1 (a)
Find Fourier series of f(x)=2Πx-x

^{2}in [0, 2Π]. Hence deduce \[ \sum^{\infty}_1 \dfrac {1}{(2n-1)^2} = \dfrac {\pi^2}{8} \] sketch the graph of f(x).
7 M

1 (b)
Find Fourier cosine series of \[ f(x)= \sin \left ( \dfrac {m\pi}{l} \right )x, \] where m is positive integer.

6 M

1 (c)
Following table gives current (A) over period (T):

Find amplitude of first harmonic:

A (amp) |
1.98 | 1.30 | 1.05 | 1.30 | -0.88 | -0.25 | 1.98 |

t (sec) |
0 | T/6 | T/3 | T/2 | 2T/3 | 5T/6 | T |

Find amplitude of first harmonic:

7 M

2 (a)
Find Fourier transformation of e-x

^{a2x2 }(-∞^{2}/3 is self reciprocal.

7 M

2 (b)
Find Fourier cosine and sine transformation of \[ f(x)= \left\{\begin{matrix}
x &0

6 M

2 (c)
Solve integral equation \[ \int^\infty_{0} f(x) \cos sx dx = \left\{\begin{matrix} 1-s &0~~
~~

7 M

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

7 M

3 (b)
Obtain solution of heat equation \[ \dfrac {\partial u}{\partial t^2} = c^2 \dfrac {\partial^2 u}{\partial t^2} \] 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 (l, y)=u (x,0)=0; \[ u (x,a)=\sin \left ( \dfrac {\pi x}{l} \right ) \]

7 M

4 (a)
The revolution (r) and time (t) are related by quadratic polynomial r=at

^{2}+bt+c. Estimate number revolution for time 3.5 units, given
Revolution |
5 | 10 | 15 | 20 | 25 | 30 | 35 |

Time |
1.2 | 1.6 | 1.9 | 2.1 | 2.4 | 2.6 | 3 |

7 M

4 (b)
Solve by graphical method, Minimize Z=20x

_{1}+10x_{2}under the constraints \[2x_1+x_2 \ge 0; \ x_1+2x_2 \le 40; \ 3x_1+x_2 \ge 0; \ 4x_1+ 3x_2 \ge 60; \ x_1,x_2 \ge 0 \]
6 M

4 (c)
A company produces 3 items A, B, C. Each unit of A requires 8 minutes, 4 minutes and 2 minutes of producing time on machine M

_{1}, M_{2}and M_{3}respectively. Similarly B requires 2, 3, 0 and C. Requires 3, 0, 1 minutes of machine M_{1}, M_{2}and M_{3}. Profit per unit of A, B and C are Rs.20, Rs.6 and Rs.8 respectively. For maximum profit, how many number of products A. B and C are to be produced? Find maximum profit. Given machine M_{1}, M_{2}and M_{3}are available for 250, 100 and 60 minutes per day.
7 M

5 (a)
By relaxation method, solve -x+6y+27z=85, 54x+y+z=110, 2x+15y+6z=72.

7 M

5 (b)
Using Newton-Raphson method derive the iteration formula to find the value of reciprocal of positive number. Hence use to find 1/e upto 4 decimals.

6 M

5 (c)
Using Power Rayleigh method find numerical largest Eigen value and corresponding Eigen vector for \[ \begin{bmatrix}
10 &2 &1 \\2 &10 &1 \\2 &-1 &10
\end{bmatrix} \] using (1, 1, 0)

^{T}as initial vector. Carry out 10 iterations.
7 M

6 (a)
Fit interpolating polynomial for f(x) using divided difference formula and hence evaluate f(z), given f(0)= -5, f(1)=-14= -125, f(8)=-21, f(10)=355.

7 M

6 (b)
Estimate t when f(t)=85 using inverse interpolation formula given:

6 M

6 (c)
A solid of revolution is formed by rotating about x-axis the area between x-axis lines x=0, x=1 and curve through the points with the following co-ordinates.

by Simpson's 3/8

by Simpson's 3/8

^{th}rule, find volume of solid formed.
7 M

7 (a)
Using the Schmidt two-level point formula solve \[ \dfrac {\partial^2 u}{\partial x^2} = \dfrac {\partial u}{\partial t} \] under the conditions u(0,t)=u(1,t)=0; t?0; u(1,0)=sin Πx 0

7 M

7 (b)
Solve the wave equation \[ \dfrac {\partial^2 u}{\partial t^2} = 4 \dfrac{\partial ^2 u}{\partial x^2} \] subject to u(0,t)=0. U(4, t)=0, u

_{1}(x,0)=0 and u(x,0)=x(4-x) by taking h=1, K=0.5
6 M

7 (c)
Solve \[ \dfrac {\partial^2 u}{\partial x^2} + \dfrac {\partial^2 u}{\partial y^2} =0 \] in the square mesh. Carry out 2 iterations.

7 M

8 (a)
State and prove recurrence relation of f-transformation hence find Z

_{T}(n), Z_{1}(n^{2}).
7 M

8 (b)
Find Z

_{T}[30^{n0}cosh n()-sin (nA+0)+n].
6 M

8 (c)
Solve difference equation un given u

_{n+2}+6u_{n+1}+9u_{n}=n2_{0}=u

_{1}=0.

7 M

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