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Matoshri college of Engineering & Research Centre, Nashik Bisection Method Newton-Raphson methods 1. Suppose we want to find a root of the polynomial x3 - 5x. Using the Bisection method and starting boundaries a = 2 and b = 4, what is the third approximation to the root obtained by the algorithm? Solve the equation ex− 4x=0 using Newton- A. 2.875 B. 2.5 B. 2.125 C. 3.0 2. Which method has slow convergence? (a) false poison (b) Secant (c) Newton-Raphson (d) Bisection 3. One root of the equation x3 + 3x2- 5x + 2 = 0 lies between: (a) –5 and –4 (b) –4 and –3 (c) 0 and 1 (d) –1 and +1 4. The root of the equation e power x=4x lies between________. A. (0, 1) C. (2, 3) B. (1, 2) D. (3, 4) 5. A root of the equation cos(x) - x * exp(x) = 0 , the first initial guess lies between. A. (0, 1) C. (-2, 3) B. (-1,-2) D. (3, 4) Regula-Falsi Method 1. _________method is faster than bi-section method. A. Secant B. Newton-Raphson C. Bisection D. Regula-falsi Secant Method 1. Next iterative value of the root of using secant method, if the initial guesses are 1 and 2, of a equation is x4-x-10 = 0 (A) 1.7142 (C) 5.5000 (B) 2.5000 (D) 5.7143 Chebyshev Method Numerical Methods & C Programming Raphson iteration. A. x=0.61906 and x=1.51213 B. x=0.35 and x=2.1 C. x=0.35740 and x=2.15329 D. Newton-Raphson iteration cannot be used since the answer oscillates between 2 and −2. 2. Use the Newton-Raphson method to solve 2x3−6x2+6x−1=0 to 4 decimal places. A. There is no solution since the curve is always increasing. B. x=0.2063. C. x=0.7351. D. Newton-Raphson cannot be used because the tangents to the curve do not cut the axes on the interval 0≤x≤1. 3. Newton-Raphson method will always converge to a solution for f(x) =0 on the interval a≤x≤b if certain conditions are met. Which of the following is not one of these conditions? A. f is continuous on the interval a≤x≤b. B. f(a) and f(b) have opposite signs. C. f′′(x) does not change sign on the interval a≤x≤b. D. f′(x) =0 on the interval a≤x≤b. 4. The function f(x) =2X3 − 2X2 − 3X + 2 has a root between 0 and 1. Which of the following conditions fail? A. f(0) and f(1) have opposite signs. B. f′(x)≠0 on 0≤x≤1. C. f′′(x) does not change sign on the interval 0≤x≤1. D. The tangents at 0 and 1 cut the axes in the interval 0≤x≤1. 6. The order of convergence of Newton-Raphson iterative algorithm is A. First order Unit – 3 B. Second order Page 1 Matoshri college of Engineering & Research Centre, Nashik C. Third order D. None of the above. 7. Newton Raphson method of solution of numerical equation is not preferred when A. The graph of f(x) is nearly horizontal where it crosses the x-axis. B. The graph of f(x) is nearly vertical where it crosses the x-axis. C. Both conditions (A) and (B) above prevail. D. None of the above. 8. The Newton-Raphson method of finding roots of nonlinear equations falls under the category of _____________ methods. (A) Bracketing (C) Random (B) Open (D) Graphical 14. If initial guess root of the equation x3–5x + 3 = 0 is 1, then first approximation for the root by Newton Raphson method is: (a) 0.5 (c) 1.0 (b) 1.5 (d) None of the above 18. Newton-Raphson method is applicable the solution of ______. A. Both algebraic and transcendental equations B. Both algebraic and transcendental and also used when the roots are complex C. Algebraic equations only D. Transcendental equations only 19. Fourth degree equations are also called _______ equations. 9. The next iterative value of the root of X2− 4 = 0 using the Newton-Raphson method, if the initial guess is 3, is A. quadratic C. linear (A) 1.5 (C) 2.167 20. In which of the following methods proper choice of initial value is very important? (B) 2.067 (D) 3.000 11. Newton Raphson method is also called as A. B. C. D. Method of chords Interval halving method Method of linear interpolation Method of tangents 12. The Iterative formula for Newton-Raphson method is: A. Xn+1 = f (Xn) C. Xn+1 = Xn – B. Xn+1 = Xn- 1 – D. Xn+1 = Xn – 13. Which iterative method requires single initial guess root? A. Bisection method B. Secant method C. Method of false position D. Newton Raphson Method B. cubic D. bi-quadratic A. Newton Raphson Method B. Bisection Method C. Iterative Method D. Regula Falsi Method 21. In the case of Newton-Raphson method the error at any stage is proportional to______. A. the error in the previous stage B. the square of the error in the previous stage C. the cubic of the error in the previous stage D. square root of the error in the previous stage 27. The root of x3 - 2x - 5 = 0 correct to three decimal places by using Newton-Raphson method is A 2.0946 C. 1.7321 B. 1.0404 D. 0.7011 28. Newton-Raphson method of solution of numerical equation is not preferred when Numerical Methods & C Programming Unit – 3 Page 2 Matoshri college of Engineering & Research Centre, Nashik A Graph of f(x1) is vertical B. Graph of x(y) is not parallel C. The graph of f(x) is nearly horizontal-where it crosses the x-axis. D. None of these 29. A root of the equation x3 - x - 11 = 0 correct to four decimals using bisection method is, A. 2.3737 B. 2.3838 C. 2.3736 D. None of these 30. Newton-Raphson method is applicable to the solution of 35. We wish to solve x2 - 2 = 0 by Newton Raphson technique. If initial guess is x0 = 1.0, subsequent estimate of x (i.e. x1) will be A. 1.414 C. 2.0 B. 1.5 D. None of these 36. Using Newton-Raphson method, find a root correct to three decimal places of the equation x3 - 3x - 5 = 0 A. 2.275 C. 2.222 B. 2.279 D. None of these N. R. Method for two variables. A. Both algebraic and transcendental Equations B. Both algebraic and transcendental and also used when the roots are complex C. Algebraic equations only D. Transcendental equations only 31. Using Newton-Raphson method, find a root correct to three decimal places of the equation sin x = 1 - x A. 0.511 C. 0.555 B. 0.500 D. None of these Curve fitting 32. The convergence of which of the following method is sensitive to starting value? A. False position B. Gauss seidal method C. Newton-Raphson method D. All of these 25. The general problem of finding equations of approximating curves which fit a given data is called. 34. Which of the following statements applies to the bisection method used for finding roots of functions? 26. The best representative curve to the given set of points for which the sum of the squares of the residuals is a minimum is known as A. Converges within a few iterations B. Guaranteed to work for all continuous functions C. Is faster than the Newton-Raphson method D. Requires that there be no error in determining the sign of the function A. curve fitting B. approximating curve C. empirical relation D. principles of least squares Numerical Methods & C Programming Unit – 3 A. curve fitting C. empirical relation B. approximating curve D. linear form Page 3