NUMERICAL METHODS MULTIPLE CHOICE QUESTIONS

1. NUMERICAL METHODS
MULTIPLE CHOICE QUESTIONS
UNIT I
1. In Regular-falsi method, the first approximation is given by
a)
)()(
)()(
1
afbf
abfbaf
x
−
−
= b)
)()(
)()(
1
afbf
aafbbf
x
−
−
= c)
)()(
)()(
1
bfaf
bafabf
x
−
−
= d)
)()(
)()(
1
bfaf
bbfaaf
x
−
−
=
2. The order of convergence of Regular-falsi method is
a) 1.235 b) 3.141 c) 1.618 d) 2.792
3. Which of the following alter name for method of false position
a) Method of chords b) Method of tangents c) Method of bisection d) Regula falsi method.
4. The order of convergence in Newton-Raphson method is
a) 2 b) 3 c) 0 d) 1
5. The Newton-Raphson algorithm for finding the cube root of N is
a) ( )nnn xNxx /
2
1
1 +=+ b) ( )2
1 /2
3
1
nnn xNxx −=+ c) ( )nnn xNxx /1 +=+ d) ( )2
1 /2
3
1
nnn xNxx +=+
6. If nx is the nth iterate, then the Newton-Raphson formula is
a)
( )
( )n
n
nn
xf
xf
xx
'
1 += − b)
( )
( )1
1
1
' −
−
− −=
n
n
nn
xf
xf
xx c)
( )
( )1
1
1
' +
+
− −=
n
n
nn
xf
xf
xx d)
( )
( )n
n
nn
xf
xf
xx
'
1 −= −
7. Newton’s iterative formula to find the value of N is
a)
( )nnn xNxx /
2
1
1 −=+
b)
( )nnn Nxxx −=+
2
1
1
c)
( )nnn xNxx /
2
1
1 +=+
d)
( )nnn Nxxx +=+
2
1
1
8. The Newton-Raphson method fails when
a) ( )xf '
is negative b) ( )xf '
is too large c) ( )xf '
is zero d) Never fails.
9. Which method is said to be direct method
a) Gauss Elimination b) Newton Raphson c) Regula Falsi d) Gauss Jacobi
10. By Gauss Elimination method the value of x and y for the equations x + y =2, 2x + 3y = 5
a) x = 2 and y = 3 b) x = 3 and y = 4 c) x = 1 and y = 1 d) x = 2 and y = 5
11. Which method is said to be indirect method
a) Regula Falsi b) Gauss Seidal c) Newton Raphson d) Gauss Jacobi
12. In Gauss elimination the given system of simultaneous equations is transformed into
a) Lower triangular matrix b) Unit matrix c) Transpose matrix d) Upper triangular matrix

2. 13. In solving simultaneous equations by Gauss-Jordan method, the coefficient matrix is reduced to
a) Unit Matrix b) Diagonal Matrix c) Null Matrix d) Square Matrix
14. Which method is said to be direct method
a) Gauss Seidal Method b) Gauss Jacobi Method c) Gauss Jordan Method d) All the above
15. Which operation can be used in Gauss Jordan Method
a) Elementary row operations b) Multiplication c) Addition d) Elementary Column operations
16. Which is the faster convergence method
a) Gauss Seidal Method b) Gauss Jacobi Method
c) Gauss Jordan Method d) Gauss Elimination Method
17. Gauss-Seidal iteration converges only if the coefficient matrix is
a) Upper Triangular b) Lower Triangular c) Diagonally dominant d) Banded Matrix
18. As soon as a new value of a variable is found by iteration, it is used immediately in the following
equations, this method is called
a) Gauss Jordan Method b) Gauss Seidal Method
c) Gauss Jacobi Method d) Gauss Elimination Method
19. The inverse of a matrix A is written as A-1
so that AA-1
=A-1
A=
a) Identity matrix b) Null matrix c) Singular matrix d) Inverse matrix
20. What type of Eigen value can be obtained using Power method
