Biju Patnaik University of Technology 2008-2nd Sem M.C.A Numerical Method - Question Paper
Total number of printed pages -6 MCA
SCM 2006Second Semester Examination - 2008
I raw
Time: 3 Hours
Answer Question No. 1 which is compulsory and any five from the rest.
The figures in the right-hand margin indicate marks.
1. Answer the following questions : 2 x 10
(a) What is an error ? What are the different characteristics and types of error ?
(b) State the rules of rounding off decimal number correct upto n significant digits.
(c) Write -4.268106 and 0.00518789 in floating point form with 4 significant digits with round off.
(d) State the basic difference between Secant method and method of false position in solving an equation.
te,
(e) What is the geometrical interpretation of Newton-Raphson method to solve an equation ?
united we stariiL.
(f) What is ill conditioning of a system of linear equations ? How can you overcome this problem ?
(g) What is an eigen value of a matrix ? Why are eigen value problems important ?
(h) Is Euler method to solve a differential equation accurate enough for practical problems ? Can it be improved ?
(i) What do you mean by single step and multi-step method to find the solution of a
differential equation ? Give one example in each case.
(j) Howto calculate error in Simpsons 1/3 rule in evaluating an integral ?
2. (a) Determine a real root of the equation
x3 5 = 80 using method of false position, correct upto three decimal places. 5
(b) What do you mean by the rate of convergence of an iterative method ? Determine the rate of convergence of Newton-Raphson method to find the solution of an equation.
5
3. (a) Solve the following system of linear equa
tions by Gauss elimination using partial pivoting: 5
2x + y + z = 1
5x + 2y + 2z = -4
3x + y + z = 5
(b) Solve the following system of linear equations by Gauss-Seidel method correct upto two decimal places: 5
10x + 2y - z =27
-3x-6y + 2z = -61.5
x + y + 5z = -21.5
4. (a) Using least square regression, find a
straight line to the following given data: 5
x:1 234 5 678 9
y : 1 1.5 2 3 4 5 8 10 13
(b) Using centered difference approximation, estimate the first and second order derivative of y = ex at x = 2 for the step size h = 0.1. 5
5. (a) Derive and estimate the error of Trape
zoidal rule for numerical integration of a function f in the range [a, b], 5
(b) Using Simpsons 3/8 rule evaluate the following integral by using 9 sub-intervals of equal width : 5
xdx.
o
6. (a) Solve the following differential equation by
using modified Eulers method for y (4.1) and y (4.2), taking h = 01 5
5x + y2 - 2 = 0 dx
(b) Solve the following differential equation for y(0.1) and y(0.2) using Runge-Kutta method of fourth order. 5
dy 1
~r=- where y (0) = 1.
dx x + y J v '
7. (a) Find all eigenvalues and eigenvectors of
the matrix 5
(b) Find the inverse of the following matrix by using LU decomposition. 5
ajfi) -0.1 -0.2 I
m 7 0 a31 tyl -0.2 io.o ;
8. (a) Evaluatethe polynomial y = x3-7x2 + 8x-
0.35 at x = 1.37. Use 3-digit arithmetic with chopping. Also evaluate the percent relative error. 5
(b) Discuss about stability and condition of a mathematical problem. Also compute and interpret the condition number for
f(x) = tanx forx=f+ai[||[- 5
SCM 2006 6 -C
|
Attachment: |
| Earning: Approval pending. |
