Test Papers / Previous Question Papers of IGNOU ET302 (A) Computer Programming & Numerical Analysis December 2005

B.Tech. Civil (Construction Management) /
B.Tech. Civil (Water Resources Engineering)
Term-End Examination

December, 2005

ET-302(A) : COMPUTER PROGRAMMING & NUMERICAL ANALYSIS

Time: 3 hours
Maximum Marks: 70

1. (a) Perform three iterations of the Regula - Falsi method to find the root of the equation x⁴ - x - 10 = 0 in the interval [1, 2]. Assume suitable initial approximations.

(b) Consider the following system of equations :
4x - y + z = 7
4x - 8y + z = - 21
-2x + y + 5z = 15
Perform only four iterations of Jacobi iteration or Gauss - Seidel itertion method for solving the equations. Assume (x_o, y_o, z_o) = (1,2,2) to start with. (7,7)

2. (a) Use the LU decomposition method to solve the system of equations :
x + y + z = 1
4x + 3y - z = 6
3x + 5y + 3z = 4

(b) Find the dominant eigenvalue and the corresponding eigenvector correct to two decimal places of the matrix

       2  -1  0
A = -1   2 -1
       0  -1  2

3. (a) Verify the equivalence of the following relations :

(i) Δ² cos (2x) = -4 sin²h cos (2x + 2h)

(ii) (Δ²/E) x³ = 6x

(iii) E (2μδ - Δ) = Δ

(b) Using Muller's method, find a root of the equation x³ - x² - x - 1 = 0 which lies between 3 and 4 correct to 3 decimal places. (6,8)

4. (a) The values of a polynomial of degree 5 are tabulated below. If f(4) is known to be in error, find its correct value.

(b) In the table below. the values of y are consecutive terms of a series of which 43 is the 6th term. Find the first and the tenth terms of the series.

5. (a) Solve the following system of linear equations with Gaussian elimination method :
2x₁ - 2x₂ + 5x₃ = 6
2x₁ + 3x₂ + x₃= 13
-x₁ + 4x₂ - 4x₃ = 3

(b) Find Lagrange's interpolating polynomial for the following data. Also obtain the value of f(2) using polynomial.

6. (a) Using second order Taylor series method upto the terms of h² solve the equation, dy/dx = 3x + y/2; y(0) = 1. Find y(0.4) taking h=0.2.

(b) Solve the differential equation
dy/dx = 1/x² - y/x - y², y(1) = -1 by Runge - Kutta method for x = 1 to x = 5/3 in steps of h = 1/3 by carrying out calculation in two steps. (7, 7)

7. (a) The sum of the squares of the first n natural numbers is given by

       n(n+1)(2n+1)
s = ----------------
              6

Write a program that will find s for n = 10 (10) 250, i.e., n = 10, 20, 30, ....., 250

(b) Using logical IF statements, write a program that calculates and prints

f(x) = {3x + 5x3 for 4.3≤x<9.1
         {6x + 8x2 for 9.1≤x<15.5

for x varying from 5.0 to 15.0 in steps of 0.5 (7, 7)

8. (a) The values of x are to be tabulated from the formula
x = (sin t e^-2t + log t) / (5t - cos t)
for t = 1.0 to 5.0 in steps of 0.1 Write a program to compute and print x for each value of t in the given range.

(b) Write a program to compute ⌊n. Test for n = 0, 1 and 3. (7,7)