Q: Which of the following is the correct general solution to the equation $$\tan \theta = 2$$?

$$\theta = \arctan(2) + k\pi$$, $$k \in \mathbb{Z}$$

Q: What is the correct formula for converting an angle $$\theta$$ from degrees to radians?

$$\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}$$

Q: Which of the following is the correct general solution to $$\sin \theta = x$$ for $$x \in [-1, 1]$$?

$$\theta = \arcsin x + 2k\pi or$$ $$\theta = \pi - \arcsin x + 2k\pi$$, $$k \in \mathbb{Z}$$

Q: Which of the following is the correct expression for $$a \cos x + b \sin x in $$terms of $$R$$ and $$\theta$$?

$$a \cos x + b \sin x = R \cos(x - \theta)$$,\ where $$R = \sqrt{a^2$ + b^2$}$$ and $$\tan \theta = \frac{b}{a}$$

Question 1

Which of the following statements correctly describes the set of irrational numbers?

Accepted Answer

The set of irrational numbers consists of all real numbers that are not rational numbers, and can be written as $$\mathbb{R} \setminus \mathbb{Q}.$$

Question 2

Given the function $$f(x) = \frac{1}{|x| - 2}$$, what is its natural domain?

Accepted Answer

$$(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$$

Question 3

Which of the following is the correct general solution to the equation $$	an 	heta = 2$$?

Accepted Answer

$$	heta = \arctan(2) + k\pi$$, $$k \in \mathbb{Z}$$

Question 4

A ladder of length $$L$$ leans against a vertical wall and slips down while keeping contact with the wall and floor. The path traced by a point at distance $$d$$ from the base is:

Accepted Answer

An arc of an ellipse

Question 5

Which of the following is the correct binomial expansion for $$(a + b)^4$?

Accepted Answer

$$a^4$$ + $$4a^3b$$ + $$6a^2b^2$ + 4ab^3$ + b^4$

Question 6

If $$|a| \u003c b$$ for $$b \geq 0$$, which of the following is equivalent?

Accepted Answer

$$-b \u003c a \u003c b$$

Question 7

Which transformation results in the graph of $$y = f(x)$$ being reflected about the y-axis?

Accepted Answer

$$y = f(-x)$$

Question 8

A sector of a circle has radius $$r$$ and subtended angle $$\alpha (in $$radians). What is the area of the sector?

Accepted Answer

$$A = \frac{1}{2} r^2$ \alpha$$

Question 9

Which of the following is the correct domain for the composite function $$f \circ g$$ where $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ - 4$$?

Accepted Answer

$$(-\infty, -2] \cup [2, \infty)$$

Question 10

Which of the following is an even function?

Accepted Answer

$$f(x) = x^2$ + 2x^4$ + 1$$

Question 11

What is the correct formula for converting an angle $$	heta$$ from degrees to radians?

Accepted Answer

$$	heta_{	ext{rad}} = 	heta_{	ext{deg}} \cdot \frac{\pi}{180}$$

Question 12

Which of the following is the correct general solution to $$\sin 	heta = x$$ for $$x \in [-1, 1]$$?

Accepted Answer

$$	heta = \arcsin x + 2k\pi or$$ $$	heta = \pi - \arcsin x + 2k\pi$$, $$k \in \mathbb{Z}$$

Question 13

Which of the following is the correct formula for the binomial coefficient $$\binom{n}{k}$$?

Accepted Answer

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Question 14

Which of the following is the correct expression for $$a \cos x + b \sin x in $$terms of $$R$$ and $$	heta$$?

Accepted Answer

$$a \cos x + b \sin x = R \cos(x - 	heta)$$,\ where $$R = \sqrt{a^2$ + b^2$}$$ and $$	an 	heta = \frac{b}{a}$$

Question 15

Which of the following is the correct formula for the sum of the first $$n$$ positive integers?

Accepted Answer

$$\sum_{r=1}^n r = \frac{n(n+1)}{2}$$

Question 16

Which of the following is the canonical form of a hyperbola?

Accepted Answer

$$\frac{x^2$}{a^2$} - \frac{y^2$}{b^2$} = 1$$

Question 17

Which of the following is the correct formula for the range of the function $$g(x) = \frac{1}{x^2$ + 1}$$?

Accepted Answer

$$(0, 1]$$

Comprehensive Study Notes for College Algebra: Numbers, Functions, Trigonometry, Induction, Binomial Theorem, and Conic Sections

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