B.Tech NUMERICAL METHODS Question paper

[NAREN]

B.Tech Supplimentary Examinations, February 2008

NUMERICAL METHODS

(Aeronautical Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks

1. (a) Find a real root of the equation f(x) = x + logx - 2 using Newton Raphson
method.
(b) Find a root of the equation x = cosx using Regula falsi method. [8+8]
2. (a) Find y(1.6) using Newton’s forward difference formula from the table
x 1 1.4 1.8 2.2
y 3.49 4.82 5.96 6.5
(b) Find y(6) if y(1)=4, y(2)=5, y(7)=5, y(=4 using Lagrange’s interpolation
formula. [8+8]
3. (a) Derive normal equations to fit the straight line y = a + bx
(b) Fit a straight line y = a + bx
x 1 2 3 4 5
y 5 7 9 10 11
[6+10]
4. (a) Find the Fourier cosine transform of 5-2x + 2e-5x.
(b) If F(s) is the complex Fourier transform of f(x), then prove that
i. F (f(x)cosax) = 1
2 [F(s+a) + F(s–a)]
ii. F[f(x - a)] = eisa F(s). [8+8]
5. (a) Find the maximum or minimum values of y
x 0 2 3 4 5
y 4 26 58 112 160
(b) Evaluate
3
R
0
dx
1+x using Weddle’s rule taking h=.5 compare with exact value.
[8+8]
6. (a) Find whether the following system of equations are consistent, if so solve them.
x1 – x2 – 2x3 = –2, 3x1 - x2 + x3 = 6, 3x1 + x2 + 4x3 = 10
(b) Solve the following tridiagonal system
x1 + 2x2 = 7, x1 – 3x2 – x3 = 4, 4x2 + 3x3 = 5 by LU decomposition. [8+8]
7. Use Runge-Kutta method to obtain an approximate solution to the differential
equation dy
dx = y -x+5 at the points x=2.1, 2.2, 2.3 with initial condition y(2)=1.
[16]
1 of 2
Code No: R05012101 Set No. 2
8. Solve the equation @2u
@x2 + @2u
@y2 = 0 in the domain of the figure 8 by Jacobi’s method
[16]

[NAREN]

B.Tech Supplimentary Examinations, February 2008

NUMERICAL METHODS

(Aeronautical Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks

1. (a) Find a real root of x3
-2x2
- 4=0 using iterative method.
(b) Find a root of the equation x3
-4x-1=0 by bisection method correct to three
decimal places. [8+8]
2. (a) Construct difference table for the following data:
x 0.1 0.3 0.5 0.7 0.9 1.1 1.3
f(x) 0.003 0.067 0.148 0.248 0.370 0.518 0.697
And find f(0.6) using a cube that fits at x=0.3, 0.5, 0.7 and 0.9 using Newton’s
forward formula
(b) Find log(337.5) from the following table using Gauss backward formula.
x 310 320 330 340 350 360
logx 2.49136 2.50515 2.51851 2.53148 2.54407 2.55630
[8+8]
3. (a) Derive normal equations to fit the straight line y = a + bx.
(b) Fit a straight line.
x 1 2 3 4 5 6 7 8 9 10
y 52.5 58.7 65.0 70.2 75.4 81.1 87.2 95.5 102.2 108.4
[10+6]
4. Obtain first Four orthogonal polynomials on fn(x) on [–1,1] with respect to the
weight function w(x)=1. [16]
5. (a) Find f' (.3)
x 0 .1 .2 .3 .4 .5 .6
f(x) 30.13 31.62 32.87 33.64 33.95 33.81 33.24
(b) Evaluate
1
R
0
e-x2
taking h = .2 using
i. Simpson’s 1
3rd
ii. Trapenzoidal rule. [8+8]
6. (a) Solve the system of equations by LU decomposition
2x – 3y + z = -3
3x + 4y + 2z = 13
2x – 3y + 4x = 0
1 of 2
Code No: R05012101 Set No. 1
(b) Find whether the following equations are consistent, if so solve them.
x + 2y - z = 3
3x - y + 2z = 1
2x - 2y + 3z = 2
x - y + z = -1. [8+8]
7. Tabulate y(.1), y(.2) and y(.3) using Taylor’s series method given that y' = y2 + x
and y(0)=1 [16]
8. Solve the equation @2u
@x2 + @2u
@y2 = 0 in the domain of the figure 8 by Jacobi’s method
[16]

