Question 1

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
2 x - y - 3 = 0 \
- 4 x + 2 y + 6 = 0
\end{array}$$

Accepted Answer

A)  $\{ ( x , 4 x - 3 ) \}$ 
B)  $\{ ( x , 3 - 2 x ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( x , 2 x - 3 ) \}$ 
A)  $\{ ( x , 4 x - 3 ) \}$ 
B)  $\{ ( x , 3 - 2 x ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( x , 2 x - 3 ) \}$

Question 2

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { l l l } 
1 & 1 & 1 \
2 & 1 & 1 \
2 & 2 & 3
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { l l l } 
- 1 & - 1 & - 1 \
- 2 & - 1 & - 1 \
- 2 & - 2 & - 3
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r } 
- 1 & 1 & 0 \
4 & - 1 & - 1 \
- 2 & 0 & 1
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c c c } 
1 & 1 & 1 \
\frac { 1 } { 2 } & 1 & 1 \
\frac { 1 } { 2 } & \frac { 1 } { 2 } & \frac { 1 } { 3 }
\end{array} \right]$$ 
D)  The inverse does not exist. 
A) 
$$\left[ \begin{array} { l l l } 
- 1 & - 1 & - 1 \
- 2 & - 1 & - 1 \
- 2 & - 2 & - 3
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r } 
- 1 & 1 & 0 \
4 & - 1 & - 1 \
- 2 & 0 & 1
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c c c } 
1 & 1 & 1 \
\frac { 1 } { 2 } & 1 & 1 \
\frac { 1 } { 2 } & \frac { 1 } { 2 } & \frac { 1 } { 3 }
\end{array} \right]$$ 
D)  The inverse does not exist.

Question 3

Graph the solution set of the system of inequalities.
-$\begin{array}{l}
y \leq-x^{2}-6 x-4 \
y \geq x^{2}+6 x+4
\end{array}$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 4

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { l } 
3 x ^ { 2 } + 3 y ^ { 2 } = 20 \
6 x ^ { 2 } + 6 y ^ { 2 } = 48
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( 2 \mathrm { i } , 2 \mathrm { i } ) , ( 2 \mathrm { i } , - 2 \mathrm { i } ) , ( - 2 \mathrm { i } , 2 \mathrm { i } ) , ( - 2 \mathrm { i } , - 2 \mathrm { i } ) \}$ 
C)  $\{ ( 2,2 ) , ( 2 , - 2 ) , ( - 2,2 ) , ( - 2 , - 2 ) \}$ 
D)  $\{ ( 2,2 ) , ( - 2 , - 2 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( 2 \mathrm { i } , 2 \mathrm { i } ) , ( 2 \mathrm { i } , - 2 \mathrm { i } ) , ( - 2 \mathrm { i } , 2 \mathrm { i } ) , ( - 2 \mathrm { i } , - 2 \mathrm { i } ) \}$ 
C)  $\{ ( 2,2 ) , ( 2 , - 2 ) , ( - 2,2 ) , ( - 2 , - 2 ) \}$ 
D)  $\{ ( 2,2 ) , ( - 2 , - 2 ) \}$

Question 5

Solve each problem.
-Mommaʹs ice cream shop sells three types of ice cream: soft-serve, chunky, and nonfat. Location I sells 27 gal of soft-serve, 100 gal of chunky, and 30 gal of nonfat ice cream each day. Location II
Sells 15 gal of soft-serve and Location III sells 60 gal of soft-serve each day. Daily sales of chunky
Ice cream are 90 gal at Location II and 120 gal at Location III. At Location II, 20 gal of nonfat are
Sold each day, and 40 gal of nonfat are sold each day at Location III. Write a 3 × 3 matrix that
Shows the sales figures for the three locations, with the rows representing the three locations.

