STUPIDSID
1 (a)
Expand f(x)=x sin x as a 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}\[\dfrac{\pi-2}{4}=\dfrac{1}{1.3}-\dfrac{1}{3.5}+\dfrac{1}{5.7}-+....\]
i)\[\dfrac{\pi}{2}=1+\dfrac{2}{1.3}-\dfrac{2}{3.5}+\dfrac{2}{5.7}\]
ii}\[\dfrac{\pi-2}{4}=\dfrac{1}{1.3}-\dfrac{1}{3.5}+\dfrac{1}{5.7}-+....\]
7 M
1 (b)
Find the half-range Fourier cosine series for the function
\[f(x)=\left\{\begin{matrix} kx, &0\leq x\leq l/2 \\k(l-x,) &l/2 where k is non-interger positive constant
\[f(x)=\left\{\begin{matrix} kx, &0\leq x\leq 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 | 2 π3 | π | 4π/3 | 5 π/3 | 2 π |
| F(x): | 1 | 1.4 | 1.9 | 1.7 | 1.5 | 1.2 | 0 |
7 M
2 (a)
Find the Fourier transform of the function f(x)=xe-2x
7 M
2 (b)
Find the Fourier sine transforms of the Functions
\[f(x)\left\{\begin{matrix} Sin x, &0
\[f(x)\left\{\begin{matrix} Sin x, &0
6 M
2 (c)
Find the inverse Fourier sine Transform of
\[F(\alpha)=\dfrac{1}{\alpha}e^{-3 \alpha} a>0\]
\[F(\alpha)=\dfrac{1}{\alpha}e^{-3 \alpha} a>0\]
7 M
3 (a)
Find the various possible solution of one dimensional wave equation \[\dfrac{\partial^2 u}{\partial x^2}=C^2\dfrac{\partial^2 u}{\partial x^2}\] by separable variable method
7 M
3 (b)
Obtain solution of heat equation \[\dfrac{\partial^2u }{\partial x^2}+\dfrac{\partial^2u }{\partial y^2}=0 \ subject\ to \ condition\ u(0,y)=u(l,y)=0,u(x,0)=0,u(x,s)=\sin \left ( \dfrac{\pi x}{l} \right )\]
7 M
4 (a)
The pressure P and Volume V of gas are related by the equation PVr = K, where r and k are constant. Fit this equation to the following set of observation (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 by 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 simple method
Maximize : Z=2x+4y, subject to the
constraint : 3x+y≤2z, 2x+3y ≤24, x≥0,y≥0
Maximize : Z=2x+4y, subject to the
constraint : 3x+y≤2z, 2x+3y ≤24, x≥0,y≥0
7 M
5 (a)
Using 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 in radians)
7 M
5 (c)
Using the power method find the largest energy value corresponding Eigen vectors of the matrix
\[\begin{bmatrix} 6 &-2 &2 \\-2 &3 &-1 \\2 &-1 &3 \end{bmatrix}\]
taking [1,1,1,]Tas the initial Eigen vectors perform 5 iterations.
\[\begin{bmatrix} 6 &-2 &2 \\-2 &3 &-1 \\2 &-1 &3 \end{bmatrix}\]
taking [1,1,1,]Tas the initial Eigen vectors perform 5 iterations.
7 M
6 (a)
From the data given in the following Table; find the number of student 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 approximates to the function described by the following table
Hence find f(0.5) and (3.1)
| X | 0 | 1 | 2 | 3 | 4 |
| f(x) | 3 | 6 | 11 | 18 | 27 |
Hence find f(0.5) and (3.1)
6 M
6 (c)
Evaluate \[\int_{0}^{1}\limits \dfrac{x}{1+x^2}dx\] by using Simpson's (3/8)th Rule, dividing interval into 3 equal parts Hence find an approximate value of log &sqrt;2.
7 M
7 (a)
Solve the one-dimensional wave equation \[\dfrac{\partial^2u }{\partial x^2}=\dfrac{\partial^2u }{\partial t^2}\]
subject to the boundary conditions u(0,t)=0, u(1,t)=0,t≥0 and the initial conditions \[u(x,0)=\sin \pix,\dfrac{\partial t}{\partial t}(x,0)=0
subject to the boundary conditions u(0,t)=0, u(1,t)=0,t≥0 and the initial conditions \[u(x,0)=\sin \pix,\dfrac{\partial t}{\partial t}(x,0)=0
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7 (b)
Consider the heat equation \[2\dfrac{\partial^2 u}{\partial x^2}=\dfrac{\partial u}{\partial t}\] under the following conditions:
i) u(0,t)=u(4,t)=0,t≥0
ii) u(x,0)=x(4-x),0 Employ the Bender-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 point of the square region shown in the following figure. The values 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 \[Z_{r}\left [\cos h\left ( \dfrac{n \pi}{2}+0 \right ) \right ]\]
7 M
8 (b)
Find \[Z_{T}^{-1}\left [ \dfrac {z^3-20z}{(z-2)^3 (z-4)} \right ]\]
6 M
8 (c)
Solve the difference equation
yn+2 -2y n+1 -3yn=3n+2n
Given y0=y1=0
yn+2 -2y n+1 -3yn=3n+2n
Given y0=y1=0
7 M
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