Decision Mathematics 2: Master Algebra, Functions, and More

Question 1

To determine the best-fit line for a given set of data using the method of least squares, which step should be taken?

Accepted Answer

Find the line that minimizes the sum of the squared vertical distances between the data points and the line.

Answer

Calculate the mean of the x and y values and draw a line through the mean point.

Answer

Plot the data and guess the best-fit line by eye.

Question 2

In a given distribution, which of the following statements about the relationship between the mean, median, and mode is true?

Accepted Answer

The three measures of central tendency (mean, median, and mode) can have different values.

Answer

The mean is always greater than or equal to both the median and mode.

Answer

The median is always greater than or equal to both the mean and mode.

Answer

The mode is always less than or equal to both the mean and median.

Question 3

Which of the following is NOT a type of square matrix?

Accepted Answer

Rectangular matrix

Answer

Symmetric matrix

Answer

Diagonal matrix

Answer

Triangular matrix

Question 4

Which scatterplot shows a positive correlation between two variables?

Accepted Answer

Points clustered in the top right and bottom left.

Answer

Points clustered in the top left and bottom right.

Answer

Points scattered randomly with no clear pattern.

Answer

Points forming a downward sloping line.

Question 5

What is statistical significance, and how is it used to make inferences about populations based on samples?

Accepted Answer

The probability of getting a result as extreme or more extreme than the one observed, assuming the null hypothesis is true.

Answer

The closeness of the sample mean to the population mean.

Answer

The margin of error in a confidence interval.

Question 6

Which of the following is a key characteristic of Decision Mathematics 2?

Accepted Answer

Extends into matrices, vectors, and statistics

Answer

Focuses exclusively on algebra and functions

Answer

Does not develop critical thinking and problem-solving skills

Answer

Is not a foundational subject for higher-level mathematics

Question 7

Determine the formula for finding the resultant vector of two vectors [a, b] and [c, d].

Accepted Answer

[a + c, b + d]

Answer

[a - c, b - d]

Answer

[(a + c)/2, (b + d)/2]

Answer

Cannot be determined without additional information

Question 8

Identify the mode of the following set of data: 2, 4, 5, 5, 7, 9, 10.

Accepted Answer

5

Answer

2

Answer

4

Answer

7

Question 9

Which of the following arguments is logically valid?

Accepted Answer

All dogs are mammals. My pet is a dog. Therefore, my pet is a mammal.

Answer

Some flowers are red. My favorite flower is red. Therefore, all flowers are red.

Answer

No cars are airplanes. My vehicle is not a car. Therefore, my vehicle is an airplane.

Answer

All birds can fly. My pet can fly. Therefore, my pet is a bird.

Question 10

Find the inverse of the matrix:
[1 2]
[3 5]

Accepted Answer

[-5 2]
[3 -1]

Answer

[5 -2]
[-3 1]

Answer

[1 -2]
[-3 5]

Answer

The matrix is not invertible.

Question 11

What is the equation of the line that passes through the points (2, -1) and (4, 3)?

Accepted Answer

y = 2x - 5

Answer

y = x - 3

Answer

y = -2x + 3

Answer

y = -x + 1

Question 12

Which of the following is the correct notation for the dot product of vectors a and b?

Accepted Answer

a . b

Answer

a + b

Answer

a * b

Answer

a / b

Question 13

Which of the following is a quadratic equation?

Accepted Answer

x^2 + 2x + 1 = 0

Answer

x^3 - 2x^2 + 3x - 4 = 0

Answer

x^2 + y^2 = 1

Answer

sin(x) = x

Question 14

Which operation subtracts corresponding elements of two matrices?

Accepted Answer

Subtraction

Answer

Addition

Answer

Multiplication

Answer

Division

Question 15

What is the average outcome predicted by the expected value of a random variable?

Accepted Answer

The average outcome

Answer

The most likely outcome

Answer

The minimum outcome

Answer

The maximum outcome

Question 16

Which characteristic defines a null hypothesis?

Accepted Answer

Assumes no relationship or difference

Answer

Always holds true

Answer

Used to establish a research hypothesis

Answer

Exclusive to research investigations

Question 17

Which of the following is not a valid operation for matrices?

Accepted Answer

Dividing a matrix by a non-zero matrix

Answer

Adding two matrices with compatible dimensions

Answer

Multiplying a matrix by a scalar

Answer

Finding the transpose of a matrix

Question 18

Which of the following probability distributions is a continuous distribution?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Uniform distribution

Question 19

Which of the following matrices has a determinant of zero?

