Free Printable Numerical Methods

1.

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

2.

Newton-Raphson method is used to find the root of the equation x2 - 2 = 0

If iterations are started from - 1, then iterations will be

a)

converge to -1

b)

converge to √2

c)

converge to -√2

d)

No converge

3.

Which of the following statements applies to the bisection method used for finding roots of functions?

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

4.

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

b)

1.5

c)

2.0

d)

None of these

5.

Using Newton-Raphson method, find a root correct to three decimal places of the equation x3 - 3x - 5 = 0

a)

2.275

b)

2.279

c)

2.222

d)

None of these

6.

In the Gauss elimination method for solving a system of linear algebraic equations, triangularzation leads to

a)

Diagonal matrix

b)

Lower triangular matrix

c)

Upper triangular matrix

d)

Singular matrix

7.

If Δf(x) = f(x+h) - f(x), then a constant k, Δk equals

a)

1

b)

0

c)

f(k)- f(0)

d)

f(x + k) - f(x)

8.

Double (Repeated) root of

4x3- 8x2- 3x + 9 = 0 by Newton-raphson method is

a)

1.4

b)

1.5

c)

1.6

d)

1.55

9.

Using Bisection method, negative root of x3 - 4x + 9 = 0 correct to three decimal places is

a)

-2.506

b)

-2.706

c)

- 2.406

d)

None of these

10.

Four arbitrary points (x1, y1), (x2, y2), (x3, y3),(x4, y4) are given in the x, y-plane. Using the method of least squares, if regressing y upon x gives the fitted line y = ax + b; and regressing y upon x gives the fitted line y + ax + b; and regressing x upon y gives the fitted line

x = cy + d, then

a)

Two fitted lines must coincide

b)

Two fitted lines need not coincide

c)

It is possible that ac = 0

d)

A must be 1/c

11.

The root of x3 - 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

12.

Newton-Raphson method of solution of numerical equation is not preferred when

a)

Graph of A(B) 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

13.

Following are the values of a function y(x) : y(-1) = 5, y(0), y(1) = 8 dy/dx at x = 0 as per Newton's central difference scheme is

a)

0

b)

1.5

c)

2.0

d)

3.0

14.

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

15.

Newton-Raphson method is applicable to 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

16.

The order of errors for the Simpson's rule for numerical integration with a step size h is

a)

h

b)

h^2

c)

h^3

d)

h^4

17.

In which of the following methods proper choice of initial value is very important?

a)

Bisection method

b)

False position

c)

Newton-Raphson

d)

Bairsto method

18.

Using Newton-Raphson method, find a root correct to three decimal places of the equation sin x = 1 - x

a)

0.511

b)

0.500

c)

0.555

d)

None of these

19.

Errors may occur in performing numerical computation on the computer due to

a)

Rounding errors

b)

Power fluctuation

c)

Operator fatigue

d)

All of these

20.

a)

2341

b)

3214

c)

1423

d)

None of these

21.

Numerical methods ..

a)

is to find the exact solutions of mathematical problems.

b)

is a study of algorithms that use numerical approximations in solving problems.

c)

should be accurate and precise enough for particular problems.

22.

A theorem that guaranteed that there is at least a root in the interval is

a)

Rolle's Theorem

b)

Extreme Value Theorem

c)

Intermediate Value Theorem

23.

Truncation error is better than round-off error. True or false?

a)

True

b)

False

24.

Secant and False Position method is to solve

a)

nonlinear equations

b)

system of linear equations

c)

quadratic equations

d)

eigen value problems

25.

Solving nonlinear equations means ..

a)

finding the root of the functions.

b)

finding the zero of the functions.

c)

finding the x value which f(x)=0.

26.

Methods of solving linear equations system consist of ..

a)

Iterative solvers

b)

Linear solvers

c)

Direct solvers

d)

Matrices Systems solvers

27.

For any given system of linear equations, what are the possible solutions?

a)

No solution

b)

Unique solution

c)

Dual solutions

d)

Infinite many solutions

28.

How many method did we had learnt in LU decomposition method?

a)

4

b)

5

c)

6

29.

Successive over Relaxation method is basically the improved version of ..

a)

Jacobi method

b)

Gauss-Seidel method

c)

Gauss elimination method

d)

Thomas method

30.

What condition number of matrices is for?

a)

To see whether or not we can trust linear system

b)

To observe whether the system giving accurate solutions

c)

To determine whether a small input changes will cause a large output changes

d)

To determine whether a small input changes will only cause a small output changes

31.

What is interpolation?

a)

estimate the missing function value over an interval

b)

determining a function that exactly represent a collection of data

c)

fitting a polynomial to a collection of data points

d)

express a straight line passing through 3 points

32.

