Number and Quantity: Master the Fundamentals of Numbers and Quantities

Question 1

Identify the vector quantity:

Accepted Answer

Velocity

Answer

Temperature

Answer

Mass

Question 2

Find the solution set for the inequality: |x - 3| < 5

Accepted Answer

{-2 < x < 8}

Answer

{x < 8}

Answer

{x > -2}

Answer

{x < 2}

Question 3

Find the complex conjugate of the number (3 + 4i).

Accepted Answer

(3 - 4i)

Answer

(4 + 3i)

Answer

(-3 - 4i)

Question 4

Find the modulus of the complex number (3+4i).

Accepted Answer

5

Answer

3

Answer

7

Answer

16

Question 5

Which of the following is NOT a fundamental property of real numbers?

Accepted Answer

Closure under division

Answer

Commutative property of addition

Answer

Associative property of multiplication

Answer

Distributive property

Question 6

Which property of real numbers guarantees that the sum of any two real numbers remains unchanged when their order is reversed?

Accepted Answer

Commutative property of addition

Answer

Associative property of multiplication

Answer

Distributive property

Answer

Closure under division

Question 7

Which of the following sets is NOT included within the real numbers?

Accepted Answer

Imaginary numbers

Answer

Integers

Answer

Rational numbers

Answer

Irrational numbers

Question 8

What is the complex conjugate of the complex number (3 + 4i)?

Accepted Answer

3 - 4i

Answer

3 + 4i

Answer

-3 + 4i

Answer

-3 - 4i

Question 9

What is a defining characteristic of complex numbers?

Accepted Answer

They have both real and imaginary parts.

Answer

They are always positive.

Answer

They can be plotted on a standard number line.

Answer

They cannot be used to solve algebraic equations.

Question 10

Which of the following is NOT a real number?

Accepted Answer

√(-1)

Answer

0

Answer

π

Answer

e

Question 11

Find the magnitude of the vector <3, 4>.

Accepted Answer

7

Answer

1

Answer

5

Answer

12

Question 12

Which of the following is a rational number?

Accepted Answer

1/3

Answer

π

Answer

√2

Answer

e

Question 13

Determine the complex conjugate of z = 3 + 2i.

Accepted Answer

3 - 2i

Answer

3 + 2i

Answer

-3 + 2i

Answer

-3 - 2i

Question 14

Simplify the expression: (3 + i)(3 - i)

Accepted Answer

10

Answer

10i

Answer

10 - 2i

Answer

10 + 2i

Question 15

Find the dot product of the vectors v = <3, 2> and w = <-1, 4>.

Accepted Answer

5

Answer

-5

Answer

10

Answer

-10

Question 16

Identify the valid complex number:

Accepted Answer

3 + 4i

Answer

3i + 4

Answer

i^2

Answer

1/i

Question 17

Identify the rational number:

Accepted Answer

0.333...

Answer

π

Answer

√2

Answer

1.414...

Question 18

Which of the following is the conjugate of the complex number 5 - 3i?

Accepted Answer

5 + 3i

Answer

5 - 3i

Answer

-5 + 3i

Answer

-5 - 3i

Question 19

Solve the equation |x - 2| = 5.

Accepted Answer

x = 3 or x = 7

Answer

x = -3 or x = 7

Answer

x = 3 or x = -7

Answer

x = -3 or x = -7

Question 20

Which of the following is the graph of the inequality x^2 + y^2 < 9?

Accepted Answer

The interior of a circle centered at the origin with radius 3

Answer

A circle centered at the origin with radius 3

Answer

A circle centered at the origin with radius 9

Answer

The exterior of a circle centered at the origin with radius 3

Question 21

Find the vertex of the parabola y = x^2 - 4x + 3.

Accepted Answer

(2, -1)

Answer

(1, 0)

Answer

(0, 3)

Answer

(-2, 1)

Question 22

Which of the following is the dot product of vectors u = (3, -2) and v = (1, 4)?

Accepted Answer

5

Answer

-5

Question 23

Solve the inequality: |x - 2| < 5

Accepted Answer

x < 7 and x > -3

Answer

x < -3 or x > 7

Question 24

Find the magnitude of the vector (0, -5).

Accepted Answer

5

Answer

-5

Question 25

Simplify the expression: (√2 - i)²

Accepted Answer

-1 + √2i

Answer

1 + √2i

Question 26

Which of the following represents the set of all complex numbers with a real part of 3?

Accepted Answer

{z ∈ ℂ | Re(z) = 3}

Answer

{z ∈ ℂ | Im(z) = 3}

Question 27

Calculate the distance between the points (2, 3) and (-1, 5).

Accepted Answer

2√5

Answer

5√2

Answer

√5

Answer

√10

Question 28

Find the unit vector in the direction of <2, -3>.

Accepted Answer

<2/√13, -3/√13>

Answer

<-2/√13, 3/√13>

Answer

<3/√13, 2/√13>

Answer

<-3/√13, -2/√13>

Question 29

Solve the linear inequality: 3x - 5 > 10

Accepted Answer

x > 15/3

Answer

x > 5

Answer

x < 5

Answer

x < 15/3

Question 30

Determine the greatest common factor (GCF) of the expressions 12x^2y and 15xy^3.

Accepted Answer

3xy

Answer

3x^2y^3

Answer

12xy^3

Answer

15x^2y

Question 31

Which of the following is NOT a subset of the real numbers?

Accepted Answer

Imaginary numbers

Answer

Integers

Answer

Rational numbers

Answer

Irrational numbers

Question 32

Find the conjugate of the complex number 3 + 4i.

