Free Printable Final Exam in Numerical Methods

1.

An infinite series that can be thought of as a polynomial with an infinite number of terms, such as 1 + x + x2 + x3 +⋯…

a)

Taylor Series

b)

Power Series

c)

Numerical Series

d)

Maclaurin Series

2.

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.

3.

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

a)

True

b)

False

4.

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.

5.

Methods of solving linear equations system consist of ..

a)

Iterative solvers

b)

Linear solvers

c)

Direct solvers

d)

Matrices Systems solvers

6.

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

7.

The methods to find the polynomials approximation are ..

a)

Lagrange interpolation

b)

Newton's Divided Difference

c)

Natural Cubic Spline

d)

Clamped Cubic Spline

8.

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

9.

These are the methods for numerical integration, except

a)

Trapezoidal rule

b)

Simpson's rule

c)

Gauss iteration

d)

Romberg integration

10.

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

11.

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

12.

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

13.

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

14.

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

15.

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

16.

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

a)

1

b)

0

c)

2

17.

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

18.

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

19.

The Newton Raphson method is also called as ____________

a)

Tangent method

b)

Secant method

c)

Chord method

d)

Diameter method

20.

It is a curve fitting that is employed where there is a significant degree of error associated with the data.

a)

Integration

b)

Regression

c)

Differentiation

d)

Optimization

21.

These are concerned with the value of a variable or a parameter that satisfies a single non-linear equation and are especially valuable in engineering design contexts where it is open impossible to explicitly solve design equations for parameters.

a)

Systems of Linear Algebraic Equations

b)

Ordinary Differential Equations

c)

Roots of Equations

d)

Integration

22.

It refers to how closely individual computer or measured values agree with each other.

a)

Precision

b)

Accuracy

c)

Truncation

d)

Round-off

23.

It refers to how closely a computed or measured value agrees with the true value of the parameter being measured.

a)

Precision

b)

Accuracy

c)

Truncation

d)

Round-off

24.

These errors happened when number having limited significant figures are used to represent exact numbers.

a)

Concatenation Errors results

b)

Truncation Errors results

c)

Round-off Errors results

d)

Relative True Errors results

25.

These are errors that do not directly related to numerical methods which occur at any stage of mathematical modeling.

a)

Gross Errors

b)

Model Errors

c)

Measurement Errors

d)

Formulation Errors

26.

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)

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

27.

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

28.

Bisection method is also known as ----- method.

a)

Newton Raphson

b)

Regula Falsi

c)

Interval Halving

d)

Newton

29.

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

a)

Interval halving method

b)

Bolzano's method

c)

Method of False Position

d)

Newton method

30.

In Trapezoidal rule, for the interval (2,5) if h=0.5 then the value of n is_________

a)

5

b)

2

c)

6

d)

4

31.

Simpson’s 1/3 rule is applicable for eight ordinates.

a)

True

b)

False

32.

By Trapezoidal rule,

\int_{x_0}^{x_n}ydx=---

a)

$\frac{h}{2}\left(y_0+2\left(y_1+y_2+...+y_{n-1}\right)+y_{n_{ }}\right)$

b)

$\frac{h}{2}\left(y_0-2\left(y_1+y_2+...+y_{n-1}\right)+y_{n_{ }}\right)$

c)

$\frac{h}{2}\left(y_0+2\left(y_1+y_2+...+y_{n-1}\right)-y_{n_{ }}\right)$

d)

$\frac{h}{3}\left(y_0+2\left(y_1+y_2+...+y_{n-1}\right)+y_{n_{ }}\right)$

33.

Numerical integration techniques can be applied to find

a)

Area

b)

Volume of a solid of revolution

c)

sum of the functional values

d)

all of these

34.

What is the main difference between Jacobi’s and Gauss-seidal?

a)

Computations in Jacobi’s can be done in parallel but not in Gauss-seidal

b)

Convergence in Seiddel method is faster

c)

Gauss seidal cannot solve the system of linear equations in three variables whereas Jacobi can

d)

Deviation from the correct answer is more in gauss seidal

35.

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

36.

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

37.

What is the size of the matrix below? Always do Rows x Columns.

a)

3 x 4

b)

4 x 3

38.

What is the size of the matrix below?

a)

0 x 1

b)

1 x 0

c)

3 x 3

d)

3 x 0

39.

Subtract

a)

3 -8
0 -1

b)

3 -4
8 -13

c)

-3 8
0 1

d)

3 -8
-8 1

40.

Add

a)

11 8
-4 2

b)

3 2
-6 -6

c)

-3 2
-4 -6

d)

10 8
-6 2

41.

Multiply

a)

20 15 -10
30 -5 0

b)

-20 15 -10
30 -5 0

c)

1 8 3
11 4 5

d)

20 -15 10
-30 5 0

42.

Which element is located in A₂₃?

a)

-3

b)

5

c)

-9

d)

2

43.

What is the location of 5?

a)

A₁₂

b)

A₂₁

c)

A₂₅

d)

A₁₃

44.

What are the dimensions of this matrix?

a)

2 x 3

b)

3 x 2

c)

6 x 1

d)

1 x 6

45.

Can the operation be performed?

a)

Yes

b)

No

c)

Sometimes

d)

only Ms. Jones knows

Final Exam in Numerical Methods

Total questions: 45

Worksheet time: 44mins

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