a) Largest eigen value b) Smallest eigen value c) Eigen vector d) Characteristic equation
21. If all the eigen values of a matrix A are distinct then the corresponding eigen vectors are
a) Linearly dependent b) Linearly independent
c) Linearly dependent or dependent d) Independent
UNIT II
1. Interpolation formulae are based on the fundamental assumption that the data can be expressed as
a) A linear function b) A quadratic function c) A polynomial function d) None of the above
2. Using Newton’s forward interpolation formula find the value of f(1.6), if
x 1 1.4 1.8 2.2
y 3.49 4.82 5.96 6.5
a) 5.54 b) 5.45 c) 5.35 d) None of these
3. Which of the following symbol is called forward difference operator
a) ∆ b) ∇ c) δ d) E
4. Interpolation is helpful is estimating
a) Missing values of a series b) An intermediary value for a given argument
c) The argument for a given entry d) All the above

3. 5. Newton’s method of divided differences is preferred when
a) When the interpolating value of the argument lies in the upper half of the series
b) The arguments are not equally spaced c) Both (a) & (b) d) None of (a) & (b)
6. The first divided differences of f(x) for the arguments 10 , xx is
a) ( )
( ) ( )
01
12
1
2 xx
xfxf
xf
x −
−
=∆ b) ( )
( ) ( )
01
01
0
1 xx
xfxf
xf
x −
−
=∆ c) ( )
( ) ( )
23
23
2
3 xx
xfxf
xf
x −
−
=∆ d)
( )
( ) ( )
34
34
3
4 xx
xfxf
xf
x −
−
=∆
7. Divided differences are independent of the __________ of the arguments.
a) Size b) Functions c) Order d) Value
8. Divided differences method can be used when the given independent variate values are
a) At equal intervals b) At unequal intervals c) Not well defined d) All the above
9. Standard notation for divided difference is a) ∆ b) ∇ c) ∆ d) D
10. Lagrange’s polynomial for interpolation can be used even if
a) The given arguments are not equally spaced b) Extrapolation is to be done
c) Inverse interpolation is to be done d) All the above
11. If (n+1) pairs of arguments and entries are given, Lagrange’s formula is
a) A polynomial of degree n in x b) A polynomial of degree n in y
c) A polynomial in x in which each term has degree n d) A polynomial with highest degree 1
12. The value of f (3) from the following table using the Lagrange formula is
x 0 1 2 4 5 6
f(x) 1 14 15 5 6 19 a) 10 b) 10.5 c) 11 d) 11.5
13. From certain experiment the following data has been obtained
x 1 3 4
y 4 12 19
Use Lagrange’s inverse formula to find the value of x for which y = 7
a) 2.124 b) 1.857 c) 2.429 d) 1.946
14. The method which gives a unique set values to the constants in the equation of the fitting curve is called
a) Graphical method b) Method of group averaging c) Method of least square d) Rough method
15. In fitting a straight line y = ax+ b, what is the formula to find the sum of the Squares of the residuals?
a) ∑ ∑∑ −−= ybxyayE 2
b) ∑ ∑∑ −+= ybxyayE 2
c) ∑ ∑∑ ++= ybxyayE 2
d)
∑ ∑∑ +−= ybxayE 2
16. In fitting the best straight line, the line must passes through
a) Paired data b) Two paired data c) Three paired data d) fixed
17. In fitting a parabola y = ax2
+ bx + c, what is the formula for finding the sum of the Squares of the
residuals?