Accepted Answer

A) 
$\left[ \begin{array} { c c r } 27 & 15 & 60 \ 100 & 90 & 120 \ 30 & 20 & 40 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r } 27 & 100 & 30 \ 15 & 90 & 20 \ 60 & 120 & 40 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } 27 & 100 & 30 \ 15 & 90 & 20 \ 60 & 40 & 120 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r r } 27 & 90 & 30 \ 15 & 100 & 20 \ 60 & 120 & 40 \end{array} \right]$ 
A) 
$\left[ \begin{array} { c c r } 27 & 15 & 60 \ 100 & 90 & 120 \ 30 & 20 & 40 \end{array} \right]$ 
B) 
$\left[ \begin{array} { r r r } 27 & 100 & 30 \ 15 & 90 & 20 \ 60 & 120 & 40 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r r } 27 & 100 & 30 \ 15 & 90 & 20 \ 60 & 40 & 120 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r r } 27 & 90 & 30 \ 15 & 100 & 20 \ 60 & 120 & 40 \end{array} \right]$

Question 6

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$\begin{aligned}
- 7 x ^ { 2 } + y ^ { 2 } & = 64 \
- 5 x ^ { 2 } + 4 y ^ { 2 } & = 100
\end{aligned}$$

Accepted Answer

Either method may be used. Elimination m

Question 7

Find the partial fraction decomposition for the rational expression.
-$$\frac { 3 x - 14 } { ( x - 4 ) ^ { 2 } }$$

Accepted Answer

A)  $\frac { 3 } { x - 4 } + \frac { - 2 } { ( x - 4 ) ^ { 2 } }$ 
B)  $\frac { 2 } { x - 4 } + \frac { - 3 } { ( x - 4 ) ^ { 2 } }$ 
C)  $\frac { 3 } { x - 4 } + \frac { 2 } { ( x - 4 ) ^ { 2 } }$ 
D)  $\frac { 6 } { x - 4 } + \frac { 4 } { ( x - 4 ) ^ { 2 } }$ 
A)  $\frac { 3 } { x - 4 } + \frac { - 2 } { ( x - 4 ) ^ { 2 } }$ 
B)  $\frac { 2 } { x - 4 } + \frac { - 3 } { ( x - 4 ) ^ { 2 } }$ 
C)  $\frac { 3 } { x - 4 } + \frac { 2 } { ( x - 4 ) ^ { 2 } }$ 
D)  $\frac { 6 } { x - 4 } + \frac { 4 } { ( x - 4 ) ^ { 2 } }$

Question 8

Solve the problem.
-Given $A = \left[ \begin{array} { l l } 2 & - 1 \ 3 & - 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 0 & - 1 \ 4 & 3 \end{array} \right]$, find $A B$ and $B A$.

Accepted Answer

A) 
$\mathrm { AB } = \left[ \begin{array} { l l } 4 & 1 \ 8 & 3 \end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } 3 & - 2 \ - 1 & 2 \end{array} \right]$ 
B) 
$\mathrm { AB } = \left[ \begin{array} { r r } - 4 & - 5 \ - 8 & - 9 \end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } - 3 & 2 \ 17 & - 10 \end{array} \right]$ 
C) 
$\mathrm { AB } = \left[ \begin{array} { r r } - 3 & 2 \ 17 & - 10 \end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } - 4 & - 5 \ - 8 & - 9 \end{array} \right]$ 
D)

$\mathrm { AB } = \left[ \begin{array} { r r } 0 & 1 \ 12 & - 6 \end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } 0 & 1 \ 12 & - 6 \end{array} \right]$ 
A) 
$\mathrm { AB } = \left[ \begin{array} { l l } 4 & 1 \ 8 & 3 \end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } 3 & - 2 \ - 1 & 2 \end{array} \right]$ 
B) 
$\mathrm { AB } = \left[ \begin{array} { r r } - 4 & - 5 \ - 8 & - 9 \end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } - 3 & 2 \ 17 & - 10 \end{array} \right]$ 
C) 
$\mathrm { AB } = \left[ \begin{array} { r r } - 3 & 2 \ 17 & - 10 \end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } - 4 & - 5 \ - 8 & - 9 \end{array} \right]$ 
D)

$\mathrm { AB } = \left[ \begin{array} { r r } 0 & 1 \ 12 & - 6 \end{array} \right] ; \mathrm { BA } = \left[ \begin{array} { r r } 0 & 1 \ 12 & - 6 \end{array} \right]$

