Multiple Choice Questions - Numerical Methods

Multiple Choice Questions - Numerical Methods - Undergraduate Level

1. VIVEKANANDA COLLEGE, TIRUVEDAKAM WEST
DEPARTMENT OF PHYSICS
MCQ Question Bank for
NUMERICAL METHODS – 06CT42 (April 2020)
Prepared by Dr. N.Meenakhisundaram
Unit - I: The Solutions of Numerical Algebraic and Transcendental Equations:
1.f(x) = 2x
3
– 9x
2
+ 12x + 6 is a polynomial of degree
a) two b) three c) one d) four

2. f(x) = a + be
x
+ c sinx + d log x is an example of
a) algebraic equation b) polynomial equation c) transcendental equation d) linear equation

3. Find the odd one out
a) Bisection method b) Bolzano’s method c) Interval halving method d) Regula-Falsi method

4. Newton Raphson method is also called
a) Method of tangents b) Method of Chords c) Bisection method d) All the above

5. Newton’s method uses
a) Euler’s algorithm b) Taylor expansion c) Interpolation formula d) Lagrange algorithm

6. Order of convergence of Newton Raphson method is
a) 1 b) 2 c) 2.5 d) 1.8
7. An iteration method the condition for the convergence of the sequence to the root is
a) xc b) '1x  c) '1x  d) '1x 
8. If )(xf is a real continuous function in ],[ba , and 0)( bfaf , then for 0xf , there is (are)
_________ in the domain ],[ba .
a) one root b) an undeterminable number of roots c) no root d) at least one root

9. For an equation like0
2
x , a root exists at 0x . The bisection method cannot be adopted to solve
this equation in spite of the root existing at0x because the function 
2
xxf
a) is a polynomial b) has repeated roots at 0x c) is always non-negative d) has a slope equal to
zero at 0x

10. Choose the algebraic equation from the following _____________________.
a) 2
10xx   b) 3 sin 2 0xx   c) log sin 2 0xx   d) 2
2 sin 0
x
e x x  

11. Choose the transcendental equation from the following _________________.
a) 3
10x b) 2
10xx   c) 1x d) 10
x
e

2. 12. Regula – Falsi method is also known as a __________________.
a) Method of tangents b) Method of chords c) Method of false position d) Method of slopes

13. Newton-Raphson method is used to find the root of the equation x
2
– 2 = 0. If iterations are
started from - 1, then iterations will
a) converge to -1 b) converge to √2 c) converge to -√2 d) not converge

14. In the Regula – False method we approximate the curve of the function fx by a
__________.
a) Tangent b) Chord c) Normal d) Pair of tangents
15. Bisection method is also known as ________________.
a) Regular false method b) Bolzano method c) Method of false position d) Method of tangents
16. One root of the equation 10
3 log 6
x
x lies between
a) 01and b) 12and c) 23and d) 34and
17. One root of 3 1 cos 0xx   lies is the interval _________________.
a)  ,
2

 b)  0,
2
 c)  ,0
2

 d)  ,
2




18. In 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 cube of the error in the previous stage d) Square root of the error in the previous stage.

19. Newton Raphson’s iteration formula is ______________________.
a) 

0
'
n
n
n
fx
xx
fx
 b) 

1
'
n
nn
n
fx
xx
fx

 c) 

1
'
n
nn
n
fx
xx
fx

 d) 
1n n n
x x f x



20. The order is a measure for the speed of ______________________.
a) iteration b) convergence c) divergence d) non convergence

Unit - II: Solutions of Simultaneous Linear Algebraic Equations
21. Gauss-Elimination method of solving Simultaneous Linear Algebraic Equation is
a) direct method b) indirect method c) iterative method d) interactive method

22. In Gauss-Elimination method the given matrix is converted in to
a) unit matrix b) upper triangular matrix c) null matrix d) lower triangular matrix

23. The fastest method of solving Simultaneous Linear Algebraic equation is
a) Gauss-Elimination method b) Gauss-Jordan method c) Gauss-Seidal method d) All the above

24. The method of obtaining the solution of the system of equations by reducing the matrix A to
____________ is known as Gauss – Jordan elimination method
a) upper triangular matrix b) lower triangular matrix c) diagonal matrix d) null matrix

