STUPIDSID
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 M
1(b)
Show that the set of functions {sin(2n+1)x}, n=0, 1, 2, ..... is orthogonal in the Interval \( \left [ 0,\dfrac{\pi}{2} \right ] \). Hence construct corresponding orthogonal set of functions.
5 M
1(c)
Find the values sine and cosine transforms of f(x)= xm-1
5 M
1(d)
For what values of x and y the given partial differential equation is hyperbolic, parabolic or elliptic \( (y+1)\dfrac{\partial^2 u}{\partial x^2}+2x\dfrac{\partial^2 u}{\partial x \partial y}+\dfrac{\partial^2 }{\partial y^2}=x+y \)
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 \( \int ^{\infty}_0 \dfrac{\sin x}{x}dx \)
Hence find the value of \( \int ^{\infty}_0 \dfrac{\sin x}{x}dx \)
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, \( x=\dfrac{\pi}{2} \), \( y=\dfrac{2x}{\pi} \)
8 M
3(a)
If \( \overrightarrow {F}=2xyzt+(x^2z+2y))\hat{j}+(x^2y)\hat{k} \) then
I) Prove that \( \overrightarrow {F} \) 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).
I) Prove that \( \overrightarrow {F} \) 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 \( \dfrac{\partial^2 u}{\partial t^2}=C^2\dfrac{\partial^2 u}{\partial x^2}.A \) string is stretched and fastened to two points l apart. Motion is started by displacing the string in the form \( y=asin\left ( \dfrac{\pi x}{l} \right ) \) 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 \( y(x,t)=a\sin\frac{\pi x}{l}\cos \frac{\pi ct}{l} \).
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 \( \dfrac{\pi}{4}=1-\dfrac{1}{3}+\dfrac{1}{5}-\dfrac{1}{7}+\cdots \)
f(x)=πx, 0≤x<1;
=0, x=1
=π(x-2), 1
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 \( \int \int _S x^3dydz+x^2ydzdx+x^2zdxdy \) 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 \( \dfrac{\partial u}{\partial t}=C^2 \dfrac{\partial^2 u}{\partial x^2} \).
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.
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), \( \left ( \dfrac{\pi}{2},0 \right ),\left ( \dfrac{\pi}{2},\dfrac{\pi}{2} \right ),\left ( 0,\dfrac{\pi}{2} \right ) \).
6 M
5(b)
Find the Fourier sine integral of the function
f(x)=x, 0 =2-x, 1 =0, x>2
f(x)=x, 0
6 M
5(c)
Solve \( \dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=0 \) 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
\[\overrightarrow {F}=(x+y)\hat{i}+(2s-z)\hat{j}+(y+z)\hat{k}\].
\[\overrightarrow {F}=(x+y)\hat{i}+(2s-z)\hat{j}+(y+z)\hat{k}\].
6 M
6(b)
Find the half range sine series of
f(x)=x, 0≤ x ≤ 2;
=4-x, 2≤ x≤ 4
f(x)=x, 0≤ x ≤ 2;
=4-x, 2≤ x≤ 4
6 M
6(c)
Find the Fourier cosine transform of \( f(x)=\dfrac{1}{1+x^2} \). Hence derive the Fourier sine transform of \( f(x)=\dfrac{x}{1+x^2} \)
8 M
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