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 e

^{s2}
7 M

3 (a)
Obtain the various possible solutions of one dimensional heat equation u

_{t}=c^{2}u_{xx}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 u

_{xx}+u_{yy}=0 given u(x,0)=0, u(x,1)=0, u(1, y)=0 and u(0,y)=u_{0}where u_{0}is a constant.
7 M

4 (a)
Using method of least square fit a curve y=ax

^{b}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+log

_{10}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

\[ A=\begin{bmatrix}25 &1 &2 \\1 &3 &0 \\2 &0 &-4 \end{bmatrix} \]

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 6

^{th}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 u

_{n}=4u_{xx}subject to u(0, t)=0; u(4, t)=0; u_{t}(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 u

_{xx}+u_{yy}=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:

y

y

_{n+2}+2y_{n+1}+y_{n}=n with y_{0}=y_{1}=0
6 M

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