The methods to find the polynomials approximation are ..

a)

Lagrange interpolation

b)

Newton's Divided Difference

c)

Natural Cubic Spline

d)

Clamped Cubic Spline

33.

The difference or derivative formula is developed based on ..

a)

the definition of differentiation

b)

the definition of limits

c)

the Taylor series

d)

the graph of the slope of a function

34.

These are the methods for numerical integration, except

a)

Trapezoidal rule

b)

Simpson's rule

c)

Gauss iteration

d)

Romberg integration

35.

What basically Gerschgorin's theorem is about?

a)

to find the eigenvalues

b)

to find the interval which contained eigenvalues

c)

to find the circle disk D in the complex plane

d)

to find the eigenvector

36.

The method to find the smallest eigenvalues are

a)

power method

b)

shifted power method

c)

inverse power method

d)

Gerschgorin's theorem

37.

The methods that based on Taylor series in solving IVP are

a)

Euler method

b)

Second order Taylor series method

c)

Improved Euler method

d)

Heun method

38.

Given above is the general formula for forth order Runge-Kutta method. What is w?

a)

w denotes the evaluation (step) of RK method

b)

w denotes the consistency of RK method

c)

w are contants, any real numbers R

39.

Where is exactly Sir June's office room?

a)

Com Lab

b)

Physics Lab

c)

Eng Lab

40.

How many chapters in BWA21303?

a)

5

b)

6

c)

7

d)

8

41.

The value of y’/x’ in terms of the angle 0 is given by _____________

a)

tanθ

b)

secθ

c)

cotθ

d)

cosecθ

42.

The Newton Raphson method is also called as ____________

a)

Tangent method

b)

Secant method

c)

Chord method

d)

Diameter method

43.

The equation f(x) is given as x2-4=0. Considering the initial approximation at x=6 then the value of x1 is given as

a)

10/3

b)

4/3

c)

7/3

d)

13/3

44.

For decreasing the number of iterations in Newton Raphson method:

a)

The value of f’(x) must be increased

b)

The value of f’’(x) must be decreased

c)

The value of f’(x) must be decreased

d)

The value of f’’(x) must be increased

45.

In Newton Raphson method f’(x) for a given point is given by the formula ____________

a)

y/x’

b)

y’/x

c)

y/x

d)

y’/x’

46.

The points where the Newton Raphson method fails are called?

a)

floating

b)

continuous

c)

non-stationary

d)

stationary

47.

The convergence of which of the following method depends on initial assumed value?

a)

False position

b)

Gauss Seidel method

c)

Newton Raphson method

d)

Euler method

48.

The equation f(x) is given as x3+4x+1=0. Considering the initial approximation at x=1 then the value of x1 is given as _______________ (VALUE SHOULD BE ROUNDED OFF )

a)

1.67

b)

1.87

c)

1.86

d)

1.85

49.

If each element of a line consists of m terms, the determinant can be expressed as the sum of ______ determinants.

a)

m-1

b)

m

c)

m+1

d)

m2

50.

A matrix B and _____ will have the same determinant.

a)

Its transpose

b)

Its inverse

c)

Its echelon matrix

d)

Its adjoint

51.

How many roots are there in the equation

x=\frac{x^4+2}{5}

?

a)

1

b)

2

c)

3

d)

4

52.

Choose the correct method(s) to determine the existence of the roots of an equation.

a)

By finding the x-intercept(s)

b)

By looking for the intersection of the two functions.

c)

By finding the changes of sign between 2 values of y.

53.

Which of the following graph best represent the real root for the equation

e^{2x}+4x-5=0

?

a)

b)

c)

d)

54.

By using Newton-Raphson method, find the value of n, where

x_{\left(n+1\right)}

is the final iteration in order to find the root (to 9 decimal places) for

f\left(x\right)=x^3+x-3

where

x_0=1.5

.

a)

6

b)

5

c)

4

d)

3

55.

Find the root for the equation $3x+\cos x=3$ between $x=0$ and $x=1$ correct to four decimal places.

a)

0.5

b)

0.7579

c)

0.758

d)

0.7469

56.

Figure shows the equation $2x^3=2-x^2$ . Find the root of the equation $2x^3+x^2-2=0$ in $\left[-2,2\right]$ .

a)

0

b)

1.414

c)

-1.414

d)

0.858

57.

Find how many intersection are there between $y=2x$ and $y=\ln x$ ?

a)

1

b)

0

c)

2

58.

Find the interval where the root of the equation $x^3=-x-1$ lies.

a)

$\left[0,0.5\right]$

b)

$\left[-1,-0.5\right]$

c)

$\left[-0.5,0\right]$

d)

$\left[0.5,1\right]$

59.

