Q: Which of the following is the general solution to $$\sin \theta = \frac{1}{2}$$?

$$\theta = \frac{\pi}{6} + 2k\pi or$$ $$\theta = \frac{5\pi}{6} + 2k\pi$$, $$k \in \mathbb{Z}$$

Q: Let $$f(x) = a \cos x + b \sin x. $$Which of the following is equivalent to $$f(x)$$?

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

Question 1

Which of the following sets is NOT an interval on the real number line?

Accepted Answer

$$(-\infty, 0) \cup (0, \infty)$$

Question 2

Let $$f(x) = \sqrt{x-2}. $$What is the natural domain of $$f$$?

Accepted Answer

$$[2, \infty)$$

Question 3

Which of the following statements about the absolute value function is TRUE?

Accepted Answer

$$|a| = \sqrt{a^2$}$$~for~all~a \in \mathbb{R}$

Question 4

Solve the inequality $$|2x - 3| \u003e 5$$ and express the solution in interval notation.

Accepted Answer

$$(-\infty, -1) \cup (4, \infty)$$

Question 5

Let $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$. What is the domain of the composite function f$$ \circ g$$?

Accepted Answer

$$[0, \infty)$$

Question 6

Which of the following functions is ODD?

Accepted Answer

$$f(x) = x^3$ - x$$

Question 7

A ladder of length $$L$$ leans against a vertical wall and slides down, keeping contact with the wall and floor. The path traced by a point one-third along the ladder from the base is:

Accepted Answer

An arc of an ellipse

Question 8

Convert the polar equation $$r = 4 \sin 	heta to $$Cartesian coordinates.

Accepted Answer

$$x^2$$ + $$y^2 = 4y$

Question 9

Find the area of a sector of a circle with radius $$r = 5$$ and central angle $$	heta = \frac{\pi}{3}$$ radians.

Accepted Answer

$$\frac{25\pi}{6}$$

Question 10

Which of the following is the general solution to $$\sin 	heta = \frac{1}{2}$$?

Accepted Answer

$$	heta = \frac{\pi}{6} + 2k\pi or$$ $$	heta = \frac{5\pi}{6} + 2k\pi$$, $$k \in \mathbb{Z}$$

Question 11

If $$f(x) = 2x + 3$$ and $$g(x) = \frac{x-1}{2}$$, what is $$f^{-1}(x$$)$$, the inverse of f$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x-3}{2}$$

Question 12

Which of the following is the binomial coefficient $$\binom{8}{3}$$?

Accepted Answer

$$56$$

Question 13

Expand $$(1 + 2x)^3$ using the binomial theorem.

Accepted Answer

$$1 + 6x + 12x^2$ + 8x^3$

Question 14

Which of the following is the correct period of the function $$f(x) = 	an x$$?

Accepted Answer

$$\pi$$

Question 15

Let $$f(x) = a \cos x + b \sin x. $$Which of the following is equivalent to $$f(x)$$?

Accepted Answer

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

Question 16

Which of the following equations represents a hyperbola in canonical form?

Accepted Answer

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

Question 17

Which of the following is the sum $$\sum_{k=1}^n k$$?

Accepted Answer

$$\frac{n(n+1)}{2}$$

Question 18

Suppose you want to prove by induction that $$2^n \u003e n^2$ for all n$$ \geq 5$$. Which of the following is the correct base case?

Accepted Answer

$$n = 5$$

Question 19

Let $$A = \{x \in \mathbb{R} \mid x^2$ \u003c 9\}$$ and $$B = (1, 4). $$What is $$A \cap B in $$interval notation?

Accepted Answer

$$(1, 3)$$

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

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