3. 25. Gauss Jordan method is __________________ method.
a) direct b) indirect c) interactive d) iterative

26. The modification of Gauss elimination method is Gauss ______ method.
a) Elimination b) Jacobi c) Jordan d) Seidal

27.
Gauss elimination and Gauss Jordan methods are _______.

a) iterative b) interpolation c) direct d) indirect

28.
In the Gauss elimination method for solving a system of linear algebraic equations,
triangularization leads to
a) diagonal matrix b) lower triangular matrix c) singular matrix d) upper triangular matrix

29.
Given [A] = then [A] is a ______________ matrix.



a) diagonal matrix b) upper triangular matrix c) singular matrix d) lower triangular matrix

30. A square matrix [A] is lower triangular if
a) ija
ij ,0 b)jia
ij ,0 c)jia
ij ,0 d) ija
ij
,0
31. The following system of equations has ____________ solution(s).
x + y = 2; 6x+6y =12
a) no b) infinite c) two d) unique

32. Division by zero during forward elimination steps in Naïve Gaussian elimination of the set of
equations [A][X]=[C] implies the coefficient matrix [A] is
a) invertible b) nonsingular c) not determinable to be singular or nonsingular d) singular

33. Consider the following system of linear equation
3x + y – z = 10, x + 5y + 2z = 18, x + 4 y + 9z = 16
If current approximation is x = 3.33, y = 2.93, z= 0.1, then the Gauss-Seidal method will give
next approximation as x = 2.39, y = 3.08 and z =
a) 1.2 b) 0.21 c) 0.14 d) 0.19

34. Consider the following system of linear equation
3x + y + z = 0, x + 4y – z = 1, 2x – y + 5z = 2
If current approximation is x = 0, y = 0.25, z= 0.45, then the Gauss-Seidal method will give next
approximation as x = – 0.23, y = 0.42 and z =
a) 0.58 b) 0.48 c) 0.7 d) 0.24

35. The goal of forward elimination steps in Naive Gauss elimination method is to reduce the coefficient
matrix to a (an) _____________ matrix. 6 2 3 9
0 1 2 3
0 0 4 5
0 0 0 6







4. a) diagonal b) identity c) lower triangular d) upper triangular

36. A square matrix [A]nxn is diagonally dominant if
a) ,
1




n
ji
j
ijii aa i = 1, 2, …, b) ,
1




n
ji
j
ijii aa i = 1, 2, …, n and ,
1




n
ji
j
ijii aa for any i = 1, 2, …, n
c) ,
1



n
j
ijii aa i = 1, 2, …, n and ,
1



n
j
ijii aa for any i = 1, 2, …, n d) ,
1



n
j
ijii aa i = 1, 2, …, n
37. Which of the following statement is wrong?
a) If two linear systems have the same solution set, then they are equivalent.
b) The augmented matrix and the coefficient matrix of a linear system have the same number of rows
c) For a linear system, the number of columns of augmented matrix is larger than the number of
columns of coefficient matrix by 1
d) The augmented matrix and coefficient matrix have the same number of columns

38. Which of the following statement is true?
a) Each elementary row operation on an augmented matrix never change the solution set of the
associated linear system b) Two matrices are equivalent if they have the same number of rows
c) If two linear systems have the same solution set, then they have the same augmented matrix
d) It two linear systems have the same coefficient matrix, then they have the same solution set

39. Which of the following row operations on a matrix may not be an elementary operation?
a) replace one row by the sum of itself and other two rows b) multiply all entries of a row by a
number c) interchange the first row and the last row d) replace one row by the difference of itself
and another row

40. To ensure that the following system of equations,
17257
52
61172
321
321
321



xxx
xxx
xxx
converges using Gauss-Seidel Method, one can rewrite the above equations as follows:
a) 





























 
17
5
6
257
121
1172
3
2
1
x
x
x b)






























 6
5
17
1172
121
257
3
2
1
x
x
x c)






























 17
5
6
1172
121
257
3
2
1
x
x
x
d) The equations cannot be rewritten in a form to ensure convergence

Unit - III: Interpolation, Central Difference Interpolation Formulae
41. Polynomial Interpolation is used to compute
a) values of argument b) integration c) differentiation d) all the above