a) ∑∑ ∑∑ −−−= ycybxyayE 2
b) ∑∑ ∑∑ +−+= ycxybyxayE 22

4. c) ∑ ∑ ∑∑ +++= ycybyxayE 22
d) ∑ ∑ ∑∑ −−−= ycxybyxayE
22
18. The nth
divided differences of a polynomial of the nth
degree are
A) constant B) variable C) equal D) unequal
UNIT III
1. The process of calculating the derivative of a function at some particular value of the
independent variable by means of a set of given values of that function is ____
a) Numerical value b) Numerical differentiation c) Numerical integration d) quadrature
2. In the Newton’s Forward difference formula what is u _________
a)
h
xx
u n−
= b) nxxu −= c)
h
xx
u n
2
)( −
= d)
h
xx
u 0−
=
3. In the second derivative using Newton’s Backward difference formula, what is the coefficient of )(3
af∇ _
a) 2
1
h
− b) 2
1
h
c)
12
11
d) 2
h−
4. In the Newton’s Backward difference formula what is v _________
a)
h
xx
v n−
= b) nxxv −= c)
h
xx
v n
2
)( −
= d)
h
xx
v 0−
=
5. In the second derivative using Newton’s Forward difference formula, what is the coefficient of )(4
af∆ ---
a)
2
1
b)
h2
11
c) 2
12
11
h
d)
12
11
6. The process of evaluating a definite integral from a set of tabulated values of the integrand f(x) is______
a) Numerical value b) Numerical differentiation c) Numerical integration d) quadrature
7.Simpson’s 1/3rd
rule is used only when __________
a) The ordinates is even b) n is multiple of 3 c) n is odd d) n is even
8. While evaluating the definite integral by Trapezoidal rule, the accuracy can be increased by taking ___
a) large number of sub-intervals b) even number of sub-intervals
c) h=4 d) has a multiple of 3
9. In application of Simpson’s 1/3rd
rule, the interval h for closer approximation should be ______
a) even b) small c) odd d) even and small
10.While applying Simpson’s 3/8 rule the number of sub intervals should be _____
a) odd b) 8 c) even d) multiple of 3
11.To calculate the value of I using Romberg’s method _____ method is used
a) Trapezoidal rule b) Simpson’s rule c) Simpson’s 1/3 rule d) Simpson’s 3/8 rule
12.By Romberg method, the value of I1 for the set if f(0) = 0.25 , f(1) = 0.20 , f(2) = 0.125 is
a) 0.3875 b) 0.3650 c) 0.3960 d) 0.4000
13.Numerical integration when applied to a function of a single variable, it is known as ___________
a) maxima b) minima c) quadrature d)quadrant

5. 14. Two Point Gaussian Quadrature formula is exact for polynomials up to degree
a) 3 b) 5 c) 2 d) 4
15.Suppose we require ∫
b
a
dxxf )( . By proper transformation, the range (a, b) is mapped into
a) ( )∞∞− , b) ( )1,1− c) ( )∞,0 d) ( )1,0
16.Three Point Gaussian quadrature formula is exact for polynomials up to degree
a) 1 b) 4 c) 3 d) 5
17.Trapezoidal and Simpson’s rules can be used to evaluate
a) Double Integrals b) Differentiation c) Multiple Integrals d) Divided difference
UNIT IV
1. Taylor series method will be very useful to give some ______ for powerful numerical methods.
a) Initial value b) finial value c) intial starting value d) Middle value
2. Find (x0,y0) ,given that y’ = x +y , y(0) = 2 using Taylor’s formula
a) (1, 2) b) (2,1) c) (0, 2) d) (2, 0)