Question 9

Find the partial fraction decomposition for the rational expression.
-$$\frac { 31 x - 70 } { 6 x ^ { 2 } - 31 x + 35 }$$

Accepted Answer

A)  $\frac { 7 } { 2 x + 7 } + \frac { 5 } { 3 x + 5 }$ 
B)  $\frac { 5 } { 3 x - 5 } + \frac { - 7 } { 2 x - 7 }$ 
C)  $\frac { 7 } { 2 x - 7 } + \frac { - 5 } { 3 x - 5 }$ 
D)  $\frac { 7 } { 2 x - 7 } + \frac { 5 } { 3 x - 5 }$ 
A)  $\frac { 7 } { 2 x + 7 } + \frac { 5 } { 3 x + 5 }$ 
B)  $\frac { 5 } { 3 x - 5 } + \frac { - 7 } { 2 x - 7 }$ 
C)  $\frac { 7 } { 2 x - 7 } + \frac { - 5 } { 3 x - 5 }$ 
D)  $\frac { 7 } { 2 x - 7 } + \frac { 5 } { 3 x - 5 }$

Question 10

Use a graphing calculator to solve the system. Express solutions with approximations to the nearest thousandth.
-$$\begin{array} { l } 
\sqrt { 2 } x + \sqrt { 7 } y - 3 = 0 \
\sqrt { 3 } x - y - \sqrt { 6 } = 0
\end{array}$$

Accepted Answer

A)  $\{ ( 1.099,0.547 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 1.670,0.443 ) \}$ 
D)  $\{ ( 1.581,0.289 ) \}$ 
A)  $\{ ( 1.099,0.547 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 1.670,0.443 ) \}$ 
D)  $\{ ( 1.581,0.289 ) \}$

Question 11

Use the given row transformation to change the matrix as indicated.
-$\left[ \begin{array} { l l } 2 & 3 \ 4 & 7 \end{array} \right] ; - 2$ times row 1 added to row 2

Accepted Answer

A) 
$\left[ \begin{array} { l l } 2 & 3 \ 1 & 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { l l } 2 & 3 \ 0 & 1 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } - 4 & - 6 \ 0 & 1 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 2 & 3 \ 0 & - 1 \end{array} \right]$ 
A) 
$\left[ \begin{array} { l l } 2 & 3 \ 1 & 1 \end{array} \right]$ 
B) 
$\left[ \begin{array} { l l } 2 & 3 \ 0 & 1 \end{array} \right]$ 
C) 
$\left[ \begin{array} { r r } - 4 & - 6 \ 0 & 1 \end{array} \right]$ 
D) 
$\left[ \begin{array} { r r } 2 & 3 \ 0 & - 1 \end{array} \right]$

Question 12

Solve the problem.
-The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the three years considered. For a function of the form $f ( x ) = a x ^ { 2 } + b x + c$ that fits this data, what is the value of $b$ ?

Accepted Answer

A)  $1.9$ 
B)  $- 1.6$ 
C)  $- 1.2$ 
D)  $2.3$ 
A)  $1.9$ 
B)  $- 1.6$ 
C)  $- 1.2$ 
D)  $2.3$

Question 13

Find the value of the determinant.
-$\left| \begin{array} { l l } 3 & 2 \ - 1 & - 1 \end{array} \right|$

Accepted Answer

A) -5 
B)  $- 1$ 
C)  5 
D)  1 
A) -5 
B)  $- 1$ 
C)  5 
D)  1

Question 14

Solve the system by substitution.
-$$\begin{array} { l } 
x + y = - 6 \
x - y = - 4
\end{array}$$

Accepted Answer

A)  $\{ ( - 5 , - 1 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 5,0 ) \}$ 
D)  $\{ ( - 6,0 ) \}$ 
A)  $\{ ( - 5 , - 1 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 5,0 ) \}$ 
D)  $\{ ( - 6,0 ) \}$

Question 15

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
- 4 x - y - 9 z = - 69 \
- 5 x + 7 y + 5 z = 36 \
- 9 x - 5 y + z = - 36
\end{array}$$