Accepted Answer

3 - 4i

Answer

3 + 4i

Answer

-3 + 4i

Answer

-3 - 4i

Question 33

Which of the following vectors is parallel to the vector <2, 3>? (Choose the vector that has the same direction.)

Accepted Answer

<4, 6>

Answer

<-4, -6>

Answer

<3, -2>

Answer

<-3, 2>

Question 34

Which of the following inequalities is equivalent to 2x + 1 > 5?

Accepted Answer

x > 2

Answer

x < 2

Answer

2x > 4

Answer

2x < 4

Question 35

Solve the system of equations: x + y = 5, x - y = 1.

Accepted Answer

x = 3, y = 2

Answer

x = 2, y = 3

Answer

x = -3, y = -2

Answer

x = -2, y = -3

Question 36

Determine the domain and range of the function f(x) = √(x - 2).

Accepted Answer

Domain: [2, ∞), Range: [0, ∞)

Answer

Domain: (-∞, 2), Range: [0, ∞)

Question 37

In vector analysis, what mathematical operation determines the magnitude of a vector?

Accepted Answer

Square root of the sum of the squares of the vector's components

Answer

Absolute value of the vector

Answer

Sum of the vector's components

Question 38

When you add the complex numbers (a + bi) and (c + di), what do you get?

Accepted Answer

(a + c) + (b + d)i

Answer

(a - c) + (b - d)i

Answer

(c - a) + (d - b)i

Question 39

Find the complex conjugate of z = 3 + 4i.

Accepted Answer

3 - 4i

Answer

3 + 4i

Question 40

Which of the following is the standard form of the complex number (1 + √3i)(1 - √3i)?

Accepted Answer

2

Answer

-2

Answer

1 + 3i

Question 41

Find the modulus (absolute value) of the complex number z = -5 + 12i.

Accepted Answer

13

Answer

5

Question 42

Find the dot product of the vectors u = (1, 2, 3) and v = (2, -1, 4).

Accepted Answer

14

Answer

10

Question 43

Which of the following is the cross product of the vectors u = (1, 1, 0) and v = (0, 1, 1)?

Accepted Answer

(0, -1, 1)

Answer

(0, 1, -1)

Question 44

Which of the following inequalities represents the set of all real numbers greater than or equal to -5?

Accepted Answer

x ≥ -5

Answer

x > -5

Question 45

Find the complex conjugate of (3 + 4i).

Accepted Answer

(3 - 4i)

Answer

(3 + 4i)

Answer

(-3 - 4i)

Answer

(-3 + 4i)

Question 46

Solve the inequality: |x - 3| < 5

Accepted Answer

-2 < x < 8

Answer

-8 < x < 2

Answer

x < -2 or x > 8

Answer

x > -8 or x < 2

Question 47

Which of the following is a unit vector in the direction of the vector (2, 3)?

Accepted Answer

(2/√13, 3/√13)

Answer

(4, 6)

Answer

(0, 0)

Answer

(1, 1)

Question 48

Calculate the dot product of the vectors (1, 2) and (3, 4).

Accepted Answer

11

Answer

5

Answer

15

Answer

26

Question 49

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 50

Calculate the volume of the sphere with radius r.

Accepted Answer

(4/3)πr^3

Answer

πr^3

Answer

4πr^2

Answer

(3/4)πr^3

Question 51

Find the solution to the linear equation: 2x - 5 = 9

Accepted Answer

7

Answer

5

Answer

9

Question 52

Find the greatest common factor (GCF) of the algebraic expressions 12x³y² and 18x²y

Accepted Answer

6x2y

Answer

6xy

Answer

12x2y2

Answer

2x3y2

Question 53

Express the rational fraction 15/24 in simplest form

Accepted Answer

5/8

Answer

3/4

Answer

2/3

Answer

1/2

Question 54

Determine the rational number that lies between 1/4 and 1/3

Accepted Answer

5/12

Answer

1/2

Answer

1/6

Answer

1/8

Question 55

Which of the following is NOT a property of real numbers?

Accepted Answer

Commutativity of subtraction

Answer

Associativity of multiplication

Answer

Distributivity of multiplication over addition

Answer

Closure under addition

Question 56

The complex number i^4 is equal to:

Accepted Answer

1

Answer

i

Answer

2

Answer

-1

Question 57

The magnitude of the vector <2, 3> is:

Accepted Answer

√13

Answer

5

Answer

2

Answer

3

Question 58

Which of the following equations has a complex solution?

Accepted Answer

x^2 + 1 = 0

Answer

x^2 + 2x + 1 = 0

Answer

x^2 - 2x + 1 = 0

Answer

x^2 - 1 = 0

Question 59

The sum of two vectors and is:

Accepted Answer

Answer

Answer

Answer

Question 60

Which of the following is a valid inequality for the real number x?

Accepted Answer

x < -2 or x > 3

Answer

x < -2 and x > 3

Answer

x > -2 and x < 3

Answer

x > -2 or x < 3

Question 61

The conjugate of the complex number 3 - 4i is:

Accepted Answer

3 + 4i

Answer

3 - 4i

Answer

-3 + 4i

Answer

-3 - 4i

Question 62

The dot product of the vectors <1, 2> and <3, 4> is:

Accepted Answer

11

Answer

20

Answer

5

Answer

15

Question 63

Which of the following is NOT a theorem related to vectors?

Accepted Answer

Pythagorean Theorem

Answer

Cauchy-Schwarz Inequality

Answer

Triangle Inequality

Answer

Parallelogram Law

Question 64

The multiplicative inverse of the complex number 2 + 3i is:

Accepted Answer

2/13 - 3/13i

Answer

-2/13 + 3/13i

Answer

2/13 + 3/13i

Answer

-2/13 - 3/13i