3. Which method requires prior calculations of higher derivatives?
a) Taylor’s b) Euler c) Adam’s d) Newton’s
4. Which of the following methods does not require starting values
a) Euler’s method b)Milne’s method c)Adam’s method d) Multi step methods
5. In the geometrical meaning of Euler’s algorithm , the curve is approximated as a
a) Straight line b)Circle c) Parabola d) Ellipse
6. yn+1 = yn + h f (xn , yn) is the iterative formula for
a) Euler’s method b)Taylor’s method c) Adam’s method d) Milne’s method
7. Which of the following formulas is a particular case of Runge-Kutta formula of the second order?
a) Taylor’s series b) Picard’s formula c) Euler’s modified d) Milne’s predictor-corrector
8. Using Euler’s method 1)0(,
2
=
−
= y
y
xy
dx
dy
the value of y(0.1) is
a) 1.1182 b) 1.1818 c) 1.1285 d) 2.2356
9. From the following which one gives the more accurate value
a)Modified Euler’s method b) Euler’s method c)Both a) and b) d) None of these
10. Single step methods are --------- a) Euler, Adam, Milne b) Euler, RK method, Milne
c) Euler, Modified Euler, RK method, Taylor d) Euler, Milne, and Taylor
11. Varies types of Runge-Kutta methods are classified according to their
a) Degree b) Order c) Rank d) Both a and b
12. The method which do not require the calculations of higher order derivatives is
a) Taylor’s method b) R-K method c) Both a) and b) d) None of these
13. Which of the following method, does not require prior calculations of higher derivatives as the

6. Taylor series method does
a) RK method b) Modified Euler method c) Simpsons d) Euler method
14. Runge-Kutta method is better than Taylor’s method because
a) it does not require prior calculations of higher derivatives b) it require at most first order derivatives
c) it require prior calculations of higher derivatives d) all the above
15. To solve the ordinary differential equation ( ) 50,sin3 2
==+ yxxy
dx
dy
, by Runge-Kutta 4th
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
16. Which of the following method is called step by step method
a) Taylor’s method b) RK method c) Milne’s method d) Newton’s method
17. For finding the value of y at xi+1 in the corrector method, the number of prior values are required
a) 1 b) 2 c) 3 d) 4
18. A predictor formula is used to predict the value of y at
a) x b) xi c) xi+1 d) yi
19. To find a better value of y1, got by predictor formula, we use
a) Adam’s predictor formula b) Corrector formula c) Improved formula d) Taylor’s formula.
20. Milne’s predictor formula is
a) ( )43224 4
3
fff
h
yy +++= b) ( )43224 4
3
fff
h
yy ++−= c) ( )43242 4
3
fff
h
yy +++= d)
( )43224 4
3
fff
h
yy +++=
21. Which of the following method is called step by step method
a) Taylor’s method b) RK method c) Adam’s method d) Newton’s method
22. -------- number of starting values is required for Adam’s method
a) 1 b) 2 c) 3 d) 4
23. )5199(
24
210101 −− +−++= ffff
h
yy is the formula for?
a) Milne’s predictor b) Milne’s corrector c) Adam’s predictor d) Adam’s corrector
24. The corrector formula is applied to
a) correct the value b) improve the value c) adjust the value d) modify the value
UNIT V
1. Laplace equation in two dimensions is of
A) hyperbolic type B) parabolic type C) circular type D) elliptic type
2. Bender-Schmidt recurrence equation is given by

7. A) ( )1,11,11,11,1,
4
1
++−++−−− +++= jijijijiji uuuuu B) ( )1,1,,1,1,
4
1
+−+− +++= jijijijiji uuuuu
C) ( )1,,1,11, ++−+ −+= jijijiji uuuu
( )1,11,11,11,1,
4
1
) ++−++−−− ++−= jijijijiji uuuuuD
3. The partial differential equation fxx – 2 fxy = 0 is
A) hyperbolic type B) parabolic type C) circular type D) elliptic type.
4. For solving one dimensional heat equation using Bender-Schmidt method the value of λ is
