Q: A triangle has sides of length 7, 24, and 25. What is the measure of the largest angle?

$\cos^{-1}\left(\frac{7^2 + 24^2 - 25^2}{2 \times 7 \times 24}\right) \approx 90^\circ$

Q: A car travels along a circular track of radius 100 meters. If it completes a quarter of the circle, what is the distance traveled?

$\frac{1}{4} \times 2\pi \times 100 = 50\pi \approx 157.08$ meters

Question 1

A pharmaceutical company models the concentration $C$ of a drug in the bloodstream over time $t$ (in hours) by the function $C(t) = 5e^{-0.4t}$. What is the concentration after 3 hours?

Accepted Answer

$C(3) = 5e^{-1.2} \approx 1.51$

Question 2

Which of the following is the correct factorization of the polynomial $f(x) = x^3 - 4x^2 + 5x - 2$?

Accepted Answer

$(x-1)(x-2)(x-1)$

Question 3

A population of bacteria doubles every 4 hours. If the initial population is 500, what will the population be after 12 hours?

Accepted Answer

$500 	imes 2^{3} = 4000$

Question 4

Which of the following is the correct inverse function of $f(x) = 3x + 7$?

Accepted Answer

$f^{-1}(x) = \frac{x-7}{3}$

Question 5

A triangle has sides of length 7, 24, and 25. What is the measure of the largest angle?

Accepted Answer

$\cos^{-1}\left(\frac{7^2 + 24^2 - 25^2}{2 	imes 7 	imes 24}ight) \approx 90^\circ$

Question 6

Which of the following is the correct solution to the equation $2^{x+1} = 32$?

Accepted Answer

$x = 4$

Question 7

A company’s revenue $R$ (in thousands of dollars) is modeled by $R(x) = -2x^2 + 20x + 100$, where $x$ is the number of units sold (in thousands). What is the maximum revenue?

Accepted Answer

$R(5) = -2(5)^2 + 20(5) + 100 = 150$

Question 8

Which of the following is the correct expression for the sum of the first $n$ terms of a geometric sequence with first term $a$ and common ratio $r$?

Accepted Answer

$S_n = a \frac{1 - r^n}{1 - r}$

Question 9

A vector $	extbf{v}$ has initial point $(2, -1)$ and terminal point $(5, 3)$. What is $	extbf{v}$ in component form?

Accepted Answer

$(3, 4)$

Question 10

Which of the following is the correct solution to the system of equations: $\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$?

Accepted Answer

$x = 2, y = 1$

Question 11

A radioactive substance decays according to $A(t) = 100e^{-0.05t}$, where $A$ is the amount remaining after $t$ years. How many years will it take for only 60 grams to remain?

Accepted Answer

$t = -\frac{1}{0.05} \ln\left(\frac{60}{100}\right) \approx 10.22$

Question 12

Which of the following is the correct formula for the area $A$ of a triangle given two sides $a$, $b$ and the included angle $C$?

Accepted Answer

$A = \frac{1}{2}ab\sin C$

Question 13

Which of the following is the correct solution to $\log_2(x) = 5$?

Accepted Answer

$x = 32$

Question 14

A car travels along a circular track of radius 100 meters. If it completes a quarter of the circle, what is the distance traveled?

Accepted Answer

$\frac{1}{4} 	imes 2\pi 	imes 100 = 50\pi \approx 157.08$ meters

Question 15

Which of the following is the correct expression for the sum $\sum_{k=1}^{5} (2k-1)$?

Accepted Answer

$1 + 3 + 5 + 7 + 9 = 25$

Question 16

Which of the following is the correct solution to the equation $|3x - 7| = 5$?

Accepted Answer

$x = 4$ or $x = \frac{2}{3}$

Question 17

Which of the following is the correct formula for the period of the function $y = 3\sin(2x)$?

Accepted Answer

$\frac{2\pi}{2} = \pi$

Question 18

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

Accepted Answer

$\sqrt{6^2 + 8^2} = 10$

Question 19

Which of the following is the correct solution to the quadratic equation $x^2 - 6x + 8 = 0$?

Accepted Answer

$x = 2$ or $x = 4$

Precalculus and College Algebra Assessment Exam Topics: Structured Study Guide

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