MCQs of Numerical Analysis-II
1: When using any discrete variable method to approximately solve an initial value
problem, there are ______ sources of error
a) One
b) Two
c) Three
d) Four
2: The Trapezoidal and Simpson's rule are Called
a) Newton-Cotes Quadrature Methods
b) Equal-Step Methods
c) Both (a) & (b)
d) None of these
3: In Simpson's Rule, the error term is known as
a) Local Error
b) Fixed Error
c) Global Error
d) Truncation Error
4: In Central Difference Formula, error term is known as
a) Local Error
b) Fixed Error
c) Global Error
d) Truncation Error
5: The mesh points and the weights are to be determined in
a) Forward Formula
b) Backward Formula
c) Center Difference Formula
d) Gaussian Quadrature
6: Polynomial in Gaussian Quadrature
a) Chebyshev
b) Hermite
c) Legendre
d) All Above
7: If step size (h) is too large then the calculation of the slope of the secant line will be
more
a) Large
b) Small
c) Accurate
d) Inaccurate
8: All the finite-difference formulae are
a) More Accurate
b) Ill-conditioned
c) Well-Known
d) Well-behaved
9: finite-difference formulae due to cancellation will produce a value of zero if h is
a) Zero
b) One
c) Small enough
d) Large enough
10: Higher-order methods for approximating the derivative, as well as methods for
higher derivatives
a) Exist
b) Not Exist
c) May be or may not be exist
d) None of these
11: The complex-step derivative formula is only valid for calculating
a) First-order derivatives
b) Second-order derivatives
c) Third-order derivatives
d) Fourth-order derivatives
12: In general, derivatives of any order can be calculated using
a) Forward Formula
b) Backward Formula
c) Center Difference Formula
d) Cauchy's integral formula
13: Differential quadrature is used to solve
a) Ordinary differential equations
b) Partial differential equations
c) Linear differential equations
d) Non-linear differential equations
14: The integrand is evaluated at a finite set of points called
a) Evaluated points
b) Integration points
c) Finite Points
d) None of these
15: Reducing the number of evaluations of the integrand reduces the number of
involved
a) Binary operations
b) Arithmetic operations
c) Both (a) & (b)
d) None of these
16: Numerical integration can be done, if the integrand is reasonably
a) Finite
b) Infinite
c) Well-known
d) well-behaved
17: Only polynomials of ______ degree are used, typically linear and quadratic
a) Low
b) High
c) One
d) Two
18: Typically interpolating functions are
a) Gaussian quadrature
b) Polynomials
c) Both (a) & (b)
d) None of these
19: The interpolating function is straight line When it involves
a) Polynomial of degree 1
b) Polynomial of degree 2
c) Polynomial of degree 3
d) Polynomial of degree 4
20: Interpolation with polynomials evaluated at equally spaced points in
a) (a, b)
b) (a, b]
c) [a, b)
d) [a, b]
21: Quadrature rules with equally spaced points have the very convenient property of
a) Nesting
b) Overlapping
c) Not Nesting
d) Not Overlapping
22: Gaussian quadrature rules
a) Nest
b) Overlap
c) Do not nest
d) Overlap
23: A Gaussian quadrature rule is typically more accurate than a
a) Simpson’s rule
b) Trapezoidal rule
c) Richardson’s rule
d) Newton–Cotes rule
24: If f(x) does not have many derivatives at all points, then Gaussian quadrature is
often
a) Sufficient
b) Insufficient
c) Global criterion
d) None of these
25: If the derivatives become large, then Gaussian quadrature is often
a) Sufficient
b) Insufficient
c) Global criterion
d) None of these
26: the sum of errors on all the intervals should be less than a real number is
a) Sufficient
b) Insufficient
c) Global criterion
d) None of these
27: Using a partial sum with remainder term is using
a) Taylor series
b) Laguerre series
c) Lagrange series
d) All Above series
28: We Use Gauss-Hermite quadrature for integrals on the
a) Positive reals
b) Negative reals
c) Whole real line
d) None of These
29: We Use Gauss-Laguerre quadrature for integrals on the
a) Positive reals
b) Negative reals
c) Whole real line
d) None of These
30: The quadrature rules are all designed to compute
a) One-dimensional integrals
b) Two-dimensional integrals
c) Three-dimensional integrals
d) None of these
31: A differential equation is an equation that relates _____ functions and their
derivatives
a) One or more
b) Two or more
c) Three or more
d) All Above
32: Linear differential equations are the differential equations that are linear in the _____
function and its derivatives
a) Known variable
b) Unknown variable
c) Known constant
d) Unknown constant
33: in general, the solutions of a differential equation cannot be expressed by
a) Opened-form expression
