STUPIDSID
1 (a)
Expand f(x)=x sin x as Fourier series in the interval (-π, π), hence deduce the following: \[ i) \ \dfrac {\pi}{2} = 1+ \dfrac {2}{1.3}- \dfrac {2}{3.5} + \dfrac {2}{5.7} \\ ii) \ \dfrac {\pi -2}{4} = \dfrac {1}{1.3} - \dfrac {1}{3.5}+ \dfrac {1}{5.7}+ \cdots \ \]
7 M
1 (b)
Find the half-range Fourier cosine series for the function. \[ f(x)= \left\{\begin{matrix}
kx &, \ 0\le x \le l/2 \\
k(l-x) &, \ l/2
6 M
1 (c)
Find the constant term and the first two harmonics in the Fourier series for f(x) given by the following table:
| x: | 0 | π/3 | 2π/3 | π | 4π/3 | 5π/3 | 2π |
| F(x): | 1.0 | 1.4 | 1.9 | 1.7 | 1.5 | 1.2 | 1.0 |
7 M
2 (a)
Find the Fourier transform of the function f(x)=xe-h|x|.
7 M
2 (b)
Find the Fourier sine transform of the \[ Function \ f(x)= \left\{\begin{matrix}
\sin x &, \ 0
6 M
2 (c)
Find the inverse Fourier sine Transform of \[ F_x (\alpha) = \dfrac {1} {\alpha} e^{-a\alpha} \ a>0. \]
7 M
3 (a)
Find various possible solution of one dimensional wave equation \[ \dfrac {\partial ^2 u}{\partial t^2} = C^2 \ \dfrac {\partial ^2 u}{\partial x^2} \] separable variable method.
7 M
3 (b)
Obtain solution of heat equation \[ \dfrac {\partial u}{\partial t}= C^2 \dfrac {\partial ^2 u}{\partial t^2} \] subject to condition u(0, t)=0, u(l, t)=0, u(x,0)=f(x).
6 M
3 (c)
Solve Laplace equation \[ \dfrac {\partial^2 u}{\partial x^2} + \dfrac {\partial^2 u}{\partial y^2}= 0
\] subject to condition u(0,y)=u, u(x,0)=0. \[
u(x,a)= \sin \left ( \dfrac {\pi x}{l} \right ) \]
7 M
4 (a)
The pressure P and volume V of a gas are related by the equation PVr=K, where r and K are constant. Find this equation to the following set of observations (in appropriate units)
| P: | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 |
| V: | 1.62 | 1.00 | 0.75 | 0.62 | 0.52 | 0.46 |
7 M
4 (b)
Solve the following LPP using the Graphical method:
Maximize: Z=3x1+4x2
Under the constraints
4x1+2x2≤80
2x1+5x2≤180
x1, x2 ≥ 0
Maximize: Z=3x1+4x2
Under the constraints
4x1+2x2≤80
2x1+5x2≤180
x1, x2 ≥ 0
6 M
4 (c)
Solve the following using simplex method,
Maximize: Z=2x+4y, subjected t the
Constraint: 3x+y ≤2z, 2x+3y≤24, x≥0, y≥0.
Maximize: Z=2x+4y, subjected t the
Constraint: 3x+y ≤2z, 2x+3y≤24, x≥0, y≥0.
7 M
5 (a)
Using the Regular-Falsi method, find a real root (correct to three decimal places) of the equation cos x=3x-1 that lies between 0.5 and 1 (Here, x is in radians).
7 M
5 (b)
By relaxation method
Solve: x+6y+27z=85, 54x+y+z=110, 2x+15y+6z=72
Solve: x+6y+27z=85, 54x+y+z=110, 2x+15y+6z=72
6 M
5 (c)
Using the power method, find the largest Eigen value and corresponding Eigen vectors of the matrix \[ A= \begin{bmatrix}
6 &-2 &2 \\-2
&3 &-1 \\2
&-1 &3
\end{bmatrix} \] taking [1, 1, 1]T as the initial Eigen vectors. Perform 5 iterations.
7 M
6 (a)
From the data given in the following Table: find the number of students who obtained
(i) Less than 45 marks
(ii) between 40 and 45 marks
(i) Less than 45 marks
(ii) between 40 and 45 marks
| Marks | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
| No. of Students | 31 | 42 | 51 | 35 | 31 |
7 M
6 (b)
Using the Lagrange's formula, find the interpolating polynomial that approximetes to the function described by the following table:
Hence find f(0.5) and f(0.31)
| X | 0 | 1 | 2 | 3 | 4 |
| f(x) | 3 | 6 | 11 | 18 | 27 |
Hence find f(0.5) and f(0.31)
6 M
6 (c)
Evaluate \[ \int^1_0 \dfrac {x} {1+x^2} dx \] by using Simpson's (3/8)th Rule, dividing the interval into 3 equal parts. Hence find an approximate value of log √2.
7 M
7 (a)
Solve the one-dimensional wave equation \[ \dfrac {\partial^2 u}{\partial x^2} = \dfrac {\partial^2 u}{\partial t^2} \] Subject to the boundary conditions u(0,t)=0, u(1, t)=0, t≤0 and the initial conditions \[ u(x,0)=\sin \pi x , \dfrac {\partial u}{\partial t} (x, 0) = 0, \ 0 < x < 1 \]
7 M
7 (b)
Consider the heat equation \[ 2\dfrac {\partial ^2 u}{\partial x^2} = \dfrac {\partial u}{\partial t} \] under the following condition.
i) u(0,t)=u (4,t)=0, t≥0.
ii) u(x,0)=x (4-x), 0 Employ the Bendre-Schmidt method with h=1 to find the solution of the equation for 0
i) u(0,t)=u (4,t)=0, t≥0.
ii) u(x,0)=x (4-x), 0
6 M
7 (c)
Solve the two-dimensional Laplace equation \[ \dfrac {\partial ^2 u}{\partial x^2}= \dfrac {\partial ^2 u}{\partial y^2}=0 \] at the interior pivotal points of the square region shown in the following figure. The value of u at the pivotal points on the boundary are also shown in the figure.
7 M
8 (a)
State and prove the recurrence relation of Z-Transformation hence find ZT(np) and \[ Z_T \left [ \cos h \left (\dfrac {n \pi}{2}+0 \right ) \right ] \]
7 M
8 (b)
Find \[ Z_{T}^{-1} \left [ \dfrac {z^3 - 20 z}{(z-2)^3 (z-4)} \right ] \]
6 M
8 (c)
Solve the difference equation.
yn+2-2yn+1-3yn=3n+2n
Given y0=y1=0.
yn+2-2yn+1-3yn=3n+2n
Given y0=y1=0.
7 M
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