42. Which among the following is correct?
a) E = 1 +  b) E = 1 -  c) E =  d) E =  - 1

5. 43. Gauss forward interpolation formula is applicable if u is ________
a) zero b) one c) between 0 and 1 d) greater than 1

44. If interpolation is required near the end of the tabular values we use
a) Newton-Gregory’s forward interpolation formula b) Newton-Gregory’s backward interpolation
formula c) Stirling formula d) Bessel formula

45. _______________ is the process of finding the most appropriate estimate for missing data.
a) finite difference b) iteration c) interpolation d) root finding

46. ______________ formula is the average of Gauss forward and Gauss backward interpolation
formula.
a) Weddle’s b) Stirling’s c) Trapezoidal d) Bessel’s

47. With usual notation, in _______________ formula the value of u lies between –1/2 and 1/2
a) Weddle’s b) Stirling’s c) Trapezoidal d) Bessel’s

48. Gauss forward interpolation formula is used to interpolate the values of y for ____________.
a) 01u b) 10u   c) 11
22
u   d) 12u
49. Gauss backward interpolation formula is used to interpolate the values of y for values of u
lying between _____________.
a) 0 and 1 b) – 1 and 0 c) – 1 and 1 d) 1 and 2

50. For interpolation with unequal intervals, we can use _____ to get the derivative value.
a. Newton Forward Interpolation Formula. b. Newton Backward Interpolation Formula.
c. Newton Forward Difference Formula. d. Lagrange’s Interpolation Formula.

51. Newton’s forward interpolation formula is used to interpolate the value of y is _____________.
a) nearer to the beginning b) nearer to the end c) nearer to the middle d) nearer to one third

52. Newton’s forward in interpolation polynomial is used to extrapolating values of y to the ____ of
the beginning.
a) right b) left c) centre d) one third

53. _____ interpolation formula is used to interpolate the value of y near the end of the set of
tabulated values.
a) Newton’s forward b) Newton’s backward c) Stirling’s d) Bessel’s

54. Newton’s backward interpolation formula is used to extrapolate the values of y to the

6. _________ of the last tabulated value.
a) right b) left c) middle d) one third

55. Newton’s forward interpolation formula is not applicable to extrapolate near the _______ value
of the table.
a) beginning b) ending c) central d) one third

56. Polynomials are the most commonly used functions for interpolation because they are easy to
a) evaluate b) differentiate c) integrate d) evaluate, differentiate and integrate

57. If h=2, find the value of y when x=5 by Newton Forward Interpolation Formula? X: 4 6 Y: 1 3
a) 0 a) 2 b) 3 d) 4

58. Gauss Forward interpolation formula involves
(a) Even differences above the central line and odd differences on the central line.
(b) Even differences below the central line and odd differences on the central line.
(c) Odd differences below the central line and even differences on the central line.
(d) Odd differences above the central line and even differences on the central line.

59. The technique for computing the value of the function inside the given argument is called
a) interpolation b) extrapolation c) partial fraction d) inverse interpolation

60. The technique for computing the value of the function outside the given argument is called
a) interpolation b) extrapolation c) partial fraction d) inverse interpolation

Unit - IV: Numerical Differentiation and Integration
61. In Interpolation techniques the value of u is given by
a) u = (x – x0) / h b) u = (x0 – h ) / x c) u = h x / x0 d) u = x x0 / h

62. Any given function y = f(x) is rearranged into collocation polynomial y = Pn(x) in the case of
a) Interpolation b) numerical differentiation c) numerical integration d) All the above

63. The general Newton-Cote’s quadrature formula reduces to trapezoidal rule by putting
a) n = 0 b) n = 1 c) n = 2 d) n = 3

64. If we put n = 3 in Newton-Cote’s formula we get
a) Trapezoidal rule b) Simpson’s one-third rule c) Simpson’s three-eighths rule d) Romberg method

65. The error is Trapezoidal rule is of order ______.
a) h b) 2
h c) 3
h d) 4
h

7. 66. The error is Simpson’s 1
3 rule is of order ______.
a) 2
h b) 3
h c) 4
h d) h