Accepted Answer

A)  $\{ ( - 3,3,6 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 3,6,3 ) \}$ 
D)  $\{ ( 3,3,6 ) \}$ 
A)  $\{ ( - 3,3,6 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( 3,6,3 ) \}$ 
D)  $\{ ( 3,3,6 ) \}$

Question 16

Determine the system of inequalities illustrated in the graph. Write inequalities in standard form.
-

Accepted Answer

A)  $x - y  7$ 
B)  $x - y  4$
$x + y  4$
$x + y > 7$ 
A)  $x - y  7$ 
B)  $x - y  4$
$x + y  4$
$x + y > 7$

Question 17

Solve the problem using matrices.
-In a study of heat transfer in a grid of wires, the temperature at an exterior node is maintained at a constant value (in ${ } ^ { \circ } \mathrm { F }$ ) as shown in the figure. When the grid is in thermal equilibrium, the temperature at an interior node is the average of the temperatures at the four adjacent nodes. For instance $\mathrm { T } _ { 1 } = \frac { 0 + 0 + 300 + \mathrm { T } _ { 2 } } { 4 }$, or $4 \mathrm {~T} _ { 1 } - \mathrm { T } _ { 2 } = 300$. Find the temperatures $\mathrm { T } _ { 1 } , \mathrm {~T} _ { 2 }$, and $\mathrm { T } _ { 3 }$ when the grid is in thermal equilibrium.

Accepted Answer

A)  $\mathrm { T } _ { 1 } = \frac { 666 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 564 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 3 } = 105 ^ { \circ } \mathrm { F }$ 
B)  $\mathrm { T } _ { 1 } = \frac { 766 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 964 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 430 } { 7 } \circ \mathrm { F }$ 
C)  $\mathrm { T } _ { 1 } = \frac { 1352 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 604 } { 7 } \circ { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 655 } { 7 } \circ \mathrm { F }$ 
D)  $\mathrm { T } _ { 1 } = \frac { 676 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 604 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 655 } { 14 } \circ \mathrm { F }$ 
A)  $\mathrm { T } _ { 1 } = \frac { 666 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 564 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 3 } = 105 ^ { \circ } \mathrm { F }$ 
B)  $\mathrm { T } _ { 1 } = \frac { 766 } { 7 } { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 964 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 430 } { 7 } \circ \mathrm { F }$ 
C)  $\mathrm { T } _ { 1 } = \frac { 1352 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 604 } { 7 } \circ { } ^ { \circ } \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 655 } { 7 } \circ \mathrm { F }$ 
D)  $\mathrm { T } _ { 1 } = \frac { 676 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 2 } = \frac { 604 } { 7 } \circ \mathrm { F } , \mathrm { T } _ { 3 } = \frac { 655 } { 14 } \circ \mathrm { F }$

Question 18

Find the partial fraction decomposition for the rational expression.
-$$\frac { - 3 x ^ { 2 } - 5 x - 42 } { ( x - 3 ) \left( x ^ { 2 } + 3 \right) }$$

Accepted Answer

A)  $\frac { 4 x + 7 } { x ^ { 2 } + 3 } + \frac { 7 } { x - 3 }$ 
B)  $\frac { 4 x + 7 } { x ^ { 2 } + 3 } + \frac { - 7 } { x - 3 }$ 
C)  $\frac { 4 x + 8 } { x ^ { 2 } + 3 } + \frac { - 7 } { x - 3 }$ 
D)  $\frac { 7 x + 4 } { x ^ { 2 } + 3 } + \frac { - 7 } { x - 3 }$ 
A)  $\frac { 4 x + 7 } { x ^ { 2 } + 3 } + \frac { 7 } { x - 3 }$ 
B)  $\frac { 4 x + 7 } { x ^ { 2 } + 3 } + \frac { - 7 } { x - 3 }$ 
C)  $\frac { 4 x + 8 } { x ^ { 2 } + 3 } + \frac { - 7 } { x - 3 }$ 
D)  $\frac { 7 x + 4 } { x ^ { 2 } + 3 } + \frac { - 7 } { x - 3 }$

Question 19

Solve the problem.
-A chair manufacturing company has two different factories. The profit function for Factory I when $x$ chairs are manufactured and sold is as follows:
$$P ( x ) = - 0.05 x ^ { 2 } + 20 x - 500$$
The profit function for Factory II when $x$ chairs are manufactured and sold is as follows:
$$P ( x ) = - 0.05 x ^ { 2 } + 30 x - 2500$$
For what number of chairs is the profit the same at both factories, and what is the profit?