b) Closed-form expression
c) Both (a) & (b)
d) None of these
34: A partial differential equation (PDE) is a differential equation that contains ____
functions and their partial derivatives
a) Known variable
b) unknown variable
c) Known multivariable
d) Unknown multivariable
35: If the function f(x) can be evaluated at values that lie to the left and right of x, then
the best two-point formula is
a) Forward Formula
b) Backward Formula
c) Center Difference Formula
d) Gaussian Quadrature
36: Which series can be used to obtain center difference formula for higher order
a) Taylor series
b) Laguerre series
c) Lagrange series
d) All Above series
37: If the function f(x) must be evaluated at values that lie on one side of x then which
formula cannot be used
a) Forward Formula
b) Backward Formula
c) Center Difference Formula
d) Gaussian Quadrature
38: Forward and backward difference formulae can be derived by differentiation of
______ interpolation polynomial
a) Taylor
b) Laguerre
c) Lagrange
d) None of these
39: The formula f'(x) = [f(x + h) - f(x)]/h - f"(ᶓ)2/h is known as forward difference
formula if
a) h=0
b) h>0
c) h<0
d) Both (b) & (c)
40: The formula f'(x) = [f(x + h) - f(x)]/h - f"(ᶓ)2/h is known as backward difference
formula if
a) h=0
b) h>0
c) h<0
d) Both (b) & (c)
41: Reducing the step size (h) will _____ improve the approximation
a) Always
b) Not always
c) May or may not always
d) None of these
42: In approximation methods, numerical differentiation is
a) Stable
b) Unstable
c) May or may not stable
d) None of these
43: the small values of step size (h) needed to reduce truncation error also cause the
round-off error to
a) Grow
b) Not grow
c) May or may not grow
d) None of these
44: The basic method involved in approximating "integration of f(x) with Limit from a to
b" is called
a) Numerical quadrature
b) Gaussian quadrature
c) Both (a) & (b)
d) None of these
45: To derive the Trapezoidal rule for approximating "integration of f(x) with Limit from
a to b", We use the ____ polynomial
a) Linear Lagrange
b) Linear Laguerre
c) Non-linear Lagrange
d) Non-linear Laguerre
46: The error term for the Trapezoidal rule involves the _____ derivative of function
a) First
b) Second
c) Third
d) Fourth
47: Simpson’s rule results from integrating over
a) (a, b)
b) (a, b]
c) [a, b)
d) [a, b]
48: Simpson’s rule results from integrating the ______ polynomial with
equally-spaced nodes.
a) First Lagrange
b) Second Lagrange
c) Third Lagrange
d) Fourth Lagrange
49: The Trapezoidal rule gives exact results when applied to any polynomial of degree
a) One or less
b) Two or less
c) Three or less
d) None of these
50: The Simpson’s rule gives exact results when applied to any polynomial of degree
a) One or less
b) Two or less
c) Three or less
d) None of these
51: The error term in Simpson’s rule involves the _____ derivative of function
a) First
b) Second
c) Third
d) Fourth
52: The error term in the interpolating polynomial of degree n involves the ______
derivative of the function being approximated
a) nth
b) (n + 1)st
c) (n + 2)nd
d) (n + 3)rd
53: Which of the following formula is exact when approximating the integral of any
polynomial of degree less than or equal to n
a) Forward Formula
b) Backward Formula
c) Center Difference Formula
d) Newton-Cotes Formula
54: All the ______ formulas use values of the function at equally-spaced points when the
formulas are combined to form the composite rules.
a) Forward Formulas
b) Backward Formulas
c) Center Difference Formulas
d) Newton-Cotes Formulas
55: Which rule approximates the integral of the function by integrating the linear
function that joins the endpoints of the graph of the function
a) Simpson’s rule
b) Trapezoidal rule
c) Richardson’s rule
d) Newton–Cotes rule
56: Gaussian quadrature chooses the points for evaluation in an optimal, rather than
equally-spaced, way
a) Optimal Way
b) Equally-spaced Way
c) Both (a) & (b)
d) None of these
57: To reduce the number of functional evaluations, more efficient method ______ can
be incorporated in place of the Newton-Cotes formulas
a) Gaussian quadrature
b) Romberg integration
c) Adaptive quadrature
d) All Above
58: The methods for approximating the solution of an initial value problem are called
a) Difference methods
b) Discrete variable methods
c) Both (a) & (b)
d) None of these
59: The solution is approximated at a set of discrete points called
a) Grid of points
b) Mesh of points
c) Both (a) & (b)
d) None of these
60: Each (P with base K)(x) has K distinct, real zeros inside the interval
a) (a, b)
b) (a, b]
c) [a, b)
d) [a, b]
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