Accepted Answer

[[0, 2],[0, 3]]

Answer

[[1, 2],[3, 4]]

Answer

[[1, 0],[0, 1]]

Answer

[[2, -1],[4, -2]]

Question 20

Which probability distribution models the number of successes in a sequence of independent Bernoulli trials?

Accepted Answer

Binomial distribution

Answer

Normal distribution

Answer

Poisson distribution

Answer

Uniform distribution

Question 21

Which of the following accurately describes the scope of Decision Mathematics 2?

Accepted Answer

Builds upon algebra and functions to include matrices, vectors, and statistics

Answer

Focuses primarily on the study of calculus

Answer

Emphasizes rote memorization of mathematical formulas

Answer

Is not applicable to real-world situations

Question 22

For the vector v = [1, 2, 3], what is the magnitude of vector v?

Accepted Answer

√21

Answer

3

Answer

6

Answer

√14

Question 23

Find the determinant of the matrix [[2, 3], [4, 5]]

Accepted Answer

10

Answer

2

Answer

-2

Answer

-10

Question 24

Identify the eigenvector of the matrix [[1, 2], [3, 4]] with the eigenvalue 2

Accepted Answer

[-1, 1]

Answer

[1, -1]

Answer

[1, 1]

Answer

[-1, -1]

Question 25

Which of the following is not a measure of central tendency?

Accepted Answer

Variance

Answer

Mean

Answer

Median

Answer

Mode

Question 26

What is the equation of the circle with center (2, -1) and radius of 3?

Accepted Answer

(x + 2)^2 + (y + 1)^2 = 9

Answer

(x - 2)^2 + (y + 1)^2 = 3

Answer

(x - 2)^2 + (y - 1)^2 = 3

Answer

(x + 2)^2 + (y - 1)^2 = 9

Question 27

Find the derivative of the function f(x) = x^3 - 2x + 1

Accepted Answer

3x^2 - 2

Answer

4x^3 - 2

Answer

3x^2 + 2

Answer

4x^3 + 2

Question 28

Which limitation is associated with decision mathematics?

Accepted Answer

Inability to handle uncertainties

Answer

Limited to problems with a small number of variables

Answer

Extensive computational power requirement

Answer

Applicable only to mathematical problems

Question 29

Which of the following is an example of a vector in space?

Accepted Answer

A list of three numbers that describe a line in space

Answer

A list of two numbers that describe a point in space

Answer

A single number that describes a quantity

Answer

A function that describes a relationship between two variables

Question 30

What is the probability of rolling a 4 on a standard six-sided die?

Accepted Answer

1/6

Answer

1/3

Answer

1/4

Answer

1/5

Question 31

Which of the following is an example of a quadratic equation?

Accepted Answer

x² + 2x + 1 = 0

Answer

x + 2 = 0

Answer

2x - 1 = 0

Answer

x³ + 3x² + 2x + 1 = 0

Question 32

What is the gradient of the line y = 2x + 1?

Accepted Answer

2

Answer

1

Answer

-1

Answer

-2

Question 33

What is the area of the triangle with vertices (0, 0), (3, 0), and (0, 4)?

Accepted Answer

12

Answer

6

Answer

18

Answer

24

Question 34

What is the derivative of the function f(x) = x³ - 2x² + 1?

Accepted Answer

3x² - 4x

Answer

3x² + 4x

Answer

x³ - 4x

Answer

x³ + 4x

Question 35

In probability, what term refers to a set of possible outcomes?

Accepted Answer

Sample space

Answer

Event

Answer

Experiment

Answer

Random variable

Question 36

Determine the equation of the line that passes through the points (1, 3) and (3, 7).

Accepted Answer

y = 2x + 1

Answer

y = x + 2

Answer

y = 2x - 1

Answer

y = x - 2

Question 37

Find the inverse function of f(x) = 2x + 1.

Accepted Answer

f^-1(x) = (x - 1)/2

Answer

f^-1(x) = x/2 - 1/2

Answer

f^-1(x) = 2x - 1

Answer

f^-1(x) does not exist

Question 38

The correlation coefficient between variables X and Y is 0.8. What does this suggest?

Accepted Answer

Strong positive correlation between X and Y

Answer

Strong negative correlation between X and Y

Answer

No correlation between X and Y

Answer

Correlation exists, but its strength is unknown

Question 39

Given the matrix A = [2 1; 3 4], calculate the value of det(A).

Accepted Answer

5

Answer

-5

Answer

10

Answer

-10

Question 40

The mean of a dataset is 10, and its standard deviation is 2. Calculate the z-score of a value equal to 14.

Accepted Answer

2

Answer

1

Answer

0

Answer

-1

Question 41

Which of the following is NOT a property of a sampling distribution?

