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

^{2}in [0, 2?]. 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 cosin 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

^{-a2x2 }(-? < x < ?) hence show that e^{ -x2/2 }is self reciprocal.
7 M

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

6 M

2 (c)
Solve integral equation \[ \int^\infty _0f(x)\cos sxdx=\left\{\begin{matrix} 1-s&0~~
~~

7 M

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

7 M

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

6 M

3 (c)
Solve Laplace equation \[ \dfrac {\partial^2 u}{\partial x^2}+\dfrac {\partial ^2 u}{\partial y^2}=0\] subjects 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. GivenRevolution | 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}?0; x_{1}+2x_{2}?40; 3x_{1}+x_{2}?0; 4x_{1}+3x_{2}?60; x_{1}, x_{2}?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. Given machine M_{1}, M_{2}, 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 reciprocal of positive number. Hence use to find 1/c upto 4 decimals.

6 M

5 (c)
Using power relay 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 (I, I, 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, f(4)=-125, f(8)=-21, f(10)=335.

7 M

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

t | 2 | 5 | 8 | 14 |

f(t) | 94.8 | 87.9 | 81.3 | 68.7 |

6 M

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

by Simpson's 3/8

x | 0 | 1/6 | 2/6 | 3/6 | 4/6 | 5/6 | 1 |

y | 0.1 | 0.8982 | 0.9018 | 0.9589 | 0.9432 | 0.9001 | 0.8415 |

by Simpson's 3/8

^{th}rule find the 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 ? x 0< x < 1, take \[ h=\dfrac {h}{1}\alpha - \dfrac {1}{6} \] Carry out 3 steps in time level.

7 M

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

_{t}(x,0)=0, u(x,0)=x(4-x) take 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_{T}(n^{2})
7 M

8 (b)
Find Z

_{T}[ e^{n?}cosh n? - sin (nA+?)+n]
6 M

8 (c)
Solve difference equation u

_{n+2}+6u_{n+1}+9u_{n}=n2^{n}given u_{0}=u_{1}=0
7 M

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