Q: A chemist mixes two solutions, one containing $$x$$ grams of sodium chloride and another containing $$y$$ grams of potassium chloride. If the total mass of the mixture is 50 grams and the mass of sodium chloride is twice that of potassium chloride, which system of equations models this scenario?

$$\begin{cases} x + y = 50 \\ x = 2y \end{cases}$$

Question 1

A chemist mixes two solutions, one containing $$x$$ grams of sodium chloride and another containing $$y$$ grams of potassium chloride. If the total mass of the mixture is 50 grams and the mass of sodium chloride is twice that of potassium chloride, which system of equations models this scenario?

Accepted Answer

$$\begin{cases} x + y = 50 \ x = 2y \end{cases}$$

Question 2

Which of the following is the correct factorization of the polynomial $$x^2 - 9$?

Accepted Answer

$$(x - 3)(x + 3)$$

Question 3

A biologist is studying the population growth of bacteria, which doubles every hour. If the initial population is $$P_0$$, which function best models the population after $$t$$ hours?

Accepted Answer

$$P(t) = P_0$ \cdot 2^t$$

Question 4

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

Accepted Answer

$$x = 2 or$$ $$x = 4$$

Question 5

Which of the following is the correct simplification of the rational expression $$\frac{x^2$ - 4}{x + 2}$$?

Accepted Answer

$$x - 2$$

Question 6

A company’s profit $$P (in $$thousands of dollars) is modeled by the quadratic function $$P(x) = -2x^2$ + 12x - 16$$, where $$x is $$the number of items sold (in thousands). What is the maximum profit, and how many items must be sold to achieve it?

Accepted Answer

Maximum profit is $$20$$ thousand dollars when $$x = 3$$ thousand items are sold.

Question 7

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 8

A patient’s medication dosage is modeled by the exponential decay function $$D(t) = 100e$$^{-0.2t}$$, where D$ is the dosage in mg and t$ is time in hours. What is the dosage after 5 hours?

Accepted Answer

$$D(5) \approx 36.79 mg$$

Question 9

Which of the following is the correct domain of the function $$f(x) = \frac{1}{x - 4}$$?

Accepted Answer

All real numbers except $$x = 4$$

Question 10

Given the function $$f(x) = 3x - 5$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x + 5}{3}$$

Question 11

A car travels from point A $$(2, 3) to $$point B $$(8, 11). $$What is the distance between the two points?

Accepted Answer

$$10$$

Question 12

Which of the following is the correct solution to the inequality $$2x - 5 \u003c 9$$?

Accepted Answer

$$x \u003c 7$$

Question 13

Which of the following is the correct result of the composition $$(f \circ g)(x) if$$ $$f(x) = x^2$ and g(x) = x + 1$?

Accepted Answer

$$(f \circ g)(x) = (x + 1)^2$

Question 14

A quadratic equation $$ax^2 + bx + c = 0$ has no real solutions. Which condition must be true?

Accepted Answer

$$b^2 - 4ac$$ \u003c 0$$

Question 15

Which of the following is the correct simplification of $$\sqrt{50}$$?

Accepted Answer

$$5\sqrt{2}$$

Question 16

A business’s revenue $$R is $$given by $$R(x) = 4x^2$ - 20x + 25. $$What is the vertex of the parabola representing the minimum revenue?

Accepted Answer

Question 17

Which of the following is the correct result when dividing $$x^3$$ - $$4x^2 + 5x - 2$ by x - 2$ using synthetic division?

Accepted Answer

Question 18

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

Accepted Answer

$$x = 32$$

Question 19

Which of the following is the correct solution to the system of equations: $$\begin{cases} x + y = 10 \ x - y = 4 \end{cases}$$?

Accepted Answer

$$x = 7$$, $$y = 3$$

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