Q: If $$\sin \theta = \dfrac{5}{13}$$ and $$\theta is in $$Quadrant II, what is $$\cos \theta$$?

$$\cos \theta = -\dfrac{12}{13}$$

Q: If $$\sin \theta = \dfrac{5}{13}$$ and $$\theta is in $$Quadrant II, what is $$\tan \theta$$?

$$\tan \theta = -\dfrac{5}{12}$$

Q: If $$\cos \theta = -\dfrac{8}{17}$$ and $$\theta is in $$Quadrant III, what is $$\sin \theta$$?

$$\sin \theta = -\dfrac{15}{17}$$

Q: If $$\cos \theta = -\dfrac{8}{17}$$ and $$\theta is in $$Quadrant III, what is $$\csc \theta$$?

$$\csc \theta = -\dfrac{17}{15}$$

Q: If $$\cos \theta = -\dfrac{8}{17}$$ and $$\theta is in $$Quadrant III, what is $$\tan \theta$$?

$$\tan \theta = \dfrac{15}{8}$$

Question 1

Solve for $$x$$: $$5^{x+2}$$ = $$3^{x-1}. $$What is the value of $$x$$?

Accepted Answer

$$x = \dfrac{2\ln 5 - \ln 3}{\ln 5 - \ln 3}$$

Question 2

Solve for $$x$$: $$2^x + 2$$^{x+1}$$ = 48. $$What is the value of $$x$$?

Accepted Answer

$$x = 5$$

Question 3

Solve for $$x$$: $$e^{2x} + 3e^x + 2 = 0$. Which of the following is a valid solution?

Accepted Answer

No real solution

Question 4

Solve for $$x$$: $$\log(x-3) + \log(x+1) = \log(10). $$What is the value of $$x$$?

Accepted Answer

$$x = 4$$

Question 5

Solve for $$x$$: $$\ln(x+4) - \ln(x) = \ln(5). $$What is the value of $$x$$?

Accepted Answer

Question 6

Solve for $$x$$: $$\log(x-1) + \log(x+2) = 1. $$What is the value of $$x$$?

Accepted Answer

$$x = 3$$

Question 7

If $$\sin 	heta = \dfrac{5}{13}$$ and $$	heta is in $$Quadrant II, what is $$\cos 	heta$$?

Accepted Answer

$$\cos 	heta = -\dfrac{12}{13}$$

Question 8

If $$\sin 	heta = \dfrac{5}{13}$$ and $$	heta is in $$Quadrant II, what is $$	an 	heta$$?

Accepted Answer

$$	an 	heta = -\dfrac{5}{12}$$

Question 9

If $$\cos 	heta = -\dfrac{8}{17}$$ and $$	heta is in $$Quadrant III, what is $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = -\dfrac{15}{17}$$

Question 10

If $$\cos 	heta = -\dfrac{8}{17}$$ and $$	heta is in $$Quadrant III, what is $$\csc 	heta$$?

Accepted Answer

$$\csc 	heta = -\dfrac{17}{15}$$

Question 11

If $$	an 	heta = \dfrac{3}{4}$$ and $$	heta is in $$Quadrant I, what is $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta = \dfrac{3}{5}$$

Question 12

If $$	an 	heta = \dfrac{3}{4}$$ and $$	heta is in $$Quadrant I, what is $$\sec 	heta$$?

Accepted Answer

$$\sec 	heta = \dfrac{5}{4}$$

Question 13

A population of bacteria doubles every 3 hours. If the initial population is 500, what will the population be after 9 hours?

Accepted Answer

$$4000$$

Question 14

A radioactive substance decays according to $$N(t) = N_0$ e$$^{-kt}$$. If N_0$ = 1000$$ and after 5 hours $$N(5) = 500$$, what is the decay constant $$k$$?

Accepted Answer

$$k = \dfrac{\ln 2}{5}$$

Question 15

Solve for $$x$$: $$272^{x-1}$$ = $$9^{x+1}. $$What is the value of $$x$$?

Accepted Answer

$$x = 3$$

Question 16

A sound has an intensity of $$I = 10$$^{-5}$$ W/m^2$. What is its decibel level L$ given by L = 10$$ \log \left( \dfrac{I}{$$I_0$$} \right)$$, where I_0$ = 10$$^{-12}$$ W/m^2$?

Accepted Answer

$$L = 80$$

Question 17

If $$\sin 	heta = \dfrac{5}{13}$$ and $$	heta is in $$Quadrant II, what is the value of $$\cos^2$ 	heta + \sin^2$ 	heta$$?

Accepted Answer

$$1$$

Question 18

If $$\cos 	heta = -\dfrac{8}{17}$$ and $$	heta is in $$Quadrant III, what is $$	an 	heta$$?

Accepted Answer

$$	an 	heta = \dfrac{15}{8}$$

Question 19

If $$	an 	heta = \dfrac{3}{4}$$ and $$	heta is in $$Quadrant I, what is $$\cos 	heta$$?

Accepted Answer

$$\cos 	heta = \dfrac{4}{5}$$

Precalculus Free-Response Practice: Exponential, Logarithmic, and Trigonometric Equations

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