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, where $$x is $$the volume of chemical A and $$y is $$the volume of chemical B?

$$\begin{cases} x + y = 100 \\ \frac{x}{y} = \frac{3}{2} \end{cases}$$

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, where $$x is $$the volume of chemical A and $$y is $$the volume of chemical B?

Accepted Answer

$$\begin{cases} x + y = 100 \ \frac{x}{y} = \frac{3}{2} \end{cases}$$

Question 2

Solve the quadratic equation $$2x^2 - 8x + 6 = 0$ for x$.

Accepted Answer

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

Question 3

Which of the following is the correct inverse of the function $$f(x) = 2x + 5$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x - 5}{2}$$

Question 4

A population of bacteria triples every 4 hours. If the initial population is 200, which exponential function models the population $$P(t)$$ after $$t$$ hours?

Accepted Answer

$$P(t) = 200 \cdot 3$$^{t/4}$$

Question 5

Which of the following is the correct solution to the inequality $$|2x - 5| \u003c 7$$?

Accepted Answer

$$-1 \u003c x \u003c 6$$

Question 6

Given the function $$f(x) = x^2$ - 4x + 3$$, which of the following describes the transformation to obtain $$g(x) = f(x - 2) + 5$$?

Accepted Answer

Shift right 2 units and up 5 units

Question 7

Which of the following is a solution to the equation $$z^2 + 4 = 0$ in the set of complex numbers?

Accepted Answer

$$z = 2i$$

Question 8

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 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

$$x = 9$$

Question 12

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 13

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

Accepted Answer

$$(-\infty, -2) \cup (3, \infty)$$

Question 14

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

Accepted Answer

$$(f \circ g)(2) = f(g(2)) = f(3) = 9$$

Question 15

Which of the following is the correct application of the properties of logarithms to simplify $$\log_3$(27x)$$?

Accepted Answer

$$\log_3$(27) + \log_3$(x) = 3 + \log_3$(x)$$

Question 16

A quadratic function models the height $$h (in $$meters) of a ball thrown upward from a building as $$h(t) = -5t^2$ + 20t + 50$$, where $$t is $$time in seconds. What is the maximum height reached by the ball?

Accepted Answer

$$h_{max} = 70$$ meters

Question 17

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

Accepted Answer

$$x = 3$$

Question 18

Which of the following is the correct result of dividing $$x^3$$ - $$2x^2 + 4x - 8$ by x - 2$?

Accepted Answer

$$x^2 + 0x + 4$

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