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

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

Q: A car is moving at $$80 km/h. $$Its wheels have a diameter of $$60 cm. $$How many revolutions per minute do the wheels make?

$$\frac{80 \times 1000}{\pi \times 0.6 \times 60}$$

Q: Which of the following is the correct expression for $$\sin^4$ \theta in $$terms of cosine functions (without even powers)?

$$\sin^4$ \theta = \frac{3}{8} - \frac{1}{2} \cos(2\theta) + \frac{1}{8} \cos(4\theta)$$

Q: Which of the following is the correct general solution to $$2\cos^2$ \theta - \sin \theta = 1$$?

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

Q: Which of the following is the correct formula for converting an angle $$\alpha$$ from degrees to radians?

$$\text{radians} = \alpha \cdot \frac{\pi}{180}$$

Question 1

Which of the following statements correctly describes the set of all real numbers $$x$$ that satisfy $$|2x + 4| \u003c 1$$?

Accepted Answer

$$-\frac{5}{2} \u003c x \u003c -\frac{3}{2}$$

Question 2

A function $$f(x) is $$defined as $$f(x) = \sqrt{x-1}. $$What is the natural domain of $$f$$?

Accepted Answer

$$(1, \infty)$$

Question 3

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

Accepted Answer

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

Question 4

Given the function $$g(x) = \frac{1}{|x| - 2}$$, which of the following describes its natural domain?

Accepted Answer

$$\mathbb{R} \setminus \{-2, 2\}$$

Question 5

Which of the following is an even function?

Accepted Answer

$$f(x) = x^2$ + 2x^4$ + 1$$

Question 6

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

Accepted Answer

$$\frac{80 	imes 1000}{\pi 	imes 0.6 	imes 60}$$

Question 7

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 8

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

Accepted Answer

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

Question 9

Which of the following is the correct expression for $$\sin^4$ 	heta in $$terms of cosine functions (without even powers)?

Accepted Answer

$$\sin^4$ 	heta = \frac{3}{8} - \frac{1}{2} \cos(2	heta) + \frac{1}{8} \cos(4	heta)$$

Question 10

Which of the following is the correct general solution to $$2\cos^2$ 	heta - \sin 	heta = 1$$?

Accepted Answer

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

Question 11

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 12

Which of the following is the correct formula for converting an angle $$\alpha$$ from degrees to radians?

Accepted Answer

$$	ext{radians} = \alpha \cdot \frac{\pi}{180}$$

Question 13

Which of the following is the correct expression for $$a \cos x + b \sin x in $$the form $$R \cos(x - 	heta)$$?

Accepted Answer

$$R = \sqrt{a^2$ + b^2$}$$, $$	heta = \arctan\left(\frac{b}{a}ight)$$

Question 14

Which of the following is the correct formula for the binomial coefficient $$\binom{n}{k}$$?

Accepted Answer

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Question 15

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

Accepted Answer

$$\frac{u^2$}{2} + v^2$ = 1$$

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

$$\pi$$

Comprehensive Study Notes: College Algebra and Trigonometry (MATH1034A)

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