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

$$R = 2,\ \theta = \frac{\pi}{3}$$

Question 1

Which of the following statements correctly describes the relationship between rational and irrational numbers?

Accepted Answer

Every real number is either rational or irrational, but not both.

Question 2

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

Accepted Answer

$$(1, \infty)$$

Question 3

Solve the inequality $$|2x + 4| \u003c 1$$ and express the solution in interval notation.

Accepted Answer

$$\left(-\frac{5}{2}, -\frac{3}{2}\right)$$

Question 4

Which of the following functions is odd?

Accepted Answer

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

Question 5

Let $$f(x) = \sqrt{x}$$ 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 6

A car is moving at a rate of $$80 km/h. $$The diameter of its wheels is $$60 cm. $$How many revolutions per minute do the wheels make?

Accepted Answer

$$711$$

Question 7

Find the general solution to $$	an 5	heta = -1.$$

Accepted Answer

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

Question 8

Express $$\cos x + \sqrt{3} \sin x in $$the form $$R \cos(x - 	heta). $$What are the values of $$R$$ and $$	heta$$?

Accepted Answer

$$R = 2,\ 	heta = \frac{\pi}{3}$$

Question 9

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 10

If $$f(x) = x^3$ + x - 1$$ and $$g(t) = t^3$ + t - 1$$ with domain $$\mathbb{R}$$, which statement is true?

Accepted Answer

The functions $$f$$ and $$g$$ are identical because their rules and domains are the same.

Question 11

Convert $$150^\circ to $$radians.

Accepted Answer

$$\frac{5\pi}{6}$$

Question 12

Which of the following is the correct period for the function $$	an 	heta$$?

Accepted Answer

$$\pi$$

Question 13

Let $$f(x) = x^2$ and g(x$$) = $$x^2. $$Which statement is true regarding their graphs?

Accepted Answer

Both graphs are symmetric about the y-axis.

Question 14

Find the constant term in the expansion of $$\left(\frac{2}{x^2$} - x^3$\$$right)^5$$.

Accepted Answer

$$80$$

Question 15

Which of the following is the canonical form of the ellipse $$x^2 + 2x$$ + $$2y^2 - 8y + 7 = 0$ after completing the square?

Accepted Answer

$$\frac{(x+1)^2$}{2} + (y-2)^2$ = 1$$

Question 16

Which of the following is a correct application of the triangle inequality for real numbers $$a$$ and $$b$$?

Accepted Answer

$$|a + b| \leq |a| + |b|$$

Question 17

Which of the following is the correct formula for the area $$A of a $$sector with radius $$r$$ and angle $$\alpha (in $$radians)?

Accepted Answer

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

Question 18

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

Accepted Answer

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

College Algebra: Numbers, Inequalities, Functions, Trigonometry, Induction, Series, and Conic Sections

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