$$\theta = -\frac{\pi}{20} + \frac{k\pi}{5}$$

$$R = \sqrt{a^2$ + b^2$}$$, $$\tan \theta = \frac{b}{a}$$

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

Question 1

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

Accepted Answer

The set of irrational numbers is $$\mathbb{R} \setminus \mathbb{Q}$$, consisting of all real numbers that are not rational.

Question 2

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

Accepted Answer

$$(1, \infty)$$

Question 3

Which of the following is the correct solution set for the inequality $$|2x + 4| \u003c 1$$?

Accepted Answer

$$(-5/2, -3/2)$$

Question 4

A car wheel has a diameter of $$60 cm $$and the car moves at $$80 km/h. $$Approximately how many revolutions per minute does the wheel make?

Accepted Answer

$$710.5$$

Question 5

Which of the following functions is odd?

Accepted Answer

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

Question 6

If $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ - 4$$, what is the domain of the composite function $$f \circ g$$?

Accepted Answer

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

Question 7

Which of the following is the correct general solution to $$	an 5	heta = -1$$?

Accepted Answer

$$	heta = -\frac{\pi}{20} + \frac{k\pi}{5}$$

Question 8

Which transformation will reflect the graph of $$y = f(x)$$ across the y-axis?

Accepted Answer

$$y = f(-x)$$

Question 9

What is the area of a sector with radius $$r$$ and angle $$\alpha (in $$radians)?

Accepted Answer

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

Question 10

Express $$a \cos x + b \sin x in $$the form $$R \cos(x - 	heta). $$What are $$R$$ and $$	heta$$?

Accepted Answer

$$R = \sqrt{a^2$ + b^2$}$$, $$	an 	heta = \frac{b}{a}$$

Question 11

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

Accepted Answer

$$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a$$^{n-k}$$ b^k$$

Question 12

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 13

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

Accepted Answer

$$x^2/a^2$ - y^2$/b^2 = 1$

Question 14

If $$f(x) = x^3$ and g(x$$) = $$x^2$$, which statement is true?

Accepted Answer

$$f is $$one-to-one, but $$g is $$not one-to-one.

Question 15

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 16

Which of the following is the correct formula for converting degrees to radians?

Accepted Answer

Multiply by $$\frac{\pi}{180}$$

Question 17

Which of the following is the correct formula for the sum of a geometric series $$a + ar + ar^2$ + \cdots + ar$$^{n-1}$$?

Accepted Answer

$$a(1 - r^n)/(1 - r)$$

Question 18

Which of the following is the correct formula for converting Cartesian coordinates $$(x, y) to $$polar coordinates $$(r, 	heta)$$?

Accepted Answer

$$r = \sqrt{x^2$ + y^2$}$$, $$	heta = \arctan\left(\frac{y}{x}ight)$$

Precalculus Study Notes: Algebra, Functions, Trigonometry, and Conic Sections

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