Practice Test 2 - AP Precalculus

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. The point $P(-\frac{5\sqrt{2}}{2}, \frac{5\sqrt{2}}{2})$ is on a circle of radius 5 centered at the origin. Which of the following could be the angle $\theta$ in standard position corresponding to point $P$?
2. A pendulum of length $2\text{ m}$ swings with $\theta(t)=0.6\cos(\pi t)$ (radians): $x(t)=2\sin\theta(t)$, $y(t)=-2\cos\theta(t)$. What is position at $t=0$?
3. A chemistry class measures a radioactive sample that steadily loses mass. The instructor emphasizes that the process is exponential with a constant decay rate, so a log transformation linearizes the data. They model mass by $m(t)=m_0(1-d)^t$ and rewrite it as $\log_{10}(m)=\log_{10}(m_0)+t\,\log_{10}(1-d)$. Here the logarithm uses base 10, and the slope equals $\log_{10}(1-d)$. The table gives measured masses in grams at integer hours. Students are asked to pick the log-linear equation that matches the measurements. Based on the scenario, which logarithmic equation represents the scenario described?
4. What is the value of $\arctan(\sqrt{3})$?
5. A photo editor scales a logo using $S=[2\ 0;\ 0\ 3]$. Based on the scenario, what changes occur to the dimensions?
6. Given that $T(m)$ is periodic with a period of 12, which of the following statements must be true?
7. Sound level is $L=10\log_{10}\!\left(\tfrac{I}{10^{-12}}\right)$ dB. If two machines differ by $15$ dB, what is the intensity ratio $\tfrac{I_2}{I_1}$?​
8. A point on a wheel follows r}(t)=\langle 3\cos(2t),\,3\sin(2t)\rangle (meters); determine the coordinates at $t=\pi/4$.
9. The effectiveness of a particular fertilizer is measured by crop yield. As the amount of fertilizer applied increases from zero, the yield increases up to a certain point, after which applying more fertilizer causes the yield to decrease. The data appears to be symmetric around the point of maximum yield. Which function type would be most appropriate for modeling the crop yield as a function of the amount of fertilizer applied?
10. Consider the functions $f(x) = 100 - 90(0.8)^x$ and $g(x) = 10 + 90(1.2)^x$. Which of the following statements correctly describes the end behavior of the functions as $x$ approaches infinity?
11. To solve the equation $\cos(2\theta) = 3\sin(\theta) - 1$, which of the following identities would be the most useful first step to express the equation in terms of a single trigonometric function?
12. With $\mathbf{r}(t)=\langle (v\cos a)t,\,(v\sin a)t-4.9t^2\rangle$, what does $\mathbf{r}(2)$ represent in context?
13. A function is defined by $f(x)=\frac{x^2+7x+12}{x^2-16}.$ Which rational expression is equivalent to $f(x)$ after simplifying?
14. A savings account balance is modeled by $f(t)=1000\cdot 1.5^t$, where $t$ is years. An investor uses the inverse to determine how long it takes to reach a desired balance. Here, $f(t)$ is the balance after $t$ years and 1000 is the initial deposit. Assume the growth factor 1.5 stays constant. Using the inverse, solve for $t$ when $f(t)=B$.
15. A city's population is approximated by $f(t)=900\cdot 2^t$, where $t$ is years. A demographer uses the inverse to determine when the population reaches a target value. Here, $f(t)$ is the population after $t$ years and 900 is the initial population. The base 2 indicates doubling per year in this model. What is the inverse of $f(t)$?
16. A culture grows as $g(t)=500\cdot 2^{t/3}$ and analysis uses $f(x)=\ln(x)$. Using the function defined, what is $f(g(6))$?​
17. Which of the following describes the horizontal asymptote of the graph of the function $f(x) = 5(0.2)^x$?
18. How many distinct solutions does the equation $2\sin(3x) = \sqrt{2}$ have in the interval $[0, 2\pi)$?
19. A rational function is a quotient of two polynomials, written $f(x)=\frac{p(x)}{q(x)}$. Zeros are the $x$-values where $f(x)=0$, which requires $p(x)=0$ and $q(x)\neq 0$. Vertical asymptotes occur at real values where $q(x)=0$ (unless the factor cancels). For example, $f(x)=\frac{x^2+1}{x^2-1}$ has denominator $x^2-1=(x-1)(x+1)$, so it is undefined at $x=\pm 1$, giving vertical asymptotes there. The numerator $x^2+1=0$ leads to $x=\pm i$, which are complex zeros, so the graph has no real x-intercepts. Based on the function described, what are the zeros of $f(x)=\frac{x^2+1}{x^2-1}$?
20. In Miami, Florida, average monthly sea-surface temperature can be modeled sinusoidally. A simplified dataset (F) is: Feb 74, Apr 77, Jun 83, Aug 86, Oct 83, Dec 76. The mean is about 80F (vertical shift), amplitude about 6F, and period 12 months. The warmest month is August (month 8), so the phase shift should place the maximum near month 8. Based on the data presented, which sinusoidal function best models the data presented?

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