STUPIDSID
1 (a)
Find the Fourier series expansion of the function f(x)=|x| in (-Π, Π), hence deduce that \[ \dfrac {\pi^2}{8} = \sum^{\infty}_{n=1} \dfrac {1}{(2n-1)^2} \]
6 M
1 (b)
Obtain the halft-range cosine series for the function;, f(x)=(x-1)2 in the interval 0?x?1 and hence show that \[ \pi^2 = 8 \left \{ \dfrac {1}{1^2}+ \dfrac {1}{3^2}+ \dfrac {1}{5^2}+ \cdots \ \cdots \right \} \]
7 M
1 (c)
Compute the constant term and first two harmonics of the Fourier series of f(x) given by.
| 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)
Obtain the Fourier cosine transform of \[ f(x)= \dfrac {1}{f+x^2} \]
6 M
2 (b)
Find the Fourier transform of \[ f(x) = \left\{\begin{matrix}
1-x^2 &for \ |x| \le 1 \\ 0
& for \ |x| > 1
\end{matrix}\right. \ and \ evaluate \ \int_c \dfrac {x \cos x - \sin x}{x^3}dx \]
7 M
2 (c)
Find the inverse Fourier sine transform of \[ \dfrac {s}{1+s^2} \]
7 M
3 (a)
Obtain the various possible solution of two dimensional Laplace's equation, uxx+uyy=0 by the method of separation of variables.
7 M
3 (b)
Solve the one-dimensional wave equation \[ C^2 \dfrac {\partial^2 u}{\partial x^2} = \dfrac {\partial^2 u}{\partial t^2}, \ 0
7 M
3 (c)
Obtain the D'Alembert's solution of the wave equation un=C2 uxx subject to the conditions \[ u(x,0)=f(x) = \ and \ \dfrac {\partial u}{\partial t} (x, 0) = 0 \]
6 M
4 (a)
Find the best values of a,b,c if the equation y=a+bx+cx2 is to fit most closely to the following observations.
7 M
4 (b)
Solve the following by graphical method to maximize z=50x+60y subject to the constraints. 2x+3y?1500. 3x+2y?1500. 0?x?400 and 0?y?400.
6 M
4 (c)
By using Simplex method. Maximum P=4x1-2x2-x3 subject to the constraints, \[ x_1 + x_2+x_3 \le 3 , \ 2x_1+2x_2+x_3 \le 4, \ x_1-x_2\le 0. x_1 \ge 0 \ and \ x_2 \ge 0. \]
7 M
5 (a)
Using Newton-Raphson method. Find a real root x sin x + cos x =0 nearer to Π carryout three iterations upto 4-decimals places.
7 M
5 (b)
Find the largest Eigen value and the corresponding Eigen vector of the matrix. \[ \begin{bmatrix}
2 &-1 &0 \\-1
&2 &-1 \\
0 &-1 &2
\end{bmatrix} \] By using the power method by taking the initial vector as [ 1 1 1]1 carryout 5-iterations.
7 M
5 (c)
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
6 M
6 (a)
A survey conducted in a slum locality reveals the following information as classified below.
\n \n
Estimate the probable number of persons in the income group 20 to 25.
| \n Income per day in \n Rupees 'X' | \n \n Under 10 | \n\n 10 - 20 | \n\n 20 - 30 | \n\n 30 - 40 | \n\n 40 - 50 | \n
| \n Number of \n Persons 'y' | \n \n 20 | \n\n 45 | \n\n 115 | \n\n 210 | \n\n 115 | \n
Estimate the probable number of persons in the income group 20 to 25.
7 M
6 (b)
Determine f(x) as a polynomials in x for the data given below by using the Newton's divided difference formula.
| x | 2 | 4 | 5 | 6 | 8 | 10 |
| f(x) | 10 | 96 | 196 | 350 | 868 | 1746 |
7 M
6 (c)
Evaluate \[ \int^1_c \dfrac {x}{1-x} dx \] by using Simpson's \[ \left (\dfrac {1}{3} \right )^{nd} \] rule by taking 6- equal strips and hence deduce an approximate value of logc2.
6 M
7 (a)
Solve the wave equation \[ \dfrac {\partial^2 u}{\partial t^2} = 4 \dfrac{\partial ^2 u}{\partial x^2} \] subject to u(0,1)=0. U(4, t)=0, u1(x,0)=0 and u(x,0)=x(4-x) by taking h=1, K=0.5 upto 4-steps.
7 M
7 (b)
Solve numerically the equation \[ \dfrac {\partial u}{\partial t} = \dfrac {\partial^2 u} {\partial x^2} \] subject to the conditions, u(0,t)=0 = u (1, t). t?0 and u(x,0)=sin Πx, 0?x?1. Carryout the computation for two levels taking h=1/3 and K=1/36.
7 M
7 (c)
Solve uxx+uyy=0 in the following square region with the boundary conditions as indicated in the Fig. Q7 (c)
6 M
8 (a)
Find the z-transform of. (i) sinh nθ (ii) cosh nθ (iii) n2.
7 M
8 (b)
Find the inverse z-transform of \[ \dfrac {2z^2 + 3z}{(z+2)(z-4)} \]
6 M
8 (c)
Solve the difference equation yn+2+6yn+1+9yn=2n with y0=y1=0 by using z-transform.
7 M
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