1 (a)
Find the Fourier series expansion of the function f(x) = |x| in (-? ?), hence deduce that \[ \dfrac {\pi^2}{8}=\sum ^{\infty}_{n=1}\dfrac {1}{(2n-1)^2} \]

6 M

1 (b)
Obtain the half range cosin series for the function, f(x)=(x-1)

^{2}in the interval 0 ? x ? 1 and hence show that \[ \pi^2 =8 \left \{ \dfrac {1}{1^2}+\dfrac {1}{3^2}+\dfrac {1}{5^2} +......\right \} \]
7 M

1 (c)
Compute the constant term and first two harmonics of the Fourier series of f(x) given by.

\[ \begin{array}{|c|c|c|} \hline\text{x} & 0& \dfrac {\pi}{3} & \frac {2\pi}{3} & \pi & \frac {4\pi} {3} & \frac {5\pi}{3} & 2\pi \\ \hline \text{f(x)} & 1.0 & 1.4&1.9&1.7&1.5&1.2&1.0 \\ \hline \end{array} \]

\[ \begin{array}{|c|c|c|} \hline\text{x} & 0& \dfrac {\pi}{3} & \frac {2\pi}{3} & \pi & \frac {4\pi} {3} & \frac {5\pi}{3} & 2\pi \\ \hline \text{f(x)} & 1.0 & 1.4&1.9&1.7&1.5&1.2&1.0 \\ \hline \end{array} \]

7 M

2 (a)
Open the Fourier cosine transform of \[ f(x)=\dfrac {1} {1+x^2} \]

6 M

2 (b)
Find the Fourier transform of \[f(x)=\left\{\begin{matrix} 1-x^2&for &|x|\le 1 \\0 &for &|x|>1 \end{matrix}\right. \ and \ evaluate \ \int_{n}\dfrac {x \cos x - \sin x}{x^3}dx \]

7 M

2 (c)
Find the inverse Fourier transform of \[ \dfrac {s}{1+s^2} \]

7 M

3 (a)
Obtain the various possible solutions of two dimensional Laplace's equation, u

_{xx}+ u_{yy}=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}=\dfrac {\partial^2 u}{\partial t^2}, 0 < x0 is constant \[ (iii) \ \dfrac {\partial u}{\partial t}(x,0)=0 \]

7 M

3 (c)
Obtain the D' Almbert's solution of the wave equation u

_{n}=C^{2}u_{xx}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+cx

^{2}is to fit most closely to the following observations.x | 1 | 2 | 3 | 4 | 5 |

y | 10 | 12 | 13 | 16 | 19 |

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. Maximize P= 4x

x

_{1}-2x_{2}-x_{3}subject to the constraints.x

_{1}+x_{2}+x_{3}?3, 2x_{1}+2x_{2}+x_{3}?4, x_{1}-x_{2}?0, x_{1}? 0 and x_{2}? 0
7 M

5 (a)
Using Newton-Raphson method find a real root of x sin x + cos x =0 nearer to ? carryout three iterations upto 4-decimal 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]

\[ \begin{bmatrix} 2&-1 &0 \\-1 &2 &-1 \\0 & -1 &2 \end{bmatrix} \] BY using the power method by taking the initial vector as [1 1 1]

^{T}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.

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)
\[ Evaluate \ \int^1_0\dfrac {x}{1+x^2}dx \ by \ using \ Simpson's \ \left ( \dfrac {1}{3} \right )^{rd} \] rule by taking 6-equal strips and hence deduce an approximate value of log

_{ e}2.
6 M

7 (a)
Solve the wave equation, \[ \dfrac {\partial^2 u}{\partial t^2}=4\dfrac {\partial^2 u}{\parial x^2} \] subjects to u(0, 1)=0, u(4, t)=0, u

_{t}(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 \[ \dfrac {\partial u}{\partial t}=\dfrac {\partial^2u}{\partial x^2} \] 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 u

_{xx}+u_{yy}=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 n? (ii) cosh n? (iii) n

^{2}
7 M

8 (b)
Find the inverse z-transform of \[ \dfrac {2z^3+3z}{(z+2)(z-4)} \]

6 M

8 (c)
Solve the difference equation y

_{n+2}+6y_{n+1}+9y_{n}=2n with y_{0}=y_{1}=0 using Z transforms.
7 M

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