1 (a)
Expand f(x)=x sin x as Fourier series in the interval (-π, π), hence deduce the following: \[ i) \ \dfrac {\pi}{2} = 1+ \dfrac {2}{1.3}- \dfrac {2}{3.5} + \dfrac {2}{5.7} \\ ii) \ \dfrac {\pi -2}{4} = \dfrac {1}{1.3} - \dfrac {1}{3.5}+ \dfrac {1}{5.7}+ \cdots \ \]

7 M

1 (b)
Find the half-range Fourier cosine series for the function. \[ f(x)= \left\{\begin{matrix}
kx &, \ 0\le x \le l/2 \\
k(l-x) &, \ l/2

6 M

1 (c)
Find the constant term and the first two harmonics in the Fourier series for f(x) given by the following table:

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)
Find the Fourier transform of the function f(x)=xe

^{-h|x|}.
7 M

2 (b)
Find the Fourier sine transform of the \[ Function \ f(x)= \left\{\begin{matrix}
\sin x &, \ 0

6 M

2 (c)
Find the inverse Fourier sine Transform of \[ F_x (\alpha) = \dfrac {1} {\alpha} e^{-a\alpha} \ a>0. \]

7 M

3 (a)
Find various possible solution of one dimensional wave equation \[ \dfrac {\partial ^2 u}{\partial t^2} = C^2 \ \dfrac {\partial ^2 u}{\partial x^2} \] separable variable method.

7 M

3 (b)
Obtain solution of heat equation \[ \dfrac {\partial u}{\partial t}= C^2 \dfrac {\partial ^2 u}{\partial t^2} \] subject to condition u(0, t)=0, u(l, t)=0, u(x,0)=f(x).

6 M

3 (c)
Solve Laplace equation \[ \dfrac {\partial^2 u}{\partial x^2} + \dfrac {\partial^2 u}{\partial y^2}= 0
\] subject to condition u(0,y)=u, u(x,0)=0. \[
u(x,a)= \sin \left ( \dfrac {\pi x}{l} \right ) \]

7 M

4 (a)
The pressure P and volume V of a gas are related by the equation PV

^{r}=K, where r and K are constant. Find this equation to the following set of observations (in appropriate units) P: | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 |

V: | 1.62 | 1.00 | 0.75 | 0.62 | 0.52 | 0.46 |

7 M

4 (b)
Solve the following LPP using the Graphical method:

Maximize: Z=3x

Under the constraints

4x

2x

x

Maximize: Z=3x

_{1}+4x_{2}Under the constraints

4x

_{1}+2x_{2}≤802x

_{1}+5x_{2}≤180x

_{1}, x_{2}≥ 0
6 M

4 (c)
Solve the following using simplex method,

Maximize: Z=2x+4y, subjected t the

Constraint: 3x+y ≤2z, 2x+3y≤24, x≥0, y≥0.

Maximize: Z=2x+4y, subjected t the

Constraint: 3x+y ≤2z, 2x+3y≤24, x≥0, y≥0.

7 M

5 (a)
Using the Regular-Falsi method, find a real root (correct to three decimal places) of the equation cos x=3x-1 that lies between 0.5 and 1 (Here, x is in radians).

7 M

5 (b)
By relaxation method

Solve: x+6y+27z=85, 54x+y+z=110, 2x+15y+6z=72

Solve: x+6y+27z=85, 54x+y+z=110, 2x+15y+6z=72

6 M

5 (c)
Using the power method, find the largest Eigen value and corresponding Eigen vectors of the matrix \[ A= \begin{bmatrix}
6 &-2 &2 \\-2
&3 &-1 \\2
&-1 &3
\end{bmatrix} \] taking [1, 1, 1]

^{T}as the initial Eigen vectors. Perform 5 iterations.
7 M

6 (a)
From the data given in the following Table: find the number of students who obtained

(i) Less than 45 marks

(ii) between 40 and 45 marks

(i) Less than 45 marks

(ii) between 40 and 45 marks

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

No. of Students | 31 | 42 | 51 | 35 | 31 |

7 M

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

Hence find f(0.5) and f(0.31)

X | 0 | 1 | 2 | 3 | 4 |

f(x) | 3 | 6 | 11 | 18 | 27 |

Hence find f(0.5) and f(0.31)

6 M

6 (c)
Evaluate \[ \int^1_0 \dfrac {x} {1+x^2} dx \] by using Simpson's (3/8)

^{th}Rule, dividing the interval into 3 equal parts. Hence find an approximate value of log √2.
7 M

7 (a)
Solve the one-dimensional wave equation \[ \dfrac {\partial^2 u}{\partial x^2} = \dfrac {\partial^2 u}{\partial t^2} \] Subject to the boundary conditions u(0,t)=0, u(1, t)=0, t≤0 and the initial conditions \[ u(x,0)=\sin \pi x , \dfrac {\partial u}{\partial t} (x, 0) = 0, \ 0 < x < 1 \]

7 M

7 (b)
Consider the heat equation \[ 2\dfrac {\partial ^2 u}{\partial x^2} = \dfrac {\partial u}{\partial t} \] under the following condition.

i) u(0,t)=u (4,t)=0, t≥0.

ii) u(x,0)=x (4-x), 0 Employ the Bendre-Schmidt method with h=1 to find the solution of the equation for 0

i) u(0,t)=u (4,t)=0, t≥0.

ii) u(x,0)=x (4-x), 0

6 M

7 (c)
Solve the two-dimensional Laplace equation \[ \dfrac {\partial ^2 u}{\partial x^2}= \dfrac {\partial ^2 u}{\partial y^2}=0 \] at the interior pivotal points of the square region shown in the following figure. The value of u at the pivotal points on the boundary are also shown in the figure.

7 M

8 (a)
State and prove the recurrence relation of Z-Transformation hence find Z

_{T}(n^{p}) and \[ Z_T \left [ \cos h \left (\dfrac {n \pi}{2}+0 \right ) \right ] \]
7 M

8 (b)
Find \[ Z_{T}^{-1} \left [ \dfrac {z^3 - 20 z}{(z-2)^3 (z-4)} \right ] \]

6 M

8 (c)
Solve the difference equation.

y

Given y

y

_{n+2}-2y_{n+1}-3y_{n}=3^{n}+2nGiven y

_{0}=y_{1}=0.
7 M

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