STUPIDSID
1 (a)
Obtain the Fourier series expansion of \[ f(x)=\left\{\begin{matrix} x,&if &0\le x \le \pi \\2\pi -x, &\if &\pi \le x \le 2\pi \end{matrix}\right. \] and hence deduce that \[ \dfrac {\pi^2}{8}=\dfrac {1}{1^2}+\dfrac {1}{3^2}+\dfrac {1}{5^2}+...... \]
7 M
1 (b)
Find the half range Fourier sine series of \[ f(x)= \left\{\begin{matrix}x, &if &0
6 M
1 (c)
Obtain the constant term and coefficients of first cosine and sine term in the expansion of y from the following table:
| x | 0 | 60° | 120° | 180° | 240° | 300<°/td> | 360° |
| y | 7.9 | 7.2 | 3.6 | 0.5 | 0.9 | 6.8 | 7.9 |
7 M
2 (a)
Find the Fourier transform of \[ f(x)=\left\{\begin{matrix} a^2-x^2&|x|\le a \\0 &|x|>a \end{matrix}\right.\] and hence deduce \[ \int^{\alpha}_0\dfrac {\sin x -x \cos x}{x^3}dx=\dfrac {\pi}{4} \]
7 M
2 (b)
Find the Fourier cosin and sine transform of f(x)=xe-ax, where a>0
6 M
2 (c)
Find the inverse Fourier transform of es2
7 M
3 (a)
Obtain the various possible solutions of one dimensional heat equation ut=c2 uxx by the method of separation of variables.
7 M
3 (b)
A tightly stretched string of length l with fixed ends is initially in equilibrium position. It is set to vibrate by giving each point a velocity \[ V_o \sin \left ( \dfrac {\pi x}{l} \right) \]. Find the displacement u(x,t).
6 M
3 (c)
Solve uxx+uyy=0 given u(x,0)=0, u(x,1)=0, u(1, y)=0 and u(0,y)=u0 where u0 is a constant.
7 M
4 (a)
Using method of least square fit a curve y=axb for the following data.
| x | 1 | 2 | 3 | 4 | 5 |
| y | 0.5 | 2 | 4.5 | 8 | 12.5 |
7 M
4 (b)
Solve the following LLP graphically:
Minimize=Z=20x+16y
Subject to 3x+y?6, x+y?4, x+3y?6 and x,y?0
Minimize=Z=20x+16y
Subject to 3x+y?6, x+y?4, x+3y?6 and x,y?0
6 M
4 (c)
Use simplex method to Maximize Z=x+(1.5)y
Subject to the constraints x+2y ?160, 3x+2y?240 and x,y?0.
Subject to the constraints x+2y ?160, 3x+2y?240 and x,y?0.
7 M
5 (a)
Using Newton-Raphson method find a real root x+log10 x=3.375 near 2.9, corrected to 3-decimal places.
7 M
5 (b)
Solve the following system of equations by relaxation method:
12x+y+z=31, 2x+8y-z=24, 3x+4y+10z=58
12x+y+z=31, 2x+8y-z=24, 3x+4y+10z=58
7 M
5 (c)
Find the largest eigen value and corresponding eigen vector of following matrix a by power method
\[ A=\begin{bmatrix}25 &1 &2 \\1 &3 &0 \\2 &0 &-4 \end{bmatrix} \]
Use X(0)=[1, 0, 0]T as the initial eigen vector.
\[ A=\begin{bmatrix}25 &1 &2 \\1 &3 &0 \\2 &0 &-4 \end{bmatrix} \]
Use X(0)=[1, 0, 0]T as the initial eigen vector.
6 M
6 (a)
In the given table below, the values of y are consecutive terms of series of which 23.6 is the 6th term, find the first and tenth terms of the series.
| x | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| y | 4.8 | 8.4 | 14.5 | 23.6 | 36.2 | 52.8 | 73.9 |
7 M
6 (b)
Construct an interpolating polynomial for the data given below using Neeton's divided difference formula.
| x | 2 | 4 | 5 | 6 | 8 | 10 |
| f(x) | 10 | 96 | 196 | 350 | 868 | 1746 |
7 M
6 (c)
\[ Evaluate \ \int^{1}_0 \dfrac {x}{1+x^2}dx \] by Weddle's rule taking 7-ordinates and hence find log.2
6 M
7 (a)
Solve the wave equation un=4uxx subject to u(0, t)=0; u(4, t)=0; ut(x, 0)=0; u(x,0)=x(4-x) by taking h=1, k=0.5 upto four steps.
7 M
7 (b)
Solve numerically the equation \[ \dfrac {\partial u}{\partial t}=\dfrac {\partial^2u}{\partial x^2} \] subject to the conditions u(0,t)=0=u(1,t),t?0 and u(x,0)=sin ?x, 0 ? x ? 1. Carryout computations for two levels taking h=1/3 and k=1/36.
7 M
7 (c)
Solve the elliptic equation uxx+uyy=0 for the following square mesh with boundary values as shown in Fig. Q7(c)
6 M
8 (a)
Find the z-transform of: i) sin h n ? ; ii) cos h n ?
7 M
8 (b)
Obtain the inverse Z-transform of \[ \dfrac {8z^2}{(2z-1)(4z-1)} \]
7 M
8 (c)
Solve the following difference equation using z-transforms:
yn+2+2yn+1+yn=n with y0=y1=0
yn+2+2yn+1+yn=n with y0=y1=0
6 M
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