1 (a)
Obtain the Fourier series for the function\[ f(x)=\left\{\begin{matrix} -\pi x&; 0\le x \le 1 \\\pi (2-x) &;1\le x\le 2 \end{matrix}\right. \] and deduce that \[ \dfrac {\pi^2}{8}=\sum^\infty_{n=1}\dfrac {1}{(2n-1)^2} \]

7 M

1 (b)
Obtain the half range Fourier sine for the function. \[ f(x)=\begin{bmatrix}1/4-x & 0<x<1/2 \\x-3/ 4&1/2<x<1 \end{bmatrix} \]

7 M

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

x | 0 | 1 | 2 | 3 | 4 | 5 |

f(x) | 4 | 8 | 15 | 7 | 6 | 2 |

6 M

2 (a)
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 hence evaluate \[ \int^\infty_0 \left ( \dfrac {x \cos x - \sin x}{x^3} \right )\cos \dfrac {x}{2}dx \]

7 M

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

7 M

2 (c)
Solve the integral equation \[ \int^{\infty}_0d(\theta)\cos \alpha \theta d\theta=\left\{\begin{matrix} 1-\alpha&;&0 \le \alpha \le 1 \\0 &; & a>1 \end{matrix}\right. \ hence \ evaluate \ \int^{\infty}_0 \dfrac {\sin^2 t}{t^2}dt. \]

6 M

3 (a)
Solve two dimesnional Laplace equation u

_{xx}+u_{yy}=0, by the method of separation of variables.
7 M

3 (b)
Solve the one dimensional heat equations \[ \dfrac {\partial u}{\partial t}=\dfrac {c^2\partial^2 u}{\partial x^2}, 0<x<\pi \] under the conditions:

(i) u(0,+)=0,u(?,t)=0

(ii) u(x,0)=u

(i) u(0,+)=0,u(?,t)=0

(ii) u(x,0)=u

_{0}sinx where u_{0}= constant ? 0.
7 M

3 (c)
Obtain the D' Almbert's solution of one dimensional wave equation.

6 M

4 (a)
Fit a curve of the form y=ae

^{bx}to the following data:x: | 77 | 100 | 185 | 239 | 285 |

y: | 2.4 | 3.4 | 7.0 | 11.1 | 19.6 |

7 M

4 (b)
Using graphical method solve the L.P.P minimize z=20x

x

_{1}=10x_{2}subject to the constraintsx

_{1}+2x_{2}?40; 3x_{1}+x_{2}?0; 4x_{1}+3x_{2}? 60; x_{1}?0; x_{2}?0
6 M

4 (c)
Solve the following L.P.P maximize z=2x

x

_{1}+ 3x_{2}+ x_{3}, subject to the constraintsx

_{1}+2x_{2}+5x_{3}?19, 3x_{1}+x_{2}+4x_{3}?25, x_{1}?0, x_{2}?0, x_{3}?0 using simplex method.
7 M

5 (a)
Using the Regular - falsi method, find the root of the equation xe

^{x}=cosx that lies between 0.4 and 0.6 Carry out four interations.
7 M

5 (b)
Using relaxation method solve the equations.

10x-2y-3z=205; -2x+10y-2z=154; -2x-y+10z=120

10x-2y-3z=205; -2x+10y-2z=154; -2x-y+10z=120

7 M

5 (c)
Using the Rayleigh's power method, find the dominant eigen value and the corresponding eigen vector of the matrix \[ A=\begin{bmatrix} 6&-2 &2 \\ -2&3 &-1 \\2 &-1 &3 \end{bmatrix} \] starting with the initial vector [1, 1, 1]

^{T}
6 M

6 (a)
From the following table, estimate the number of students who have obtained the marks between 40 and 45:

Marks | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |

Number of student | 31 | 42 | 51 | 35 | 31 |

7 M

6 (b)
Using Lagrange's formula, find the interpolating polynomial that approximate the function described by following table:

Hence find f(3).

x | 0 | 1 | 2 | 5 |

f(x) | 2 | 3 | 12 | 147 |

Hence find f(3).

7 M

6 (c)
A curve is drawn to pass through the points given by the following table:

Using Weddle's rule, estimate the area bonded bt the curve, the x-axis and the lines x=1, x=4.

x | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |

y | 2 | 2.4 | 2.7 | 2.8 | 3 | 2.6 | 2.1 |

Using Weddle's rule, estimate the area bonded bt the curve, the x-axis and the lines x=1, x=4.

6 M

7 (a)
Solve the Laplace's equation u

:IMAGE-

_{xx}+u_{yy}=0, given that;:IMAGE-

7 M

7 (b)
\[ Solve \ \dfrac {\partial^2u}{\partial t^2}=4 \dfrac {\partial^2 u}{\partial x^2} \] subject to u(0,t)=0; u(4,t)=0; u(x,0)=x (4-x). Take h=1, k=0.5

7 M

7 (c)
Solve the equation \[ \dfrac {\partial u}{\partial t}=\dfrac {\partial^2 u}{\partial x^2} \] subject to the conditions u(x,0)=sinx, 0?x?1; u(0, t)=u(1, t)=0 using Schmidt's method. Carry out computations for two levels, taking h-1/3, k=1/36.

6 M

8 (a)
Find the Z-transform of : \[ i) \ (2n-1)^2 \\ ii) \ \cos \left (\dfrac {n\pi}{2}+\pi/4 \right ) \]

7 M

8 (b)
Obtain the inverse Z-transform of \[ \dfrac {4x^2-2z}{z^3-5z^2+8z-4} \]

7 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.
6 M

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