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

^{2}, -π ≤ x ≤ π. Hence deduce that [ dfrac {1}{1^2}-dfrac {1}{2^2}+dfrac {1}{3^2}- cdots cdots = dfrac {pi^2}{12} ] Is the above deduced series convergent? (Answer in Yes or No)
7 M

1 (b)
Define :

i) Half range Fourier sine series of f(x)

ii) Complex form of Fourier series of f(x)

Find the half range cosine series of f(x)=x in 0

i) Half range Fourier sine series of f(x)

ii) Complex form of Fourier series of f(x)

Find the half range cosine series of f(x)=x in 0

7 M

1 (c)
Obtain a

_{0}, a_{1}, b_{1}in the Fourier expansion of y, using harmonic analysis for the data given.x | 0 | 1 | 2 | 3 | 4 | 5 |

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

6 M

2 (a)
Find the Fourier transform of [ egin {align*} f(x)&=1-x^2 & for & |x|le 1 \ &=0 &for & |x|>1 end{align*} ] Hence evaluate [ int^{infty}_{0}dfrac {x cos x -sin x}{x^3}cos left ( dfrac {x}{2} ight )dx ]

7 M

2 (b)
Find the Fourier sine transform of [ dfrac {e^{-ax}}{x} ]

7 M

2 (c)
Find the Fourier cosine transform of [ egin{align*} f(x)&=4x, & for &04 end{align*} ]

6 M

3 (a)
i) Write down the two dimensional heat flow equation (p d e) in steady state (or two dimesional) Laplace's equation. Just mention.

ii) Solve one dimensional heat equation by the method of separation of variables.

ii) Solve one dimensional heat equation by the method of separation of variables.

7 M

3 (b)
Using D' Alembert's method, solve one dimensional wave equation.

7 M

3 (c)
A string is stretched and fastened to two points/apart. Mention is started by displacing the string in the form of y=a sin (πx/l) from which it is released at time t=0. Show that the displacement of any point at a distance x from one end at time t is, [ y(x,t)=a sin left (dfrac {pi x}{l} ight )cos left ( dfrac {pi ct}{l} ight ) ] Start the answer assuming the solution to be

y=(C

y=(C

_{1}cos(px)+C_{2}sin(px))(C_{3}cos(cpt)+C_{4}sin(cpt))
6 M

4 (a)
Fit a linear law, P=mW+C, using the data

P | 12 | 15 | 21 | 25 |

W | 50 | 70 | 100 | 120 |

6 M

4 (b)
Find the best values of a and b by fitting the law V=at

Use base 10 for algorithm for computations.

^{b}using method of least square for the data,V (ft/min) | 350 | 400 | 500 | 600 |

t (min) | 61 | 26 | 7 | 26 |

Use base 10 for algorithm for computations.

7 M

4 (c)
Using simplex method,

Maximize Z=5x

Subject to, x

Maximize Z=5x

_{1}+3x_{2}Subject to, x

_{1}+x_{2}≤2; 5x_{1}+2x_{2}≤10; 3x_{1}+8x_{2}≤12; x_{1}, x_{≥0.}
7 M

5 (a)
Use Newton-Raphson method, to find the real root of the equation 3x=(cos x)+1 Take x

_{0}=0.6. Perform two iterations.
6 M

5 (b)
Apply Gauss-Seidel iteration method to solve equations.

20x+y-2z=17

3x+20y-z=-18

2x-3z+20z=25

Assume initial approximation to be x=y=z=0. Perform three interations.

20x+y-2z=17

3x+20y-z=-18

2x-3z+20z=25

Assume initial approximation to be x=y=z=0. Perform three interations.

7 M

5 (c)
Using Rayleigh's power method to find the largest eigen values and the corresponding eigen vector of the matrix.

[ A=egin{bmatrix}6 &-2 &2 \-2 &3 &-1 \2 &-1 &3 end{bmatrix} ] Take [1 0 0]

[ A=egin{bmatrix}6 &-2 &2 \-2 &3 &-1 \2 &-1 &3 end{bmatrix} ] Take [1 0 0]

^{T}as the initial approximation. Perform four iterations.
7 M

6 (a)
Use appropriate interpolating formula to compute y(82) and y(98) for the data

x | 80 | 85 | 90 | 95 | 100 |

y | 5026 | 5674 | 6362 | 7088 | 7854 |

7 M

6 (b)
i) For the point (x

ii) If f(1)=4, f(3)=32, f(4)=55, f(6)=119: find interpolating polynomial by newton's divided difference formula.

_{0}, y_{0}) (x_{1}, y_{1}) (x_{2}, y_{2}) mension Lagrage's interpolation formula.ii) If f(1)=4, f(3)=32, f(4)=55, f(6)=119: find interpolating polynomial by newton's divided difference formula.

7 M

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

^{rd}rule ii) Simpson's 3/8^{th}rule iii) Weddle's rule usingx | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

f(x)=1/1+x^{2} | 1 | 0.5 | 0.2 | 04 | 0.0588 | 0.0385 | 0.027 |

6 M

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

_{t}(x,0)=0 and u(x, 0)=x(4-x) by taking h=1, k=0.5 upto four steps.
7 M

7 (b)
Solve two dimensional Laplace equation at the pivotal or nodal points of the mesh shown in Fig. Q7(b). To find the initial values assume u

_{4}=0. Perform three iterations including computation of initial values.

7 M

7 (c)
Solve the equation [ dfrac {partial^2 u}{partial t^2}=4 dfrac {partial^2 u}{partial x^2} ] subjects to the conditions u(x, 0)=sinπx, 0 ≤ x ≤ 1; u(0,t)=u(1,t)=0. Carry out computations for two levels, taking h=1/3, k=1/36.

6 M

8 (a)
Find the z-transform of, [ dfrac {n}{3^n}+2^nn^2+4cos (n heta)+4^n+8 ]

7 M

8 (b)
State and prove i) initial value theorem ii) Final value theorem of z-transforms.

7 M

8 (c)
Using the z-transform solve

u

u

_{n+2}+4u_{n+1}+3un_{n}=3^{n}with u_{0}=0, u_{1}=1.
6 M

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