1 (a)
Expand f(x)=x sin x as a 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}\[\dfrac{\pi-2}{4}=\dfrac{1}{1.3}-\dfrac{1}{3.5}+\dfrac{1}{5.7}-+....\]

i)\[\dfrac{\pi}{2}=1+\dfrac{2}{1.3}-\dfrac{2}{3.5}+\dfrac{2}{5.7}\]

ii}\[\dfrac{\pi-2}{4}=\dfrac{1}{1.3}-\dfrac{1}{3.5}+\dfrac{1}{5.7}-+....\]

7 M

1 (b)
Find the half-range Fourier cosine series for the function

\[f(x)=\left\{\begin{matrix} kx, &0\leq x\leq l/2 \\k(l-x,) &l/2 where k is non-interger positive constant

\[f(x)=\left\{\begin{matrix} kx, &0\leq x\leq 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 | 2 π3 | π | 4π/3 | 5 π/3 | 2 π |

F(x): | 1 | 1.4 | 1.9 | 1.7 | 1.5 | 1.2 | 0 |

7 M

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

^{-2x}
7 M

2 (b)
Find the Fourier sine transforms of the Functions

\[f(x)\left\{\begin{matrix} Sin x, &0

\[f(x)\left\{\begin{matrix} Sin x, &0

6 M

2 (c)
Find the inverse Fourier sine Transform of

\[F(\alpha)=\dfrac{1}{\alpha}e^{-3 \alpha} a>0\]

\[F(\alpha)=\dfrac{1}{\alpha}e^{-3 \alpha} a>0\]

7 M

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

7 M

3 (b)
Obtain solution of heat equation \[\dfrac{\partial^2u }{\partial x^2}+\dfrac{\partial^2u }{\partial y^2}=0 \ subject\ to \ condition\ u(0,y)=u(l,y)=0,u(x,0)=0,u(x,s)=\sin \left ( \dfrac{\pi x}{l} \right )\]

7 M

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

^{r}= K, where r and k are constant. Fit this equation to the following set of observation (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 by 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 simple method

Maximize : Z=2x+4y, subject to the

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

Maximize : Z=2x+4y, subject to the

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

7 M

5 (a)
Using 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 in radians)

7 M

5 (c)
Using the power method find the largest energy value corresponding Eigen vectors of the matrix

\[\begin{bmatrix} 6 &-2 &2 \\-2 &3 &-1 \\2 &-1 &3 \end{bmatrix}\]

taking [1,1,1,]

\[\begin{bmatrix} 6 &-2 &2 \\-2 &3 &-1 \\2 &-1 &3 \end{bmatrix}\]

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 student 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 approximates to the function described by the following table

Hence find f(0.5) and (3.1)

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

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

Hence find f(0.5) and (3.1)

6 M

6 (c)
Evaluate \[\int_{0}^{1}\limits \dfrac{x}{1+x^2}dx\] by using Simpson's (3/8)

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

7 (a)
Solve the one-dimensional wave equation \[\dfrac{\partial^2u }{\partial x^2}=\dfrac{\partial^2u }{\partial t^2}\]

subject to the boundary conditions u(0,t)=0, u(1,t)=0,t≥0 and the initial conditions \[u(x,0)=\sin \pix,\dfrac{\partial t}{\partial t}(x,0)=0

subject to the boundary conditions u(0,t)=0, u(1,t)=0,t≥0 and the initial conditions \[u(x,0)=\sin \pix,\dfrac{\partial t}{\partial t}(x,0)=0

7 M

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

i) u(0,t)=u(4,t)=0,t≥0

ii) u(x,0)=x(4-x),0 Employ the Bender-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 point of the square region shown in the following figure. The values 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 \[Z_{r}\left [\cos h\left ( \dfrac{n \pi}{2}+0 \right ) \right ]\]
7 M

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

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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