67. Putting n ______ in Newton-Cote’s quadrature formula, we get Weddle’s rule.
a) 2 b) 4 c) 6 d) 3
68. Physically, integrating ()
b
a
f x dx means finding the
a) area under the curve from a to b b) area to the left of point c) area to the right of point b
d) area above the curve from a to b
69.     
0
0 2 4 1 3
24
3
n
x
n
x
h
f x dx y y y y y y                 
 is called _______.
a) Trapezoidal rule b) Simpson’s 1
3 rule c) Simpson’s 3
8 rule d) Weddle’s rule
70. 
0
2 3 2
2
0 0 0
1
2
2 3 2
n
x
x
n n n
f x dx h ny y y
 
            

 is called _______.
a) Trapezoidal rule b) Newton’s Cote’s formula c) Simpson’s 1
3 rule d) Weddle’s rule
71. Trapezoidal rule is derived from ______formula.
a) Newton-Cotes b) Newton's forward interpolation c) Newton's backward interpolation
d) Inverse Lagrange's

72. The degree of y(x) in Trapezoidal Rule is _______.
A. 1 B. 2 C. 3 D. 6

73. The degree of y(x) in Simpson's one third rule is____.
A. 1 B. 2 C. 4 D. 6

74. The degree of y(x) in Simpson's three eight rule is ______.
A. one B. two C. three D. six

75. In Simpson’s (1/3)
rd
Rule the number of intervals ______.
A. odd B. even C. multiple of 3 D. multiple of 6

76. 2 3 4
0 0 02
1 11
12
y y y
h

            

 ________
a) 0
xx
dy
dx



 b) 0
2
2
xx
dy
dx



 c) dy
dx d) 1

8. 77. 24
1222
11
yy
hh


          

 __________
a) 0
2
2
xx
dy
dx



 b) 0
xx
dy
dx



 c) ds
dt d) 1

78. Angular velocity = _________.
a) 2
2
d
dt
 b) d
dt
 c) ds
dt d) 1

79. Angular acceleration = ____________.
a) 2
2
d
dt
 b) d
dt
 c) 2
2
ds
dt d) 0

80. 2 3 4
2
1 11
12
n n n
y y y
h

           

 _____________.
a) n
xx
dy
dx



 b) 2
2
n
xx
dy
dx



 c) Zero d) 1

Unit - V Numerical Solutions of Ordinary Differential Equations

81. The differential equation 50 ,322
2
 yxyx
dx
dy is
a) linear b) nonlinear c) linear with fixed constants d) undeterminable to be linear or nonlinear

82. A differential equation is considered to be ordinary if it has
a) one dependent variable b) more than one dependent variable c) one independent variable
d) more than one independent variable

83. To solve the ordinary differential equation
50 ,sin53
2
 yxy
dx
dy
by Euler’s method, you need to rewrite the equation as

a) 50 ,5sin
2
 yyx
dx
dy b)  50 ,5sin
3
1
2
 yyx
dx
dy c) 50 ,
3
5
cos
3
1
3









 y
y
x
dx
dy

9. d) 50 ,sin
3
1
 yx
dx
dy

84. In which of the following method, we approximate the curve of solution by the tangent in each
interval.
a) Picard’s method b) Runge-Kutta method c) Newton’s method d) Euler’s method

85. A differential equation is considered to be ordinary if it has
a) one dependent variable b) more than one dependent variable c) one independent variable
d) more than one independent variable

86. A partial differential equation requires
a) exactly one independent variable b) two or more independent variables
c) more than one dependent variable d) equal number of dependent and independent variables

87. The partial differential equation is classified as

a) elliptic b) parabolic c) hyperbolic d) none of the above

88 .To solve the ordinary differential equation
50 ,sin3
2
 yxxy
dx
dy
by the Runge-Kutta 2
nd
order method, you need to rewrite the equation as
a)50 ,sin
2
 yxyx
dx
dy b) 50 ,sin
3
1
2
 yxyx
dx
dy c)50 ,
3
cos
3
1
3









 y
xy
x
dx
dy
d) 50 ,sin
3
1
 yx
dx
dy

89.

