Root of equation & Error approximation

Assertion and Reasoning 35 Pages

Savitribai Phule Pune University

Mechanical Engineering

Contributed by

Aarav Patel

Multiple Choice Questions (MCQ)
Unit I Root of equation & Error approximation

Bisection Method
1. Suppose we want to find a root of the polynomial x
3
- 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?

A. 2.875 B. 2.125
B. 2.5 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 x
3
+ 3x
2
- 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) B. (1, 2)
C. (2, 3) D. (3, 4)

5. A root of the equation cos(x) - x * exp(x) = 0 , the first initial guess lies between.

A. (0, 1) B. (-1,-2)
C. (-2, 3) D. (3, 4)

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Newton-Raphson methods

5.Solve the equation e
x
− 4x=0 using Newton-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.

6. Use the Newton-Raphson method to solve 2x
3
−6x
2
+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.

7. 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.

8. The function f(x) =2X
3
− 2X
2
− 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.
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9. The order of convergence of Newton-Raphson iterative algorithm is

A. First order B. Second order
C. Third order D. None of the above.

10 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.

11. The Newton-Raphson method of finding roots of nonlinear equations falls under the category
of _____________ methods.

(A) Bracketing (B) Open
(C) Random (D) Graphical

12. The next iterative value of the root of X
2
− 4 = 0 using the Newton-Raphson method, if the
initial guess is 3, is

(A) 1.5 (B) 2.067
(C) 2.167 (D) 3.000

13. Newton Raphson method is also called as
A. Method of chords
B. Interval halving method
C. Method of linear interpolation
D. Method of tangents

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14. 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 –

15. Which iterative method requires single initial guess root?
A. Bisection method
B. Secant method
C. Method of false position
D. Newton Raphson Method

16. If initial guess root of the equation x
3
–5x + 3 = 0 is 1, then first approximation for the root by
Newton Raphson method is:

(a) 0.5 (b) 1.5
(c) 1.0 (d) None of the above

17. 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

18. Fourth degree equations are also called _______ equations.
A. quadratic B. cubic
C. linear D. bi-quadratic

Page 4
19. In which of the following methods proper choice of initial value is very important?
A. Newton Raphson Method
B. Bisection Method
C. Iterative Method
D. Regula Falsi Method

20. 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

21. The root of x
3
- 2x - 5 = 0 correct to three decimal places by using Newton-Raphson method
is
A 2.0946 B. 1.0404
C. 1.7321 D. 0.7011

23.The Newton-Raphson method of finding roots of nonlinear equations falls under the category
of _____________ methods.
(A) bracketing
(B) open
(C) random
(D) graphical

24.The Newton-Raphson method of finding roots of nonlinear equations falls under the category
of _____________ methods.
(E) bracketing
(F) open
(G) random
(H) graphical

Page 5
25.The Newton-Raphson method of finding roots of nonlinear equations falls under the category
of _____________ methods.
(I) bracketing
(J) open
(K) random
(L) graphical

26.The Newton-Raphson method formula for finding the square root of a real number
R
from
the equation
0
2
=− Rx
is,
(A)
2
1
i
i
x
x =
+

(B)
2
3
1
i
i
x
x =
+

(C)








+=
+
i
ii
x
R
xx
2
1
1

(D)








−=
+
i
ii
x
R
xx 3
2
1
1

27.The next iterative value of the root of
04
2
=−x
using the Newton-Raphson method, if the
initial guess is 3, is
(A) 1.5
(B) 2.067
(C) 2.167
(D) 3.000

Page 6
28.The root of the equation
0)( =xf
is found by using the Newton-Raphson method. The
initial estimate of the root is
3
0
=x
,
( )
53 =f
. The angle the line tangent to the function
)(xf

makes at
3=x
is
°57
with respect to the x-axis. The next estimate of the root,
1
x
most nearly
is
(A) –3.2470
(B) −0.2470
(C) 3.2470
(D) 6.2470

29.The root of
4
3
=x
is found by using the Newton-Raphson method. The successive iterative
values of the root are given in the table below.
Iteration
Number
Value of Root
0
2.0000
1
1.6667
2
1.5911
3
1.5874
4
1.5874

The iteration number at which I would first trust at least two significant digits in the
answer is
(A) 1
(B) 2
(C) 3
(D) 4

30.The ideal gas law is given by

RTpv =

where
p
is the pressure,
v
is the specific volume,
R
is the universal gas constant,
and
T
is the absolute temperature. This equation is only accurate for a limited range
Page 7
of pressure and temperature. Vander Waals came up with an equation that was
accurate for larger ranges of pressure and temperature given by

( )
RTbv
v
a
p =−






+
2

where
a
and
b
are empirical constants dependent on a particular gas. Given the value of
08.0=R
,
592.3=a
,
04267.0=b
,
10=p
and
300=T
(assume all units are
consistent), one is going to find the specific volume,
v
, for the above values. Without
finding the solution from the Vander Waals equation, what would be a good initial guess
for
v
?
(A) 0
(B) 1.2
(C) 2.4
(D) 3.6

31
f(a) < 0, f(b) > 0 and if x
0
∈ (a, b)is first approximation with f(x
0
) < 0 then in bisection
method,
(a) x
0
is to be replaced by a (b) ais to be replaced by x
0

(c) bis to be replaced by x
0
(d) x
0
is to be replaced by b

32
For real root of an equation x
3
– 2x – 5 = 0, the root lies between
(a) 0 and 1 (b) 2 and 3 (c) 1 and 2 (d)none of them

33
From the following _______ method is not iterative method.
(a) False position (b) Bisection (c) Lagranges (d)none of them

Page 8
34
For the function f(x): x
3
– 2x – 5 = 0 if the root of equation lies between (2, 3) and if at i
th

iteration c= 2.5 then next approximation by bisection method gives c =
(a)
3+2.75
2
(b)
2 2.5
2
+
(c)
3+2.5
2
(d) none of them
35
If in a method of successive approximation, the root of equation lies between 1 and 2,
( )
2
1
1
=
−
gx
x
, and initial guess is 1.25 then next approximation is
(a) 0.5625 (b) 1.2177 (c) 1.7777 (d)none of them

36
From the following _______ method is the best method to obtain root of equation f(x) = 0.
(a) False position (b) Bisection (c) Newton’s Raphson (d)none of them

37
Absolute error is defined as
(a) Present Approximation – Previous Approximation
(b) True Value – Approximate Value
(c) abs (True Value – Approximate Value)
(d) abs (Present Approximation – Previous Approximation)

38
The number 0.01850 x 10
3
has ________ significant digits
(a) 3 (b) 4 (c) 5 (d) 6

39
For an equation like x
2
= 0, a root exists at x = 0. The bisection method cannot be adopted
to solve this equation in spite of the root existing at x = 0 because the function f(x) =x
2

(a) is a polynomial (b) has repeated roots at x= 0
(c) is always non-negative (d) has a slope equal to zero at x= 0
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40
If for a real continuous function f(x), f(a)f(b)<0, then in the range of [a,b] for f(x)=0, there
is (are)
(a) one root (b) an undeterminable number of roots
(c) no root (d) at least one root

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Root of equation & Error approximation

Root of equation & Error approximation

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