dfrac{\pi}{2},0 \right ),\left ( \dfrac{\pi}{2},\dfrac{\pi}{2} \right ),\left ( 0,\dfrac{\pi}{2} \right ) . 6 marks

5(b) Find the Fourier sine integral of the function
f(x)=x, 0<x&lt;1 <br="">   =2-x, 1<x&lt;2 <br="">   =0, x>2</x&lt;2></x&lt;1>
6 marks

5(c) Solve ∂2u∂x2+∂2u∂y2=0∂2u∂x2+∂2u∂y2=0 \dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=0 for 0<x&lt;π, 0<y&lt;π="" ,="" with="" conditions="" given:="" u(0,="" y)="u(π," π)="0,u(x," 0)="sin^&lt;sup">2x.</x&lt;π,&gt;<> 6 marks

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
<mover><mi>F</mi><mo>→</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>i</mi><mo stretchy="false">^</mo></mover></mrow><mo>+</mo><mo stretchy="false">(</mo><mn>2</mn><mi>s</mi><mo>−</mo><mi>z</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>j</mi><mo stretchy="false">^</mo></mover></mrow><mo>+</mo><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">^</mo></mover></mrow>[/itex]" role="presentation" style="font-size: 125%; text-align: center; position: relative;">F=(x+y)^i+(2sz)^j+(y+z)^k<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>F</mi><mo>→</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>i</mi><mo stretchy="false">^</mo></mover></mrow><mo>+</mo><mo stretchy="false">(</mo><mn>2</mn><mi>s</mi><mo>−</mo><mi>z</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>j</mi><mo stretchy="false">^</mo></mover></mrow><mo>+</mo><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mover><mi>k</mi><mo stretchy="false">^</mo></mover></mrow>[/itex]<script type="math/tex; mode=display" id="MathJax-Element-16">\overrightarrow {F}=(x+y)\hat{i}+(2s-z)\hat{j}+(y+z)\hat{k}</script>.
6 marks

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

6(c) Find the Fourier cosine transform of f(x)=11+x2f(x)=11+x2 f(x)=\dfrac{1}{1+x^2} . Hence derive the Fourier sine transform of f(x)=x1+x2f(x)=x1+x2 f(x)=\dfrac{x}{1+x^2} 6 marks