Newton-Raphson method will fail for the following reasons, except

a)

$f'\left(x_o\right)\$ is approaching zero

b)

$f'\left(x_o\right)\$ increases too rapidly

c)

$x_o$ is too far from the root

d)

$f\left(x\right)\$ is approaching zero

60.

Given that the equation $2x^3-4x^2-1=0$ has a root in the interval [2, 3]. Using Newton-Raphson method, the root correct to 2 decimal places is

a)

2.13

b)

2.11

c)

2.09

d)

2.15

61.

State methods to construct solution of linear equation.

a)

Gaussian Elimination Method

b)

Lu Decomposition Doolittle Method

c)

Lu Decomposition Crout Method

d)

Newton Raphson Method

62.

What is the elimination coefficient for the matrix given if we want to eliminate the value at column 1 and row 2.

a)

1

b)

4

c)

$\frac{1}{4}$

d)

$\frac{5}{4}$

63.

This is the formula for LU Doolittle Method.

a)

True

b)

False

64.

Transform the simultaneous equations into matrix form Ax=B, is the first step for the LU Crout Method

a)

True

b)

False

65.

Given the simultaneous equations. Find $l_{11},\ l_{21},\ l_{31}$ by using LU Crout Method.

a)

$l_{11}=\ 3\ ,\ l_{21}=2\ ,\ l_{31}=-1$

b)

$l_{11}=3\ ,\ l_{21}=7,\ l_{31}=3$

c)

$l_{11}=2\ ,\ l_{21}=-1,\ l_{31}=0$

d)

$l_{11}=-1,\ l_{21}=6\ ,\ l_{31}=2$

66.

State the solution of polynomial equation.

a)

LU Doolittle Method

b)

LU Crout Method

c)

Fixed Point Iteration Method

d)

Newton Raphson Method

67.

Formula given is the formula to find initial value for fixed point iteration method.

a)

True

b)

False

68.

Which method use the formula to find a root.

a)

LU decomposition Method

b)

Gaussian Elimination Method

c)

Newton Raphson Method

d)

Fixed Point Iteration Method

69.

Given the simultaneous equations. Find $u_{11},\ u_{12},\ u_{13}$ by using LU Doolittle Method.

a)

$u_{11}=\ 3\ ,\ u_{12}=2\ ,\ u_{13}=-1$

b)

$u_{11}=3\ ,\ u_{12}=7,\ u_{13}=3$

c)

$u_{11}=2\ ,\ u_{12}=-1,\ u_{13}=0$

d)

$u_{11}=-1,\ u_{12}=6\ ,\ u_{13}=2$

70.

In Newton Raphson Method, we must integrate the function given before find a root.

a)

True

b)

False

71.

False Position method is used to solve

a)

quadratic equations

b)

system of linear equations

c)

eigen value problems

d)

nonlinear equations

72.

Regula Falsi method is also known as ------

a)

Interval halving method

b)

Bolzano's method

c)

Method of False Position

d)

Newton method

73.

The iteration formula for Newton-Raphson method is

a)

$x_{n+1}=x_n+\frac{f\left(x_n\right)}{f'\left(x_n\right)}$

b)

$x_{n+1}=x_n-\frac{f\left(x_n\right)}{f'\left(x_n\right)}$

c)

$x_{n+1}=x_n-\frac{f'\left(x_n\right)}{f\left(x_n\right)}$

d)

$x_{n+1}=x_n+\frac{f'\left(x_n\right)}{f\left(x_n\right)}$

74.

The following graphs show the situations where Newton-Raphson method may not work, except

a)

b)

c)

d)

75.

What line is used to find the next iteration in Newton-Raphson method?

a)

Secant

b)

Normal

c)

Chord

d)

Tangent

76.

In Newton-Raphson method, if we want to give our answer correct to 3 decimal places then we need to use how many decimal places in the calculation of the iterations?

a)

3 decimal places

b)

2 decimal places

c)

4 decimal places

d)

any number of decimal places will do

77.

If we want to give answer correct to 3 significant figures in the Newton-Raphson method, what the the stopping criteria?

a)

when two successive iterations give the same answers correct to 4 significant figures.

b)

when two successive iterations give the same answers correct to 2 significant figures.

c)

when two successive iterations give the same answers correct to 3 significant figures.

d)

when two successive iterations give the same answers correct to 1 significant figures.

78.

Given that the equation $2x^3-4x^2-1=0$ has a root in the interval [2, 3]. Using Newton-Raphson method, the root correct to 2 decimal places is

a)

2.13

b)

2.11

c)

2.09

d)

2.15

79.

Which of the following iteration formula can not be derived from $x^2-6x+2=0$

a)

$x=\frac{x^2+2}{6}$

b)

$x=\sqrt{6x-2}$

c)

$x=6-\frac{2}{x}$

d)

$x=\sqrt{3\left(x-2\right)}$

80.