A) 2
ah
k
B) 2
ak
h
C)
ah
k
D)
ak
h
5. In solving the Laplace equation 0=+ yyxx uu , the diagonal five point formula is
A) [ ]1,11,11,21,1,
4
1
+−−−−+++ +++= jijijijiji uuuuu B) [ ]1,11,12,11,1,
4
1
+−−++−++ +++= jijijijiji uuuuu
C) [ ]1,11,11,1,,
4
1
+−−−−+ +++= jijijijiji uuuuu D) [ ]1,11,11,11,1,
4
1
−−+−−+++ +++= jijijijiji uuuuu
6. In solving the Laplace equation 0=+ yyxx uu , the standard five point formula is
A) [ ]1,11,11,21,1,
4
1
+−−−−+++ +++= jijijijiji uuuuu B) [ ]1,1,,1,1,
4
1
+−+− +++= jijijijiji uuuuu
C) [ ]1,11,11,1,,
4
1
+−−−−+ +++= jijijijiji uuuuu D) [ ]1,11,11,11,1,
4
1
−−+−−+++ +++= jijijijiji uuuuu
7. The partial differential equation ),(2
2
2
2
yxf
y
u
x
u
=
∂
∂
+
∂
∂
is called
A) Poisson Equation B) Heat Equation C) Wave Equation D) Laplace Equation
8. The two dimensional heat equation in steady state
0=+ yyxx uu
is
A) Parabolic B) Hyperbolic C) Elliptic D) Circle
9. The partial differential equation 032 2
22
2
2
=
∂
∂
+
∂∂
∂
+
∂
∂
y
u
yx
u
x
u
is
A) Hyperbolic B) Elliptic C) Parabolic D) Rectangular Hyperbola
10. The formula used to solve poisson equation is
A) ),(4 ,1,1,,1,1 jhihfuuuuu jijijijiji =−+++ +−+− B) ),(4 2
,1,1,1,1,1 jhihfhuuuuu jijijijiji =−+++ ++++−
C) ),(4 2
,1,1,,11,1 jhhfhuuuuu jijijijiji =−+++ +−+−− D)
),(4 2
,1,1,,1,1 jhihfhuuuuu jijijijiji =−+++ +−+−
11. The partial differential equation 0=+ yyxx uu is called
A) Laplace Equation B) Heat Equation C) Wave Equation D) Poisson Equation
12. The partial differential equation 0=+ yyxx uu is called
A) Wave Equation B) Heat Equation
C) Two dimensional heat equation D) One dimensional heat equation
13. In one dimensional heat equation 2
2
2
x
u
t
u
∂
∂
=
∂
∂
α ,the value of 2
α is
A) 22
c
k
ρ
B) 22
2
c
k
ρ
C) 2
c
k
ρ
D)
c
k
ρ
14. In solving the parabolic equation xxt uu 2
α= the value of λ in Bender Schmidt formula is
A) 2 B) 1/2 C) 0 D) -1

8. 15. What is the value of λ under which Crank – Nicholson formula
A) 1 B) -1 C) 2 D) ½
16. The simplest form of the explicit formula to solve xxtt uu 2
α= , can be got if we select λ as___________
A) 10 ≤≤ λ B)
4
1
0 ≤≤ λ C) 10 ≤〈λ D)
2
1
0 ≤〈λ
17. The partial differential equation fxx – 2 fxy + fyy= 0 is
A) hyperbolic type B) parabolic type C) circular type D) elliptic type.
18. The partial differential equation xxtt uu 2
α= is
A) Wave Equation B) Two dimensional heat equation
C) One dimensional heat equation D) Laplace Equation
19. For Solving numerically the hyperbolic equation xxtt ucu 2
= , the starting solution is provided by the
boundary condition
A) u(o,t) = 0 B) u(l,t) = 0 C) 0)0,( =xut D) u(x,0) = f(x)
20. The partial differential equation 0=− xxy ff is
A) circular B) parabolic e C) Hyperbolic D) elliptic
21. The one dimensional wave equation xxtt ucu 2
= is
A) circular B) parabolic C) elliptic D) Hyperbolic
22. The finite difference formula used to solve the hyperbolic equation 02
2
2
2
2
=
∂
∂
−
∂
∂
t
u
x
u
C is
A) 1,,1,1
22
,
22
1, )()1(2 −+−+ −+++= jijijijiji uuuauau λλ
B) 1,,1,1
22
,
22
1, )()1(2 −+−+ −++−= jijijijiji uuuauau λλ
C) 1,,1,1
22
,
22
1, )()1( −+−+ +++−= jijijijiji uuuauau λλ
D) 1,,1,1
22
,
22
1, )()1(2 −+−+ −−+−= jijijijiji uuuauau λλ
23. The explicit form used to solve the hyperbolic equation is
A) 1,,1,11, −−++ −+= jijijiji uuuu B) 1,,1,11, −−++ +−= jijijiji uuuu
C) 1,,1,11, −+++ −+= jijijiji uuuu D) 1,,1,11, −−−+ −+= jijijiji uuuu
24. The partial differential equation 03 2
22
2
2
=
∂
∂
+
∂∂
∂
+
∂
∂
y
u
yx
u
x
u
is
A) Hyperbolic B) Elliptic C) Parabolic D) Rectangular Hyperbola
25. What is the value of k to solve xxu
t
u
2
1
=
∂
∂
by Bender – Schmidt method with h = 1 if h & k are the
increments of x and t respectively?
A) 1/ 2 B) 3/ 2 C ) 1 / 4 D) 2 / 3