1(a)
Find the Fourier series for the function f(x) = x(2π -x)in 0≤x≤2π. Hence deduce that \(\dfrac{\pi }{8}=1+\dfrac{1}{3^{2}}+\dfrac{1}{5^{2}}+\dots \dots\)

7 M

1(b)
Find the half-range cosine series for the function f(x)=(x-1)

^{2}in 0
6 M

1(c)
Obtain the constant term and the co-efficient of the 1

^{st}sine and cosine terms in the Fourier series of y as given in the following table.x | 0 | 1 | 2 | 3 | 4 | 5 |

y | 9 | 18 | 24 | 28 | 26 | 20 |

7 M

2(a)
Solve the integral equation :\[ \int ^{\infty }_{0} f(\theta (cos \alpha \theta d\theta =\left\{\begin{matrix}
1-\alpha ,0\leq \alpha \leq 1 & \\
0,\alpha > 1 &
\end{matrix}\right.\text{Hence evalute }\int ^{\infty}_{0}\dfrac{sin^{2}t}{t^{2}}dt.\]

7 M

2(b)
Find the Fourier transform of\( f(x)=e^{|x|}.\).

6 M

2(c)
Find the infinite Fourier cosine transform of e

^{-x2.}
7 M

3(a)
Solve two dimensional Laplace equatio u

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

3(b)
Obtain the D' Alembert's solution of he wave equation u

_{n}= C^{2}u_{xx}subject to the conditions u(x,o) = f(x) and \( \dfrac{\partial u}{\partial t}(x,0)=0.)
6 M

3(c)
Solve the boundary value problem \(\dfrac{\partial u}{\partial t}=c^{2}\dfrac{\partial ^{2}u}{\partial x^{2}} , \ \ \ \ \ \ \ \ 0<x<l\) subject to the conditions \(\dfrac{\partial u}{\partial x}(0,t)=0;\
\dfrac{\partial u}{\partial x}(l,t)=0;\
u(x,0)=x\)

7 M

4(a)
Find the equation of the best fit straight line for the following data and hence estimate value of the dependent variable corresponding to the value of the independent variable with 30.

x | 5 | 10 | 15 | 20 | 25 |

y | 16 | 19 | 23 | 26 | 30 |

7 M

4(b)
Solve by graphical method:

Max Z= x + 1.5y

Subject to the constraints x+2y≤160

3x+2y≤240

x≥0;y≥0.

Max Z= x + 1.5y

Subject to the constraints x+2y≤160

3x+2y≤240

x≥0;y≥0.

6 M

4(c)
Solve by simplex method :

max z = 3x +5y

subject to 3x +2y≤ 18

x≤ 4

y≤ 6

x,y≥0.

max z = 3x +5y

subject to 3x +2y≤ 18

x≤ 4

y≤ 6

x,y≥0.

7 M

5(a)
Using the method of false position, find a real root of the equation x log

_{10}x-1.2=0, correct to 4 decimal places.
7 M

5(b)
By relaxation method, solve:

10x+2y+z=9; x+10y-z=-22; -2x+3y+10Z=22.

10x+2y+z=9; x+10y-z=-22; -2x+3y+10Z=22.

6 M

5(c)
Find the largest Eigen value and the corresponding Eigen vector for the matrix

\[ \begin{bmatrix} 6 & -2 & 2\\ -2 & 3 & -1\\ 2 & -1 & 3 \end{bmatrix}\] using Rayleigh's power method, taking x

\[ \begin{bmatrix} 6 & -2 & 2\\ -2 & 3 & -1\\ 2 & -1 & 3 \end{bmatrix}\] using Rayleigh's power method, taking x

_{0}=[1 1 1]^{t}. Perform 5 iterations.
7 M

6(a)
Find the cubic ploynomial by using Newton's forward interpolation which takes the following values.

x | 0 | 1 | 2 | 3 |

y | 1 | 2 | 1 | 10 |

Hence evaluate f(4).

7 M

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

x | 0 | 1 | 2 | 5 |

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

Hence find f(3).

6 M

6(c)
Evaluate \( \int_{4}^{5.2}log_{e}\, x \, dx\) using Weddler's rule by taking 7 ordinates.

7 M

7(a)
Solve u

!MAGE:

_{xx}+ u_{yy}= 0 in the following square Mesh. Carry out two interations.!MAGE:

7 M

7(b)
The transverse displacement of a point at a distance x from one end to any point 't' of a vibrating string satisfies the equation : \( \dfrac{\partial^{2} u}{\partial ^{2}}=25\dfrac{\partial^{2} u}{\partial t^{2}}\) with boundary condition u(0, t)= u(5,t)=0 and initial condition u(x,o)=\( \left\{\begin{matrix}20x for 0\leq x\leq 1 & \\ 5(5-x) for 1\leq x\leq 5& \end{matrix}\right.\)and u

_{t}(x,0)=0 solve by taking h=1, k=0.2 upto t=1.
6 M

7(c)
Find the solution of the equation u

_{xx}=2u_{t}when u(0,t) =0 and u(4,t) =0 and u(x,0)= x(4-x) taking h=1, Find values upto t=5.
7 M

8(a)
Find the Z- transformation of the following : i)\(3n-4 sin\dfrac{\pi }{4}+5a^{2}\) ii) \( \dfrac{a^{n}e-a}{n!}\).

7 M

8(b)
Find the inverse Z - transfrmation of \( \dfrac{4z^{2-2z}}{z^{3}+5z^{2}+8z-4}\).

6 M

8(c)
Solve the difference equation : y

_{n+2}+ 6y_{n+1}+ 9y_{n}= 2^{n}; given y_{0}= y_{1}=0 using Z- transformation.
7 M

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