[NAREN]

B.Tech Supplimentary Examinations, February 2008

NUMERICAL METHODS

(Aeronautical Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks

1. (a) Find a real root of the equation x4
- x - 10=0 by bisection method.
(b) Find p3 17 using Newton- Raphson method. [8+8]
2. (a) Find y(54) from the following table using Newton’s forward difference formula.
x 50 60 70 80
y 205 225 248 274
(b) Find Lagrange’s interpolation formula to find f(5) given that f(1)=2, f(2)=4,
f(3)=8, f(4)=16, f(7)=128. [8+8]
3. (a) Fit a straight line from the following table.
x 0 1 2 3 4
y 1 5 10 22 38
(b) Fit a parabola y=a+bx+cx2.
x 20 40 60 80 100 120
y 5.5 9.1 14.9 22.8 33.3 46.0
[8+8]
4. Use the Fast Fourier transform to determine the trigonometric interpolating poly-
nomial of degree 8 on x2cosx on [-p,p]
5. (a) Find f'(5) from the following table representing y=f(x)
x 0 2 3 4 7 9
y 4 26 58 112 466 922
(b) Evaluate
/2
R
0
sin xdx using
i. Trapezoidal rule
ii. Simpson’s 1
3rd rule by dividing the range into 10 intervals [8+8]
6. (a) Solve the following tridiagonal system
x1 - 2x2=2, 2x1 + 3x2 + 3x3=5, x2 – 3x3 – x4 = 8, x3 – 5x4 = 3 by a
simple method
(b) Find whether the following set of equations are consistent, if so solve them.
5x + 3y + 7z = 4; 3x + 26y + 2z = 9; 7x + 2y + 10z = 5. [8+8]
1 of 2
Code No: R05012101 Set No. 3
7. Using Runga-kutta method with h=0.5 first compute y(0.5), y(1), y(1.5) given that
dy
dx = x+y
2 ; y (0) = 2 then compute y(2) using Milne’s predictor corrector method.
[16]
8. Solve the equation @2u
@x2 + @2u
@y2 = 0 in the domain of the figure 8 by Gauss Seidel
method [16]

[NAREN]

B.Tech Supplimentary Examinations, February 2008

NUMERICAL METHODS

(Aeronautical Engineering)
Time: 3 hours Max Marks: 80
Answer any FIVE Questions
All Questions carry equal marks

1. (a) Find a real root of x3 - 4x - 9=0 using Regula falsi method.
(b) Find a real root of the equation x3 - 9x + 1=0 by bisection method. [8+8]
2. (a) Prove that (1 -
) (1 - ?) = 1.
(b) Express the polynomial x3 - 2x2 + x - 1 in terms of factorial notation.
(c) Find f(2) given that f(0)=5, f(1)=6,f(3)=50 f(4)=105 using Langrange’s inter-
polation formula. [5+5+6]
3. Fit a curve of the form y = A1ex + A2e2x for the following data.
x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
y 1.175 1.336 1.51 1.698 1.904 2.129 2.376 2.646 2.942
[16]
4. Use the Fast Fourier Transform to compute the trignometric interpolating polyno-
mial of degree 4 on [-p, p] for the function f(x) = p(x-p) by Fast fourier transforms
[16]
5. (a) Evaluate
5.2
R
4
log xdx taking h=.2 using Trapezoidal rule
(b) Using Simpson’s (1/3) rule evaluate
b
R
0
ydx [8+8]
x: 0 1 2 3 4 5 6
y: 0 0.4 4.9 25.4 81.8 195.2 373.2
6. (a) Test for consistency the set of equations and solve them if they are consistent.
x + 2y + 2z = 2, 3x - 2y - z = 5, 2x - 5y + 3z = -4, x + 4y + 6z = 0
(b) Solve the system of equation
x1 - 4x2 + x3 = 5, 5x1 + x2 – x3 = 2, 2x1 + 5x2 + x3 = 7 by LU
decomposition. [8+8]
7. Find y(.1) and y(.2) using Picard’s method given that dy
dx = 1 - 2xy, y(0)=0 [16]
8. Solve the equation uxx + uyy = 0 in the domain of the figure 8 by Gauss Seidel
method [16]