Q: A chemist is preparing a solution by mixing two chemicals. The total volume must be 100 mL, and the ratio of chemical A to chemical B must be 3:2. Which system of linear equations models this scenario, and what are the volumes of chemicals A and B?

$$\begin{cases} x + y = 100 \\ \frac{x}{y} = \frac{3}{2} \end{cases}$$; $$x = 60 mL$$, $$y = 40 mL$$

Question 1

A chemist is preparing a solution by mixing two chemicals. The total volume must be 100 mL, and the ratio of chemical A to chemical B must be 3:2. Which system of linear equations models this scenario, and what are the volumes of chemicals A and B?

Accepted Answer

$$\begin{cases} x + y = 100 \ \frac{x}{y} = \frac{3}{2} \end{cases}$$; $$x = 60 mL$$, $$y = 40 mL$$

Question 2

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

Accepted Answer

$$x = 1,\ x = 3$$

Question 3

A population of bacteria doubles every 4 hours. If the initial population is 500, which exponential function models the population $$P(t)$$ after $$t$$ hours?

Accepted Answer

$$P(t) = 500 \cdot 2$$^{t/4}$$

Question 4

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

Accepted Answer

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

Question 5

Solve the rational equation for $$x$$: $$\frac{2}{x} + 1 = \frac{5}{x}.$$

Accepted Answer

$$x = 1$$

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

$$9$$

Question 8

Which of the following is the correct solution set for the inequality $$|2x - 3| \u003c 5$$?

Accepted Answer

$$-1 \u003c x \u003c 4$$

Question 9

A company’s profit $$P(x) in $$thousands of dollars is modeled by $$P(x) = -2x^2$ + 12x - 16$$, where $$x is $$the number of units sold in thousands. What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $$-2(3)^2 + 12(3) - 16 = 2$ thousand dollars at x = 3$ thousand units

Question 10

Which of the following is the correct partial fraction decomposition of $$\frac{5x + 3}{(x + 1)(x - 2)}$$?

Accepted Answer

$$\frac{2}{x + 1} + \frac{3}{x - 2}$$

Question 11

Which of the following is the correct solution to the equation $$\log_2$(x - 1) = 3$$?

Accepted Answer

Question 12

Which of the following is the correct solution to the equation $$\log_2$(x - 1) = 3$$?

Accepted Answer

$$x = 9$$

Question 13

Which of the following is the correct result of dividing $$x^3$$ + $$2x^2 - x + 4$ by x - 1$?

Accepted Answer

$$x^2 + 3x + 2$$ + \frac{6}{x - 1}$$

Question 14

Which of the following is the correct solution set for the inequality $$\frac{x + 2}{x - 3} \u003e 0$$?

Accepted Answer

$$x  3$$

Question 15

Which of the following is the correct simplified form of $$\sqrt{-16} in $$terms of complex numbers?

Accepted Answer

$$4i$$

Question 16

If $$f(x) = 2x^2$ - 3x + 1$$, what is the result of the transformation $$f(x - 2) + 5$$?

Accepted Answer

$$2(x - 2)^2$ - 3(x - 2) + 6$$

Question 17

Which of the following is the correct property of logarithms to simplify $$\log_b(MN)$$?

Accepted Answer

$$\log_b(MN) = \log_b M + \log_b N$$

Question 18

A rational function has a vertical asymptote at $$x = 2$$ and a horizontal asymptote at $$y = 1. $$Which of the following could be its equation?

Accepted Answer

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

Question 19

Which of the following is the correct solution to the equation $$2^{x + 1} = 16$?

Accepted Answer

$$x = 3$$

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