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
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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
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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
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