Q: Which of the following is the correct reduced row echelon form (REF) for the augmented matrix $\begin{bmatrix} 1 & 2 & | & 5 \\ 3 & -1 & | & 4 \end{bmatrix}$?

$\begin{bmatrix} 1 & 0 & | & 2 \\ 0 & 1 & | & 1.5 \end{bmatrix}$

Question 1

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

Accepted Answer

$x = 2 \pm \sqrt{2}$

Question 2

A population of bacteria doubles every 3 hours. If the initial population is 200, which equation models the population $P$ after $t$ hours?

Accepted Answer

$P = 200 \cdot 2^{t/3}$

Question 3

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

Accepted Answer

$x = 4, x = -\frac{2}{3}$

Question 4

Given the system of equations: $\begin{cases} x + 2y = 5 \ 3x - y = 4 \end{cases}$, what is the solution?

Accepted Answer

$x = 2, y = 1.5$

Question 5

Which of the following is the correct line of best fit for the data points $(1,2), (2,4), (3,6)$?

Accepted Answer

$y = 2x$

Question 6

Solve the equation $\log_2(x - 1) = 3$.

Question 7

Which of the following is the correct reduced row echelon form (REF) for the augmented matrix $\begin{bmatrix} 1 & 2 & | & 5 \ 3 & -1 & | & 4 \end{bmatrix}$?

Accepted Answer

$\begin{bmatrix} 1 & 0 & | & 2 \ 0 & 1 & | & 1.5 \end{bmatrix}$

Question 8

If $f(x) = x^2 - 4x + 7$, what is the vertex of the parabola?

Accepted Answer

$(2, 3)$

Question 9

A radioactive substance decays according to $A(t) = 100e^{-0.05t}$. How much remains after 10 years?

Accepted Answer

$A(10) = 100e^{-0.5} \approx 60.65$

Question 10

Solve for $x$: $\sqrt{x + 5} = 4$.

Accepted Answer

$x = 11$

Question 11

Which of the following is a complex zero of $x^2 + 4 = 0$?

Accepted Answer

$x = 2i$

Question 12

Which equation represents a rational function?

Accepted Answer

$f(x) = \frac{2x + 1}{x - 3}$

Question 13

If $f(x) = 3x - 7$ and $g(x) = x^2$, what is $(f \circ g)(2)$?

Accepted Answer

$f(g(2)) = 3(4) - 7 = 5$

Question 14

Which of the following is the solution to $2^{x} = 16$?

Accepted Answer

$x = 4$

Question 15

A chemist mixes two solutions: one with 30% acid and one with 50% acid. If she wants 10 liters of a 40% acid solution, how many liters of each should she use?

Accepted Answer

5 liters of each solution

Question 16

Which of the following is the correct solution to $\frac{2x}{x-1} = 4$?

Question 17

Which of the following best describes the end behavior of $f(x) = -2x^3 + 5x^2 - x + 7$ as $x 	o \infty$?

Accepted Answer

$f(x) 	o -\infty$

Question 18

If $\log_{10}(x) = 2$, what is $x$?

Accepted Answer

$x = 100$

Question 19

Which of the following is the correct solution to $e^{2x} = 7$?

Accepted Answer

$x = \frac{1}{2} \ln 7$

Question 20

A business’s profit $P$ is modeled by $P = 5000e^{0.03t}$, where $t$ is years since opening. What is the profit after 5 years?

Accepted Answer

$P = 5000e^{0.15} \approx 5811.31$

College Algebra I: Final Exam Review (Chapters 1–6, 12) – Study Notes

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