A theorem that guaranteed that there is at least a root in the interval is

a)

Rolle's Theorem

b)

Extreme Value Theorem

c)

Intermediate Value Theorem

81.

This is an example of a

a)

System of Quadratic Equations

b)

Reduced Row Echelon Form

c)

Augmented Matrix

d)

A Canine Doing a Backflip

82.

What is an appropriate first row command to solve this by Gaussian Elimination?

a)

2R2 + R2 -> R2

b)

-2R1 + R2 -> R2

c)

R3 +R2 --> R3

d)

2R1 + R2 --> R2

83.

Aina is simplifying the matrix using Gaussian Elimination. Did she complete the step correctly?

a)

No, she should have changed the 2 to a zero and multiplied -2 by R2 and added R1

b)

No, she wanted to change the 3 to a zero so she should have multiplied R2 and added it to R3

c)

Yes. When you need a zero you multiply by the number's reciprocal.

d)

No. Just punch in the calculator. Who cares about Carl Gauss

84.

Ammar wrote a matrix to represent his system of equations. What was his mistake

a)

The z's are not aligned correctly

b)

There should be a one in the top row for zero

c)

There should be a zero in the second column of the third representing the y

d)

Nothing. This is correct.

85.

Which of the following is the correct representation for the system of equations?

a)

A

b)

B

c)

C

d)

D

86.

Farah solved the matrix on the left. Ibrahim solved the matrix on the right. Who is correct?

a)

Farah, because she multiplied by the reciprocal.

b)

Ibrahim, because he got the top left number to be a one.

c)

Neither. They needed to take care of the one and make it a zero first.

d)

Both. Each step is legal and takes care of the top left term

87.

By using graphical method, there is a real root between [a,b] for $\ln\left(x-2\right)+x^2-4=0$ Hence, state the values of a and b.

a)

$a=1,\ b=4$

b)

$a=2,\ b=3$

c)

$a=0,\ b=2$

d)

$a=0,\ b=4$

88.

Is there a real root for $x^2-\sqrt[]{x}-36=0$ between [6, 6.5]?

a)

Yes

b)

No

89.

Show that there is a real root for $x^3+1=x^2-9$ by sketching graphs.

State its approximate value.

a)

$x_0=1$

b)

$x_0=-4$

c)

$x_0=-2$

d)

$x_0=\frac{1}{2}$

90.

The equation $e^x=4\sin x$ has a root between x=1 and x=2. Hence, by using Newton-Raphson method, find the real root correct to 3 s.f.

a)

$2.05$

b)

$1.78$

c)

$2.61$

d)

$1.37$

91.

By taking 0.2 as the first approximation, evaluate the real root of the equation $x^2-\frac{1}{x}+4=0$ correct to 3 s.f.

a)

0.246

b)

0.635

c)

0.153

d)

0.724

92.

Show that the equation $2x^3+x^2=33$ has a root in the interval $2<x<2.5$ .

Find this root correct to 3 s.f.

a)

2.17

b)

2.45

c)

2.39

d)

2.28

93.

By taking $x=2$ as the first approximation, calculate using Newton-Raphson method, the third approximation to $7^{\frac{1}{3}}$ (3 s.f.)

a)

1.91

b)

1.92

c)

1.89

d)

1.90

94.

Estimate $\int_0^8\sqrt[]{1+x^2}dx$ by using trapezoidal rule with 5 ordinates correct to 3 d.p. What is this value?

a)

33.946; approximated value

b)

33.946; absolute value

c)

32.246; approximated value

d)

32.246; absolute value

95.

Evaluate $\int_0^1\ \left(6xe^{3x^2+1}\right)dx$ correct to 4 d.p.

(Note: We'll be using this value later in Q11)

a)

52.4290

b)

51.8799

c)

51.8273

d)

52.0164

96.

Estimate $\int_0^1\ \left(6xe^{3x^2+1}\right)dx$ by using trapezoidal rule with 5 subintervals correct to 4 d.p.

(Note: We'll be using this value later in Q11)

a)

59.1659

b)

57.7352

c)

58.0016

d)

56.7529

97.

By using the answers found from Q9 and Q10, compute the error.

a)

7.2861

b)

5.9253

c)

7.9014

d)

6.3881

98.

Use the trapezoidal rule to estimate $\int_0^2g\left(x\right)dx$ from the data given:

a)

13.9962

b)

16.9473

c)

17.4530

d)

15.8836

99.

(12 + 22 + 32 + … + 102) is equal to:

a)

380

b)

385

c)

390

d)

392

100.

The sixth term of the sequence 2, 6, 11, 17… is:

a)

24

b)

30

c)

32

d)

36

Numerical Methods

Total questions: 100

Worksheet time: 50mins

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