$V = \pi r^2 h = \pi (2)^2 (5) = \pi (4)(5) = 20\pi$ cm$^3$

$(2 + 3i)(2 - 3i) = 4 - 9i^2 = 4 + 9 = 13$

$\frac{6 + 8i}{2 - i} = 4 + 2i$

$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (3)^2 (4) = \frac{1}{3} \pi (9)(4) = \frac{36}{3} \pi = 12\pi$ cm$^3$

Question 1

Which of the following is the correct formula for the volume of a sphere?

Accepted Answer

$V = \frac{4}{3} \pi r^3$

Question 2

Given the quadratic equation $2x^2 - 4x + 3 = 0$, what is the value of the discriminant, and what does it tell you about the solutions?

Accepted Answer

$b^2 - 4ac = (-4)^2 - 4(2)(3) = 16 - 24 = -8$, so there are two complex solutions.

Question 3

A rectangular garden has a length of 10 meters and a width of 6 meters. What is its perimeter?

Accepted Answer

$P = 2l + 2w = 2(10) + 2(6) = 20 + 12 = 32$ meters

Question 4

Which of the following is the correct definition of the imaginary unit $i$?

Accepted Answer

$i = \sqrt{-1}$

Question 5

If $x^2 = 25$, what are the solutions for $x$?

Accepted Answer

$x = 5$ and $x = -5$

Question 6

Which of the following is the correct formula for the quadratic formula?

Accepted Answer

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Question 7

A right triangle has legs of length 3 and 4. What is the length of the hypotenuse?

Accepted Answer

$c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Question 8

Which of the following is the correct expression for the difference of two squares?

Accepted Answer

$a^2 - b^2 = (a + b)(a - b)$

Question 9

If $x$ is a real number and $|x| < 3$, which of the following is the correct interval notation for the solution set?

Accepted Answer

$(-3, 3)$

Question 10

What is the sum of the complex numbers $3 + 2i$ and $5 - 4i$?

Accepted Answer

$8 - 2i$

Question 11

Which of the following is the correct result of multiplying $(x + 2)(x - 3)$ using the FOIL method?

Accepted Answer

$x^2 - x - 6$

Question 12

A circular cylinder has a radius of 2 cm and a height of 5 cm. What is its volume?

Accepted Answer

$V = \pi r^2 h = \pi (2)^2 (5) = \pi (4)(5) = 20\pi$ cm$^3$

Question 13

If $a = 2 + 3i$ and $b = 2 - 3i$, what is $a \cdot b$?

Accepted Answer

$(2 + 3i)(2 - 3i) = 4 - 9i^2 = 4 + 9 = 13$

Question 14

Which of the following is the correct solution to the equation $|x - 4| = 7$?

Accepted Answer

$x = 11$ and $x = -3$

Question 15

Which of the following is the correct result of dividing $6 + 8i$ by $2 - i$?

Accepted Answer

$\frac{6 + 8i}{2 - i} = 4 + 2i$

Question 16

If $x^2 + 6x + 9 = 0$, what is the solution for $x$?

Accepted Answer

$x = -3$

Question 17

Which of the following is the correct interval notation for the solution set of $x \geq -2$?

Accepted Answer

$[-2, \infty)$

Question 18

A cone has a radius of 3 cm and a height of 4 cm. What is its volume?

Accepted Answer

$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (3)^2 (4) = \frac{1}{3} \pi (9)(4) = \frac{36}{3} \pi = 12\pi$ cm$^3$

Question 19

Which of the following is the correct solution to the equation $2x - 5 < 9$?

Accepted Answer

$x < 7$

College Algebra: Essential Concepts and Problem-Solving Strategies

Study Guide - Practice Questions