Q: Given the matrix $$A = \begin{pmatrix} 2 & 3 \\ 4 & 5 \\ 6 & 7 \end{pmatrix}$$, what is its order and what is the element in the second row, first column?

Order: $$3 \times 2$$, Element: $$4$$

Q: Given $$B = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$$, what is the order of $$B$$ and what is the element in the first row, third column?

Order: $$2 \times 3$$, Element: $$3$$

Question 1

Which of the following functions has a horizontal asymptote at $$y = 2$$?

Accepted Answer

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

Question 2

Identify the vertical asymptote(s) of the function $$f(x) = \frac{5}{x^2$ - 4}.$$

Accepted Answer

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

Question 3

Which of the following is a polynomial function?

Accepted Answer

$$f(x) = 3x^4$ - 2x^2$ + 7$$

Question 4

Suppose $$f(x) = x^3$ - 4x + 1. $$According to the Intermediate Value Theorem, which interval must contain a root?

Accepted Answer

$$[0, 1]$$

Question 5

Evaluate the exponential expression $$3^4.$$

Accepted Answer

$$81$$

Question 6

Which of the following points lies on the graph of $$y = 2^x$$?

Accepted Answer

$$(2, 4)$$

Question 7

Rewrite the logarithmic equation $$\log_3$ x = 4 in $$exponential form and solve for $$x.$$

Accepted Answer

$$x = 81$$

Question 8

Convert the exponential equation $$2^x = 16 to $$logarithmic form and solve for $$x.$$

Accepted Answer

$$x = \log_2$ 16 = 4$$

Question 9

Solve for $$x$$: $$\log_{10} x = 2.$$

Accepted Answer

$$x = 100$$

Question 10

Expand the logarithmic expression $$\log_2$ (8x)$$ using properties of logarithms.

Accepted Answer

$$\log_2$ 8 + \log_2$ x$$

Question 11

Expand $$\log_3$ \left(\frac{x^2$}{y}\right)$$ using properties of logarithms.

Accepted Answer

$$2\log_3$ x - \log_3$ y$$

Question 12

Condense the expression $$\log_5$ x + 2\log_5$ y$$ into a single logarithm.

Accepted Answer

$$\log_5$ (x y^2$)$$

Question 13

Solve for $$x$$: $$4^x = 64.$$

Accepted Answer

$$x = 3$$

Question 14

Solve for $$x$$: $$5^{2x} = 125$.

Accepted Answer

$$x = 1.5$$

Question 15

A population of bacteria doubles every 3 hours. If there are initially 500 bacteria, how many will there be after 9 hours?

Accepted Answer

$$4000$$

Question 16

A radioactive substance decays according to $$N(t) = N_0$ e$$^{-kt}$$. If N_0$ = 1000$$ and $$k = 0.2$$, what is $$N(5)$$?

Accepted Answer

$$N(5) = 1000 e$$^{-1}$$ \approx 367.9$$

Question 17

Solve the system by substitution: $$x + y = 5$$, $$x - y = 1.$$

Accepted Answer

$$x = 3, y = 2$$

Question 18

Solve the system: $$2x + 3y = 7$$, $$x - y = 1.$$

Accepted Answer

$$x = 2, y = 1$$

Question 19

Given the matrix $$A = \begin{pmatrix} 2 & 3 \ 4 & 5 \ 6 & 7 \end{pmatrix}$$, what is its order and what is the element in the second row, first column?

Accepted Answer

Order: $$3 	imes 2$$, Element: $$4$$

Question 20

Given $$B = \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{pmatrix}$$, what is the order of $$B$$ and what is the element in the first row, third column?

Accepted Answer

Order: $$2 	imes 3$$, Element: $$3$$

Question 21

Compute $$A + B$$ where $$A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$$ and $$B = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}.$$

Accepted Answer

$$\begin{pmatrix} 6 & 8 \ 10 & 12 \end{pmatrix}$$

Question 22

Evaluate the determinant of $$\begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix}.$$

Accepted Answer

$$(2 	imes 5) - (3 	imes 4) = 10 - 12 = -2$$

Question 23

Evaluate the determinant of $$\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}.$$

Accepted Answer

$$(1 	imes 4) - (2 	imes 3) = 4 - 6 = -2$$

Question 24

Evaluate the determinant of the matrix $$\begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{pmatrix}$$ using expansion by minors.

Accepted Answer

$$0$$

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