Q: Which of the following is the correct general solution to $$\tan 5\theta = -1$$?

$$\theta = -\frac{\pi}{20} + k\frac{\pi}{5}$$, $$k \in \mathbb{Z}$$

Q: Express $$a \cos x + b \sin x in $$the form $$R \cos(x - \theta). If$$ $$a = 1$$ and $$b = \sqrt{3}$$, what are $$R$$ and $$\theta$$?

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

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

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

Question 1

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

Accepted Answer

The set of irrational numbers is denoted by $$\mathbb{Q}'$$ and consists of all real numbers that are not rational numbers.

Question 2

A pharmaceutical company needs to store a chemical at temperatures between $$2^	ext{o}C $$and $$8^	ext{o}C$$, inclusive. Which interval notation correctly represents the set of allowable temperatures?

Accepted Answer

$$[2, 8]$$

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

A medical device rotates at 120 revolutions per minute. What is its angular speed in radians per second?

Accepted Answer

$$4\pi$$ radians/sec

Question 6

A sector of a circle has a radius of $$3$$ meters and subtends an angle of $$60^	ext{o}. $$What is the area of the sector?

Accepted Answer

$$\frac{3\pi}{2} m$^2$$

Question 7

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

Accepted Answer

$$	heta = -\frac{\pi}{20} + k\frac{\pi}{5}$$, $$k \in \mathbb{Z}$$

Question 8

Express $$a \cos x + b \sin x in $$the form $$R \cos(x - 	heta). If$$ $$a = 1$$ and $$b = \sqrt{3}$$, what are $$R$$ and $$	heta$$?

Accepted Answer

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

Question 9

A ladder of length $$L$$ leans against a wall and slips down, keeping contact with the floor and wall. The path traced by a point at distance $$d$$ from the base (not at the endpoints) is:

Accepted Answer

An ellipse

Question 10

Which of the following is the binomial coefficient $$\binom{7}{3}$$?

Accepted Answer

$$35$$

Question 11

Expand $$(a + b)^4$ using the binomial theorem. What is the coefficient of a^2$b^2$?

Accepted Answer

$$6$$

Question 12

Which of the following is the correct canonical form for the equation $$x^2 + 2x$$ + $$2y^2 - 8y + 7 = 0$?

Accepted Answer

$$u^2/2$$ + $$v^2 = 1$

Question 13

A curve in the plane is the graph of a function if and only if:

Accepted Answer

Each vertical line intersects the curve at most once.

Question 14

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 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

A point in the plane has Cartesian coordinates $$(x, y) = (-3, \sqrt{3}). $$What are its polar coordinates?

Accepted Answer

$$(2\sqrt{3}, \frac{5\pi}{6})$$

Question 17

Which of the following is a correct statement about the principle of mathematical induction?

Accepted Answer

If a statement is true for $$n = 1$$ and whenever it is true for $$n = k$$, it is also true for $$n = k + 1$$, then it is true for all $$n \in \mathbb{N}.$$

Question 18

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

Accepted Answer

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

College Algebra: Core Concepts, Functions, Trigonometry, Induction, and Conic Sections

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