Q: Which of the following is the correct half-angle formula for $$\sin \frac{\theta}{2}$$?

$$\sin \frac{\theta}{2} = \pm \sqrt{\frac{1-\cos \theta}{2}}$$

Q: Which of the following is the correct double-angle formula for $$\cos 2\theta$$?

$$\cos 2\theta = 2\cos^2$ \theta - 1$$

Q: A point $$P on $$the unit circle has coordinates $$(x, y). If$$ $$P$$ corresponds to an angle $$\theta$$, which of the following is true?

$$x = \cos \theta$$ and $$y = \sin \theta$$

Question 1

Which of the following best describes the domain of the function $$f(x) = \frac{1}{x-3}$$?

Accepted Answer

$$(-\infty, 3) \cup (3, \infty)$$

Question 2

A company models its profit $$P (in $$thousands of dollars) as a function of the number of units sold $$x by$$ $$P(x) = -2x^2$ + 40x - 100. $$What is the maximum profit, and how many units must be sold to achieve it?

Accepted Answer

Maximum profit is $$300 at$$ $$x = 10$$

Question 3

Which of the following is the correct equation for the graph of a circle centered at $$(2, -3)$$ with radius $$5$$?

Accepted Answer

$$(x-2)^2$$ + $$(y+3)^2 = 25$

Question 4

Given $$f(x) = 2x^2$ - 3x + 1$$ and $$g(x) = x - 4$$, what is $$(f \circ g)(2)$$?

Accepted Answer

$$21$$

Question 5

A population of bacteria triples every 4 hours. If the initial population is 200, which function models the population $$P$$ after $$t$$ hours?

Accepted Answer

$$P(t) = 200 \cdot 3$$^{t/4}$$

Question 6

Which of the following is the correct inverse of the function $$f(x) = 5x - 7$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x+7}{5}$$

Question 7

A Ferris wheel has a radius of 20 meters and its center is 25 meters above the ground. If a rider starts at the lowest point, which function models the height $$h(t)$$ above the ground after $$t$$ seconds, assuming one full revolution every 40 seconds?

Accepted Answer

$$h(t) = 25 + 20 \sin\left(\frac{\pi t}{20} - \frac{\pi}{2}\right)$$

Question 8

Which of the following is the solution set for $$2\sin x = 1 in $$the interval $$[0, 2\pi)$$?

Accepted Answer

$$\left\{ \frac{\pi}{6}, \frac{5\pi}{6} \right\}$$

Question 9

If $$	an 	heta = 3$$ and $$	heta is in $$the second quadrant, what is $$\sin 	heta$$?

Accepted Answer

$$\frac{3}{\sqrt{10}}$$

Question 10

Which of the following is the correct half-angle formula for $$\sin \frac{	heta}{2}$$?

Accepted Answer

$$\sin \frac{	heta}{2} = \pm \sqrt{\frac{1-\cos 	heta}{2}}$$

Question 11

A projectile is launched from the ground with an initial velocity of $$v_0 = 50$ m/s at an angle of 30$$^\circ$$ above the horizontal. What is the maximum height reached by the projectile? (Use g = 9.8$ m/s^2$)

Accepted Answer

$$31.9 m$$

Question 12

Which of the following is the correct equation for the parabola with vertex at $$(1, -2)$$ and focus at $$(1, 0)$$?

Accepted Answer

$$(x-1)^2 = 8(y+2$$)$$

Question 13

Given the function $$f(x) = \log_2$(x-1)$$, what is the domain of $$f$$?

Accepted Answer

$$(1, \infty)$$

Question 14

Which of the following is the correct sum-to-product identity for $$\sin A + \sin B$$?

Accepted Answer

$$2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$$

Question 15

A right triangle has legs of length 7 and 24. What is the length of the hypotenuse?

Accepted Answer

$$25$$

Question 16

Which of the following is the correct equation for an ellipse centered at the origin with a major axis of length 10 along the $$x-$$axis and a minor axis of length 6 along the $$y-$$axis?

Accepted Answer

$$\frac{x^2$}{25} + \frac{y^2$}{9} = 1$$

Question 17

If $$\cos 	heta = -\frac{1}{2}$$ and $$	heta is in $$the third quadrant, what is $$	heta$$?

Accepted Answer

$$\frac{4\pi}{3}$$

Question 18

Which of the following is the correct double-angle formula for $$\cos 2	heta$$?

Accepted Answer

$$\cos 2	heta = 2\cos^2$ 	heta - 1$$

Question 19

A point $$P on $$the unit circle has coordinates $$(x, y). If$$ $$P$$ corresponds to an angle $$	heta$$, which of the following is true?

Accepted Answer

$$x = \cos 	heta$$ and $$y = \sin 	heta$$

Pre-Calculus Math (MATH-2412) Syllabus and Course Structure

Study Guide - Practice Questions

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