STUPIDSID
1(a)
Expand F(X)=√1−cosx,,0<X<2π in a Fourier series.Hence evaluate 11.3+13.5+15.7+...
7 M
1(b)
Find the half - range sine series of [f(x)=e^{x}=e ] in (0,1).
6 M
1(c)
In a machine the displacement y of a given for a certain angle x as follows:
Find the constant term and the two harmonics in Fourier series expansion of y.
| x | 0 | 30 | 60 | 90 | 120 | 150 | 180 | 210 | 240 | 270 | 300 | 330 |
| y | 7.9 | 8 | 7.2 | 5.6 | 3.6 | 1.7 | 0.5 | 0.2 | 0.9 | 2.5 | 4.7 | 6.8 |
Find the constant term and the two harmonics in Fourier series expansion of y.
7 M
2(a)
Find Fourier transforme−|x| and hence evaluate∫\oe0cosxt1+t2dt.
7 M
2(b)
Find Fourier sine transform ofF(x)={x,0<x≤12−x,1≤x<2.0,x>2.
6 M
2(c)
Solve the integral equation∫\oe0f(x)λxdx=e−λ.
7 M
3(a)
Find various possible solution of one-dimensional heat equation by separable variable method.
10 M
3(b)
A rectangular plate with insulated surface is 10cm wide and so long compared to its width that it may be considered infinite in length without introducing an appreciable error.If the temperature of the short edgey−0 is given by
u−20x0≤X≤5−20(10−X),5≤X≤10
and the two long edges X=00,X=10 as well as the other short edge are kept at 0C,find the temperature u(x,y)
u−20x0≤X≤5−20(10−X),5≤X≤10
and the two long edges X=00,X=10 as well as the other short edge are kept at 0C,find the temperature u(x,y)
10 M
4(a)
Fit a curve of the form y−aebx to the data:
| x | 1 | 5 | 7 | 9 | 12 |
| y | 10 | 15 | 12 | 15 | 21 |
7 M
4(b)
use graphical method to solve the following LPP:
minimizeZ−20x1+30x2Subject to 2x1+2x2≤20;3x1,x2≥0.
minimizeZ−20x1+30x2Subject to 2x1+2x2≤20;3x1,x2≥0.
6 M
4(c)
Solve the following LPP by using simplex method.
maximizeZ−3x1−2x2+5x3Subject to x11 2x2 x3≤4303x1−2x3 ≤ 460x1 ≥ x2≥ 0.
maximizeZ−3x1−2x2+5x3Subject to x11 2x2 x3≤4303x1−2x3 ≤ 460x1 ≥ x2≥ 0.
7 M
5(a)
Useing the Gauss-Seidal iterative method to solve the system of linear equations.27x+6y+z=85;6x+15y-2z=72;x-y+54z=110.Carry out 3 iterations by taking the initial approximation to the solution as (2,3,2). Consider four decimal places at each stage for each variable.
7 M
5(b)
Using the Newton-Raphson method,find the real root of the equation xsinx+cosx-0 near to xπ,carry out four iterations (x in radians).
6 M
5(c)
Find the largest Eigen value and the corresponding Eigen vector of the matrix
A=(41−123−1−215)\] by power method. Take (100)as the initial vector.Perform 5 iterations.
A=(41−123−1−215)\] by power method. Take (100)as the initial vector.Perform 5 iterations.
7 M
6(a)
Find f(0.1) by using Newton's forward interpolation formula and f(4.99)by using newton's backward interpolation formula form the data:
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| f(x) | -8 | 0 | 20 | 58 | 120 | 212 |
7 M
6(b)
find the interpolating polynomial f(x) by using Newton's divided difference interpolation formula form the data;
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| f(x) | 3 | 2 | 7 | 24 | 59 | 118 |
6 M
6(c)
Evaluate ∫20exdx using Weddle's rule.Taking six equal sub intervals,compare the result with exact value.
7 M
7(a)
Solve ∂2∂x2+∂2u∂y2−0 in the following square mesh.carry out two iterations.
7 M
7(b)
Solve the Poisson's equation▽2u=8x2y2 for the square mesh given below with u-0 on the boundary and mesh length,h-1.
6 M
7(c)
Evaluate the pivotal values of ∂2∂xt2=16∂2u∂x2taking h-1 upto t-1.25. The boundary conditions are u(0,t),u(5,t)=0,∂u∂t(x,0)=0,u(x,0)=x2(5−x).
7 M
8(a)
Find the Z-transforms of i)(12)n+(13)n
3ncosπn4
3ncosπn4
7 M
8(b)
State and prove initial value theorem in Z-transforms.
6 M
8(c)
Solve the difference equation
u_{n+2}-2u_{n+1}+u_{n};u_{n}=2,u_{1}=1\].
u_{n+2}-2u_{n+1}+u_{n};u_{n}=2,u_{1}=1\].
7 M
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