Test Papers / Previous Question Papers of IGNOU CS71 Computer Oriented Numerical Techniques December 2005

BACHELOR IN COMPUTER APPLICATIONS
Term-End Examination

December, 2005

CS71 (S) : COMPUTER ORIENTED NUMERICAL TECHNIQUES

Time: 3 hours
Maximum Marks: 75

1. (a) Write down the Newton - Raphson's formula to compute the cuberoot of 10. Starting from the initial value x₀ = 2.0 find 3√10 correct upto three decimals. (5)

(b) Show that a root of the equation x³ - 2x - 5 = 0 lies between 2 and 3 Perform the next two iterations of Regula - Falsi method towards computing the root. (5)

(c) Find Δ³x³. (3)

(d) Compute f(6) using Lagrange's formula on the following data : (3)

(e) From the values tabulated find the point x, where the function f has maximum value : (3)

Use the forward difference formula.

(f) Show three iterations of Gauss - Jacobi method for solving the equations : (6)
5x₁ + x₂ + 2x₃ = 10
3x₁ + 8x₂ + x₃ = 13
x₁ + x₂ + 4x₃ = 10

(g) Given dy/dx = y-x/y+x, y(0) = 1. Find, by Euler's method, y(0.1), y(0.2) and y(0.3) taking h=0.1. (5)

2. (a) The approximate root of the equation x³ - x - 1 = 0 is 1.3. Show that the iterative scheme x_n+1, = g(x_n) will not converge if we take g(x) = x³ - 1, but wilt converge for g(x) = (1 + x)^1/3. Hence find the root, correct upto three decimals. (5)

(b) From the following data compute f(1) and f(10) using Newton's forward or backward difference formulas, whichever is appropriate. (5)

(c) Using divided difference, show that the following data represents a third degree polynomial. Obtain this polynomial. Hence, find the vatue of f(5). (5)

3. (a) Solve the following simultaneous equations by the Gaussian elimination method. (5)
x₂+ 2x₃= 1
3x₁ +x₂ +x₃= 4
x₁ + 2x₂ + 3x₃ = 3

(b) Evaluate the integral I = ∫²₀xe^-xdx
by (i) Trapezoidal Rule, (ii) Simpson's Rule. Divide the interval into four subintervals. (6)

(c) Show that the divided difference f[x_o, x₁, x₂] is independent of the order of the arguments x₀, x₁ and x₂ i.e., f [x_o, x₁, x₂] = f [x₁, x₀, x₂] etc. (4)

4. (a) Find the solution of the following simultaneous equations by the Gauss - Seidel method correct upto the nearest integer. (8)
2x + y + 5z = 8
4x + y + 2z = 8
x + 5y + z = 1 2

(b) Given dy/dx = x² + y², y(0) = 1. obtain the first five terms of Taylor's series for computing y(x_o + h). Hence compute y(0.5) taking h = 0.5. (7)

5. (a) Compute y(0.2) by the Runge - Kutta fourth oder method for the differential equation (5)
dy/dx = ½ (x+y), y(0) = 1.
Take h=0.2

(b) Determine the root of the function x²-√x-2 = 0 correct to three decimal places using the (i) bisection method.
(ii) secant method. (10)