Q: A savings account earns interest compounded continuously at an annual rate of 4%. If you deposit $$\1000$, what is the amount A$ in the account after 5 years?

$$A = 1000 \cdot e$$^{0.04 \times 5}$$

Q: Which of the following matrices is the inverse of $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$?

$$A^{-1}$$ = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}$$

Question 1

Which of the following is the correct vertex form of a quadratic function, and what does the vertex represent in the context of the graph?

Accepted Answer

$$y = a(x - h)^2$ + k$$, where $$(h, k) is $$the vertex of the parabola

Question 2

A company models its profit $$P(x) in $$thousands of dollars by the polynomial $$P(x) = -2x^3$ + 9x^2$ + 12x - 5$$, where $$x is $$the number of units sold (in thousands). If $$x = 1 is a $$zero of $$P(x)$$, which method can be used to factor $$P(x)$$, and what is the next step?

Accepted Answer

Use synthetic division with $$x = 1 to $$divide $$P(x)$$, then factor the resulting quadratic

Question 3

Given the polynomial $$f(x) = x^4$ - 5x^2$ + 4$$, which theorem helps you determine the possible number of positive real zeros?

Accepted Answer

Descartes' Rule of Signs

Question 4

A rational function is given by $$f(x) = \frac{2x^2$ - 3x + 1}{x^2$ - 4}. $$What are the equations of the vertical asymptotes?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 5

Which of the following statements is true about the inverse of a function $$f(x)$$?

Accepted Answer

A function has an inverse if and only if it is one-to-one

Question 6

A population of bacteria doubles every 3 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/3}$$

Question 7

Solve for $$x$$: $$\log_3$(x) + \log_3$(x-2) = 2$$

Accepted Answer

$$x = 6$$

Question 8

A savings account earns interest compounded continuously at an annual rate of 4%. If you deposit $$\1000$, what is the amount A$ in the account after 5 years?

Accepted Answer

$$A = 1000 \cdot e$$^{0.04 	imes 5}$$

Question 9

Given the system of equations: $$\begin{cases} 2x + 3y = 7 \ 4x - y = 5 \end{cases}$$, what is the value of $$x$$ using the elimination method?

Accepted Answer

$$x = 2$$

Question 10

Which of the following matrices is the inverse of $$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$$?

Accepted Answer

$$A^{-1}$$ = \begin{bmatrix} -2 & 1 \ 1.5 & -0.5 \end{bmatrix}$$

Question 11

Which of the following is the geometric definition of a parabola?

Accepted Answer

The set of all points equidistant from a fixed point (focus) and a fixed line (directrix)

Question 12

Given the ellipse $$\frac{x^2$}{9} + \frac{y^2$}{4} = 1$$, what is its eccentricity?

Accepted Answer

$$e = \frac{\sqrt{5}}{3}$$

Question 13

A sequence is defined by $$a_n = 5 + 3n. $$What type of sequence is this, and what is the common difference?

Accepted Answer

Arithmetic sequence, common difference $$d = 3$$

Question 14

Find the sum of the first 6 terms of the geometric series: $$2 + 6 + 18 + \ldots$$

Accepted Answer

$$728$$

Question 15

Expand $$(x + 2)^3$ using the binomial theorem.

Accepted Answer

$$x^3$$ + $$6x^2 + 12x + 8$

Question 16

A box contains 5 red balls and 7 blue balls. If two balls are drawn at random without replacement, what is the probability that both are red?

Accepted Answer

$$\frac{5}{66}$$

Question 17

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 18

If $$f(x) = 2x^3$ - 3x^2$ + x - 5$$, what is the remainder when $$f(x) is $$divided by $$x - 2$$?

Accepted Answer

$$-7$$

Question 19

Which of the following best describes direct variation?

Accepted Answer

A relationship where $$y = kx$$ for some constant $$k$$

Question 20

A medical test has a 95% accuracy rate. If 3 independent tests are performed, what is the probability that all three tests are accurate?

Accepted Answer

$$0.857$$

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