Accepted Answer

A)  100 chairs; $\$ 1000$ 
B)  200 chairs; $\$ 1500$ 
C)  0 chairs; $\$ 0$ 
D)  300 chairs; $\$ 2000$ 
A)  100 chairs; $\$ 1000$ 
B)  200 chairs; $\$ 1500$ 
C)  0 chairs; $\$ 0$ 
D)  300 chairs; $\$ 2000$

Question 20

Provide an appropriate response.
-Which one of the following is a description of the graph of the inequality?
$$( x + 7 ) ^ { 2 } + ( y + 2 ) ^ { 2 } < 9$$

Accepted Answer

A)  The region inside a circle with center $( - 7 , - 2 )$ and radius 3 
B)  The region outside a circle with center $( 7,2 )$ and radius 3 
C)  The region inside the circle with center $( 7,2 )$ and radius 3 
D)  The region outside a circle with center $( - 7 , - 2 )$ and radius 3 
A)  The region inside a circle with center $( - 7 , - 2 )$ and radius 3 
B)  The region outside a circle with center $( 7,2 )$ and radius 3 
C)  The region inside the circle with center $( 7,2 )$ and radius 3 
D)  The region outside a circle with center $( - 7 , - 2 )$ and radius 3

Solve the system. If the system has infinitely many solutions, write the solution set with x arbitrary. - 2x-y-3=0 -4x+2y+6=0

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { l l l } 1 & 1 & 1 \\2 & 1 & 1 \\2 & 2 & 3\end{array} \right]$

Graph the solution set of the system of inequalities. - y\leq--6x-4 y\geq+6x+4 $Graph the solution set of the system of inequalities. - \begin{array}{l} y \leq-x^{2}-6 x-4 \\ y \geq x^{2}+6 x+4 \end{array}$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - 3+3=20 6+6=48

Which method should be used to solve the system? Explain your answer, including a description of the first step. - -7+ =64 -5+4 =100

Find the partial fraction decomposition for the rational expression. - $\frac { 3 x - 14 } { ( x - 4 ) ^ { 2 } }$

Solve the problem. -Given $A = \left[ \begin{array} { l l } 2 & - 1 \\ 3 & - 2 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 0 & - 1 \\ 4 & 3 \end{array} \right]$ , find $A B$ and $B A$ .

Find the partial fraction decomposition for the rational expression. - $\frac { 31 x - 70 } { 6 x ^ { 2 } - 31 x + 35 }$

Use a graphing calculator to solve the system. Express solutions with approximations to the nearest thousandth. - x+y-3=0 x-y-=0

Use the given row transformation to change the matrix as indicated. - $\left[ \begin{array} { l l } 2 & 3 \\ 4 & 7 \end{array} \right] ; - 2$ times row 1 added to row 2

Find the value of the determinant. - $\left| \begin{array} { l l } 3 & 2 \\ - 1 & - 1 \end{array} \right|$

Solve the system by substitution. - x+y=-6 x-y=-4

Solve the system by using the inverse of the coefficient matrix. - -4x-y-9z=-69 -5x+7y+5z=36 -9x-5y+z=-36

Determine the system of inequalities illustrated in the graph. Write inequalities in standard form. -

Find the partial fraction decomposition for the rational expression. - $\frac { - 3 x ^ { 2 } - 5 x - 42 } { ( x - 3 ) \left( x ^ { 2 } + 3 \right) }$

Provide an appropriate response. -Which one of the following is a description of the graph of the inequality? $( x + 7 ) ^ { 2 } + ( y + 2 ) ^ { 2 } < 9$

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Analytic Geometry

Further Topics in Algebra

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Exam 9: Systems and Matrices