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

$$\theta = \dfrac{\pi}{12} + \dfrac{k\pi}{3}$$, $$k \in \mathbb{Z}$$

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 for all real numbers $$a$$?

Accepted Answer

$$|a| = \sqrt{a^2$}$$

Question 4

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

Accepted Answer

$$[-1, 4]$$

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 an odd function?

Accepted Answer

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

Question 7

A circle has a radius of $$5 cm. $$What is the length of the arc subtended by a central angle of $$2$$ radians?

Accepted Answer

$$10 cm$$

Question 8

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

Accepted Answer

$$\dfrac{5\pi}{6}$$

Question 9

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

Accepted Answer

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

Question 10

Express $$3\cos x + 4\sin x in $$the form $$R\cos(x - 	heta)$$, where $$R \u003e 0$$ and $$0 \u003c 	heta \u003c \dfrac{\pi}{2}.$$

Accepted Answer

$$5\cos(x - \arctan(\dfrac{4}{3}))$$

Question 11

Which of the following is the correct expansion of $$(2x - 3)^3$ using the Binomial Theorem?

Accepted Answer

$$8x^3$$ - $$36x^2 + 54x - 27$

Question 12

Let $$A = \{x \in \mathbb{R} \mid x^2$  -1\}. $$What is $$A \cap B in $$interval notation?

Accepted Answer

$$(-1, 3)$$

Question 13

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

Accepted Answer

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

Question 14

If $$	an 	heta = 2$$, what is $$\sin 	heta in $$terms of $$\cos 	heta$$?

Accepted Answer

$$2\cos 	heta$$

Question 15

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

Accepted Answer

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

Question 16

A point in the plane has polar coordinates $$(r, 	heta) = (4, \dfrac{3\pi}{4}). $$What are its Cartesian coordinates?

Accepted Answer

$$(-2\sqrt{2}, 2\sqrt{2})$$

Question 17

Which of the following statements about the function $$f(x) = x^2$ is TRUE?

Accepted Answer

$$f is $$not one-to-one on $$\mathbb{R}$$

Question 18

Let $$f(x) = 2x + 3$$ and $$g(x) = x^2$. What is (f$$ \circ g)(x)$$?

Accepted Answer

$$2x^2 + 3$

Question 19

Which of the following is the correct general solution to $$	an 3	heta = 1$$?

Accepted Answer

$$	heta = \dfrac{\pi}{12} + \dfrac{k\pi}{3}$$, $$k \in \mathbb{Z}$$

Comprehensive Study Notes: College Algebra and Trigonometry (MATH1034A)

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