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

BACHELOR IN COMPUTER APPLICATIONS
Term-End Examination

December, 2005

CS71 : COMPUTER ORIENTED NUMERICAL TECHNIQUES

Time: 3 hours
Maximum Marks: 75

1. (a) Write down the Newton - Raphson formula to find √10. Starting from the initial value x₀ = 3, find the next two estimates, Compute upto two decimals. (3)

(b) Show that a root of the equation x³ + x - 6 = 0 lies between 1 and 2. Perform the next two Iterations of Regula - Falsi method towards computing the root. Compute upto two decimal places. (5)

(c) If f(x) = 1/x, find Δ³f(x). (3)

(d) Show, by the divided difference table that a polynomial of degree three may be made to pass through the points (0, 1) (1, 1), (3, 25), (4, 61) and (6,211). You do not need to find the actual polynomial. (3)

(e) Find the cube root of 20 by Lagrange's method from the following data : (3)

(f) Show, only three iterations of Gauss-Jacobi method for solving the equations,
3x + 5y + 2z = 15,
2x + 9 + 4z = 8,
5x + 2y + z = 10. (6)

(g) Evaluate the integral ∫^π/2_-π/2cos x dx
(i) by Trapezoidal Rule,
(ii) by Simpson's 1/3 Rule.
Divide the interval into four equal subintervals. (4)

(h) Given dy/dx = x + y, y(0) = 1. find by Euler's method, y(0.2), y(0.4) and y(0.6), taking h=0.2. (3)

2. (a) Find the root of the equation f(x) = x² - ln x - 2 = 0 by Newton-Raphson's method correct upto two places of decimals only taking x₀ = 1.5. (5)

(b) From the following data, compute f(5) and f(9) using Newton's forward and backward difference formulae, respectively : (5)

(c) The distance of a car at different times is given below :

Using Newton's forward formula find the time when

3. (a) Solve the following simultaneous equations by Gaussiane limitation method: (5)
x₁ + 2x₂ + 3x₃ = 14
3x₁ + x₂ + 5x₃ = 20
2x₁ + 5x₂ + 2x₃ = 18

(b) Find, by the bisection method, the interval of length 0.125 in which the least positive root of the equation x₃ - 4x - 9 = 0 lies. (5)

(c) Evaluate Δ²/E (x²). (5)

4. (a) Solve the following simultaneous equations by the Gauss - Seidel method nearest to a whole number : (6)
2x₁ + 10x₂ + x₃ = 51
x₁ + 2x₂ + 10x₃ = 61
10x₁ + x₂ + 2x₃ = 44

(b) Find the magnitude of the maximum error in computing ∫⁴₂log_ex dx by
(i) Trapezoidal Rule
(ii) Simpson's Rule
where h = 0.5. (4)

(c) Prove that the convergence of Neuwton - Raphson's method is quadratic. (5)

5. (a) Given dy/dx = x + y where y(0) = 1, compute y at x = 0.1 and x = 0.2 by Runge - Kutta fourth order method. Take h = 0.1. Compute upto two decimal places only. (8)

(b) Find the condtion number of
(i) f(x) = 1/1-x
(ii) f(x) = 1/√x
Further, are these functions ill-conditioned ? Give reasons for your answer. (4)

(c) If f(x) = ax² + bx + c (a, b, c, ∈ R) then show that f[1, 2, 3] = a. (3)