Accepted Answer

Identical in shape to the population distribution

Answer

Normally distributed for large sample sizes

Answer

Mean equal to the population mean

Answer

Standard deviation equal to the population standard deviation divided by the square root of the sample size

Question 42

Which type of matrix has the same number of rows and columns?

Accepted Answer

Square matrix

Answer

Vector

Answer

Symmetric matrix

Answer

Diagonal matrix

Question 43

The formula for the determinant of a 2x2 matrix |a b| is:

Accepted Answer

(a)(d) - (b)(c)

Answer

a + b

Answer

a - b

Answer

a * b

Question 44

Which of the following is NOT a characteristic of vectors?

Accepted Answer

They have length and width

Answer

They possess both magnitude and direction

Answer

They can be added and subtracted

Answer

They can be multiplied by scalars

Question 45

The dot product of two vectors results in a:

Accepted Answer

Scalar quantity

Answer

Vector quantity

Answer

Unit vector

Answer

Zero vector

Question 46

The standard deviation of a data set represents:

Accepted Answer

The spread of the data

Answer

The central tendency of the data

Answer

The skewness of the data

Answer

The kurtosis of the data

Question 47

The inverse of the function f(x) = x^2 is:

Accepted Answer

f^-1(x) = √x

Answer

f^-1(x) = x^2

Answer

f^-1(x) = -x^2

Answer

f^-1(x) does not exist

Question 48

What is the mode of the following data set: {5, 7, 4, 5, 6, 7, 8, 9, 5}?

Accepted Answer

5

Answer

4

Answer

7

Answer

9

Question 49

Use matrix inversion to solve the following system of equations: 2x + 3y = 13, x - y = 4

Accepted Answer

x = 3, y = 1

Answer

x = 1, y = 3

Answer

x = 2, y = 2

Question 50

Which property is characteristic of vectors?

Accepted Answer

They possess both magnitude and direction.

Answer

They are always parallel to the coordinate axes.

Answer

They can be represented by complex numbers.

Question 51

What formula is used to calculate the angle between two vectors?

Accepted Answer

cos(theta) = (a.b) / (|a|*|b|)

Answer

tan(theta) = (a.b) / (|a|*|b|)

Answer

sin(theta) = (a.b) / (|a|*|b|)

Answer

cos(theta) = (a+b) / (|a|+|b|)

Question 52

What does the mean of a data set represent?

Accepted Answer

The average value.

Answer

The median value.

Answer

The mode value.

Answer

The range value.

Question 53

What does the standard deviation of a data set measure?

Accepted Answer

The spread of the data.

Answer

The skewness of the data.

Answer

The kurtosis of the data.

Answer

The central tendency of the data.

Question 54

What is a key characteristic of a normal distribution?

Accepted Answer

It is bell-shaped and symmetrical.

Answer

It is skewed to the left.

Answer

It is skewed to the right.

Answer

It has a uniform distribution.

Question 55

Within what range does the probability of an event occurring always fall?

Accepted Answer

Between 0 and 1.

Answer

Greater than 1.

Answer

Equal to -1.

Answer

Less than 0.

Question 56

Which of the following represents a vector?

Accepted Answer

[3, -2, 5]

Answer

x^2 + y^2

Answer

2x + 3

Answer

sin(x)

Question 57

Which of the following is a measure of central tendency?

Accepted Answer

Mean

Answer

Variance

Answer

Range

Answer

Standard deviation

Question 58

Determine the probability of rolling a 6 on a standard six-sided die.

Accepted Answer

1/6

Answer

1/3

Answer

1/2

Answer

1/4

Question 59

Find the inverse function of f(x) = 2x + 1.

Accepted Answer

f^-1(x) = (x - 1)/2

Answer

f^-1(x) = 2/x + 1

Answer

f^-1(x) = x^2 - 1

Answer

f^-1(x) = 1 - 2x

Question 60

Which equation represents a quadratic equation?

Accepted Answer

x^2 - 5x + 6 = 0

Answer

x + y = 3

Answer

sin(x) = x

Question 61

Which of the following is a factor that can shift the equilibrium point in a linear optimisation problem?

Accepted Answer

The objective function

Answer

The feasible region

Answer

The constraints

Answer

The decision variables

Question 62

Which type of matrix has a determinant of 0?

Accepted Answer

Singular matrix

Answer

Symmetric matrix

Answer

Diagonal matrix

Answer

Identity matrix

Question 63

Which of the following is a measure of the variability of a dataset?

Accepted Answer

Standard deviation

Answer

Mean

Answer

Median

Answer

Mode

Question 64

Which of the following is NOT a common technique used in decision-making?

Accepted Answer

Guesswork

Answer

Cost-benefit analysis

Answer

Risk assessment