Gujarat Technological University 2010 M.C.A Computer Oriented Numerical Methods - Question Paper
Seat No. Enrolment No.
GUJARAT TECHNOLOGICAL UNIVERSITY
MCA. Sem-II Examination July 2010
Subject code: 620005 Subject Name: Computer Oriented Numerical Methods Date: 07 /07 /2010 Time: 11.00 am - 01.30 pm
Total Marks: 70
1. Attempt all questions.
2. Figures to the right indicate full marks.
3. Use of calculators is allowed provided they are silent and battery operated.
4. Intermediate calculation steps and results are to be shown, even while using calculator.
Q.1 (a) Explain the following terms : Blunders, Formulation Errors, Data Uncertainty. 05 Explain Total Numerical Error. How can one control numerical errors ?
(b) Graphically derive Newton-Raphson method to find the root of the equation 05 f(x)=0. Also, explain graphically the pit-falls of the Newton-Raphson method.
(c) Economize ex series to obtain four significant digit accuracy. 04
Q.2 (a) Can Birge-Vieta method be used to find roots of any f(x)=0 ? Find the root of the 07 equation x3 + 2x2 + 10x - 20 = 0 correct upto three significant digits using Birge-Vieta method (Hint : Take r0 = 1).
(b) Give graphical representation of the Successive Approximation method to find 07 the root of the equation f(x)=0, for cases of divergence as well as convergence.
OR
(b) Use Bisection method to find the smallest positive root of the following equation 07 x4 - x - 10 = 0, correct upto four significant digits.
07
|
Q.3 (a) From the following table, find P when t = 142 C and 175 C, using appropriate 07 Newtons Interpolation formula._____ | ||||||||||||
|
|
curve to the following d |
ata by t |
he method of least | |||
|
x : |
1 |
2 |
3 |
4 |
5 |
|
y : |
7.1 |
27.4 |
62.1 |
110.0 |
161.0 |
OR
Q.3 (a) Derive the formula for Newtons Divided Difference Interpolating Polynomial. 07 (b) Obtain the cubic spline approximations for the function f(x)=0 from the 07 following data :
|
x : |
-1 |
0 |
1 |
2 |
|
y : |
-1 |
1 |
3 |
35 |
|
Q.4 (a) The values of pressure and specific volume of super heated steam are as 07 follows : | ||||||||||||
| ||||||||||||
|
Find the rate of change of pressure with respect to volume when V = 2 and V = 8. |
Evaluate Jlog(1 + x2)dx using
0
(i) Trapezoidal rule
(ii) Simpsons rule, taking h = 0.2 for both cases
OR
Q.4 (a) The velocity v of a particle at distances from a point on its linear path is given 07 below :
| ||||||||||||||||||||
|
Estimate the time taken by the particle to traverse the distance of 20 metres, using Simpsons rule. (b) For the following pairs of x and y, find numerically the first and second order 07 derivatives at x = 1.9. |
|
x : |
1.0 |
1.2 |
1.4 |
1.6 |
1.8 |
2.0 |
|
y : |
0 |
0.128 |
0.544 |
1.296 |
2.432 |
4.000 |
Q.5 (a) Find numerically the largest eigen value and the corresponding eigen vectors of 07 the following matrix, using the Power method :
"1 - 3 2"
4 4 -1
6 3 5
(b)
07
Q.5 (a)
07
Vdx = (x + y)e'
(b)
07
equation
with
Given the following differential equation
/ dx x2 x
Compute y(1.1), y(12) and y(1.3) using Runge-Kutta third order method and obtain y(1.4) using Milne-Simpsons predictor corrector method.
OR
State the necessary and sufficient condition for the convergence of Gauss-Seidel method for solving a system of simultaneous linear equations. Hence, solve the following system of equations, using Gauss-Seidel method, correct upto four decimal places.
30x - 2y + 3z = 75 2x + 2y + 18z = 30 x + 17y - 2z = 48
differential
Given the following
y(-0.1) = 0.9053. Compute y(0), y(0.1) and y(0.2) using Runge-Kutta second order method and obtain y(0.3) using Adam-Bashforth-Moultons predictor corrector method.
*************
2
|
Attachment: |
| Earning: Approval pending. |