and using a step size of h=0.3, the value of y(0.9) using Euler’s method is most nearly

a) -35.318 b) -36.458 c)-658.91 d)-669.05

90. Given 53.0,3
2
 yey
dx
dy
x , and using a step size of 3.0h , the best estimate of 9.0
dx
dy
Runge-Kutta 4
th
order method is most nearly
a) -1.6604 b) -1.1785 c) -0.45831 d) 2.7270

91. Numerical solutions of linear algebraic equations can be obtained by

10. a) Euler’s modified method b) Runge -Kutta Method c) Euler’s method d) Newton’s method

92. Given y(2) most nearly is

a) 0.17643 b) 0.29872 c) 0.32046 d) 0.58024

93. The form of the exact solution to
50 ,32 

yey
dx
dy
x
a) xx
BeAe


5.1 b) xx
BxeAe


5.1 c) xx
BeAe


5.1 d) xx
BxeAe


5.1



94. The following nonlinear differential equation can be solved exactly by separation of variables.
 10000 ,8110
26




dt
d
The value of most nearly is

a) −99.99 b) 909.10 c) 1000.32 d) 1111.10

95. Given
53.0 ,sin53
2
 yxy
dx
dy
and using a step size of h=0.3, the value of using Euler’s method y(0.9) is most nearly
a) −36.318 b) -35.318 c) -658.91 d) -669.05

96. The velocity (m/s) of a body is given as a function of time (seconds) by
  0 ,1ln200  ttttv
Using Euler’s method with a step size of 5 seconds, the distance in meters traveled by the
body from t = 2 to t = 12 seconds is most nearly
a) 3133. 1 b) 3939.7 c) 5638 d) 39.39

97. Given
53.0 ,sin53
2
 yxy
dx
dy
and using a step size of h = 0.3, the value of y =0.9 using the Runge-Kutta 2
nd
order is most nearly

a) –429.74 b) –4936.7 C) -4297.4 D) -493.67

98. The velocity (m/s) of a body is given as a function of time (seconds) by
  0 ,1ln200  ttttv
Using Runge-Kutta 2
nd
order method with a step size of 5 seconds, the distance in meters traveled by
the body from t = 2 to t = 12 seconds is most nearly 2 3 sin 2 , 0 6
dy
y x y
dx
   100

11. a) 3133. 1 b) 3939.7 c) 5638 d) 3904.9

99. To solve the ordinary differential equation
50 ,sin3
2
 yxxy
dx
dy
by the Runge-Kutta 4
th
order method, you need to rewrite the equation as
a)50 ,sin
2
 yxyx
dx
dy b) 50 ,sin
3
1
2
 yxyx
dx
dy c)50 ,
3
cos
3
1
3









 y
xy
x
dx
dy
d) 50 ,sin
3
1
 yx
dx
dy

100. The velocity (m/s) of a parachutist is given as a function of time (seconds) by
),17.0tanh(8.55)( ttv 0t
Using Runge-Kutta 4
th
order method with a step size of 5 seconds, the distance in meters traveled
by the body from t = 2 to t =12 seconds is estimated most nearly as

a) 343. 43 b) 428.97 c) 429.05 d) 393.9

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12. ANSWERS:
UNIT I UNIT II UNIT III UNIT V UNIT V
1.B 21.A 41.D 61.A 81.A
2.C 22.B 42.A 62.D 82.C
3.D 23.C 43.C 63.B 83.B
4.A 24.C 44.B 64.C 84.D
5.B 25.A 45.C 65.B 85. C
6.B 26.C 46.B 66.C 86. B
7.C 27.C 47.B 67.C 87.A
8.D 28.D 48.A 68.A 88. B
9.C 29.B 49.B 69.B 89. A
10.A 30.A 50.D 70.B 90.A
11.D 31.B 51.A 71.A 91. C
12.B 32.C 52.B 72.A 92. C
13.B 33.A 53.B 73.B 93. A
14.B 34.C 54.A 74.C 94. B
15.B 35.D 55.B 75.B 95. B
16.C 36.B 56.D 76.B 96. A
17.B 37.D 57.B 77.A 97. C
18.B 38.A 58.C 78.B 98. D
19.B 39.B 59.A 79.A 99. B
20.B 40.B 60.B 80.B 100. C


Prepared By
Dr. N. Meenakshisundaram
Assistant Professor of Physics
Vivekananda College, Madurai - 625234