Question 1

Graph the solution set of the system of inequalities.
-

Accepted Answer

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

Question 2

Solve the system.
-$$\begin{array} { l } 
6 x + 6 y - z = 32 \
x + 9 y - 4 z = - 2 \
6 x + y + z = 35
\end{array}$$

Accepted Answer

A)  $\{ ( 5,1,4 ) \}$ 
B)  $\{ ( - 5,1,10 ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( 5,4,1 ) \}$ 
A)  $\{ ( 5,1,4 ) \}$ 
B)  $\{ ( - 5,1,10 ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( 5,4,1 ) \}$

Question 3

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

Accepted Answer

A)  $- 2 x + y \leq 4$
$- 3 x + 4 y \geq 8$ 
B)  $2 x + y \geq 4$
$3 x + 4 y \leq 8$ 
C)  $2 x + y \leq 4$
$3 x + 4 y \geq 8$ 
D)  $- 2 x + y \geq 4$
$- 3 x + 4 y \leq 8$ 
A)  $- 2 x + y \leq 4$
$- 3 x + 4 y \geq 8$ 
B)  $2 x + y \geq 4$
$3 x + 4 y \leq 8$ 
C)  $2 x + y \leq 4$
$3 x + 4 y \geq 8$ 
D)  $- 2 x + y \geq 4$
$- 3 x + 4 y \leq 8$

Question 4

Solve the linear programming problem.
-A summer camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The average monthly salary of a counselor is $\$ 2400$, and the average monthly salary of an aide is $\$ 1100$. The camp can accommodate up to 35 staff members and needs at least 20 to run properly. The camp must have at least 10 aides and may have up to 3 aides for every 2 counselors. How many counselors and how many aides should the camp hire to minimize cost?

Accepted Answer

A)  13 counselors and 16 aides 
B)  13 counselors and 17 aides 
C)  8 counselors and 12 aides 
D)  19 counselors and 23 aides 
A)  13 counselors and 16 aides 
B)  13 counselors and 17 aides 
C)  8 counselors and 12 aides 
D)  19 counselors and 23 aides

Question 5

Find the indicated matrix.
-Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & 2 \ 0 & 2 \end{array} \right]$. Find $3 \mathrm {~A}$.

Accepted Answer

A) 
$\left[\begin{array}{ll}
0 & 5 \
3 & 5
\end{array}\right]$ 
B) 
$\left[\begin{array}{rr}
-9 & 6 \
0 & 2
\end{array}\right]$ 
C) 
$\left[\begin{array}{rr}
-9 & 6 \
0 & 6
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr}
-9 & 2 \
0 & 2
\end{array}\right]$ 
A) 
$\left[\begin{array}{ll}
0 & 5 \
3 & 5
\end{array}\right]$ 
B) 
$\left[\begin{array}{rr}
-9 & 6 \
0 & 2
\end{array}\right]$ 
C) 
$\left[\begin{array}{rr}
-9 & 6 \
0 & 6
\end{array}\right]$ 
D) 
$\left[\begin{array}{rr}
-9 & 2 \
0 & 2
\end{array}\right]$

Question 6

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { r r r r } 
2 & - 4 & 0 & 2 \
6 & - 8 & 0 & 4 \
6 & - 16 & 0 & 6 \
0 & 0 & - 5 & 0
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r r } 
- 1 & \frac { 3 } { 4 } & 0 & 3 \
\frac { 1 } { 2 } & 0 & 0 & - \frac { 1 } { 2 } \
0 & - \frac { 1 } { 4 } & 0 & - \frac { 1 } { 2 } \
0 & 0 & - \frac { 1 } { 5 } & 0
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r r } 
- 1 & \frac { 1 } { 2 } & 0 & 0 \
\frac { 3 } { 4 } & 0 & - \frac { 1 } { 4 } & 0 \
0 & 0 & 0 & - \frac { 1 } { 5 } \
3 - \frac { 1 } { 2 } & - \frac { 1 } { 2 } & 0
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c c c c } 
\frac { 1 } { 2 } & - \frac { 1 } { 4 } & 0 & \frac { 1 } { 2 } \
\frac { 1 } { 6 } & - \frac { 1 } { 8 } & 0 & \frac { 1 } { 4 } \
\frac { 1 } { 6 } & - \frac { 1 } { 16 } & 0 & \frac { 1 } { 6 } \
0 & 0 & - \frac { 1 } { 5 } & 0
\end{array} \right]$$ 
D) 
$\text { The inverse does not exist. }$ 
A) 
$$\left[ \begin{array} { r r r r } 
- 1 & \frac { 3 } { 4 } & 0 & 3 \
\frac { 1 } { 2 } & 0 & 0 & - \frac { 1 } { 2 } \
0 & - \frac { 1 } { 4 } & 0 & - \frac { 1 } { 2 } \
0 & 0 & - \frac { 1 } { 5 } & 0
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r r r } 
- 1 & \frac { 1 } { 2 } & 0 & 0 \
\frac { 3 } { 4 } & 0 & - \frac { 1 } { 4 } & 0 \
0 & 0 & 0 & - \frac { 1 } { 5 } \
3 - \frac { 1 } { 2 } & - \frac { 1 } { 2 } & 0
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c c c c } 
\frac { 1 } { 2 } & - \frac { 1 } { 4 } & 0 & \frac { 1 } { 2 } \
\frac { 1 } { 6 } & - \frac { 1 } { 8 } & 0 & \frac { 1 } { 4 } \
\frac { 1 } { 6 } & - \frac { 1 } { 16 } & 0 & \frac { 1 } { 6 } \
0 & 0 & - \frac { 1 } { 5 } & 0
\end{array} \right]$$ 
D) 
$\text { The inverse does not exist. }$

Question 7

Find the partial fraction decomposition for the rational expression.
-$\frac { 4 } { x ^ { 2 } \left( x ^ { 2 } + 7 \right) }$

Accepted Answer

A)  $\frac { 4 } { 7 x ^ { 2 } } + \frac { - 4 } { 7 \left( x ^ { 2 } + 7 \right) }$ 
B)  $\frac { - 4 } { 7 x ^ { 2 } } + \frac { 4 } { 7 \left( x ^ { 2 } + 7 \right) }$ 
C)  $\frac { - 4 } { x ^ { 2 } } + \frac { 4 } { \left( x ^ { 2 } + 7 \right) }$ 
D)  $\frac { 4 } { x ^ { 2 } } + \frac { - 4 } { \left( x ^ { 2 } + 7 \right) }$ 
A)  $\frac { 4 } { 7 x ^ { 2 } } + \frac { - 4 } { 7 \left( x ^ { 2 } + 7 \right) }$ 
B)  $\frac { - 4 } { 7 x ^ { 2 } } + \frac { 4 } { 7 \left( x ^ { 2 } + 7 \right) }$ 
C)  $\frac { - 4 } { x ^ { 2 } } + \frac { 4 } { \left( x ^ { 2 } + 7 \right) }$ 
D)  $\frac { 4 } { x ^ { 2 } } + \frac { - 4 } { \left( x ^ { 2 } + 7 \right) }$

Question 8

Find the cofactor of the indicated element.
-$\text { a33 }$
$\left|\begin{array}{rrr}
3 & 3 & 2 \
3 & 5 & 4 \
-1 & -1 & 4
\end{array}\right|$

Accepted Answer

A)  6 
B)  $- 2$ 
C)  2 
D)  $- 6$ 
A)  6 
B)  $- 2$ 
C)  2 
D)  $- 6$

Question 9

Provide an appropriate response.
-Describe the elements of a $3 \times 3$ zero matrix.

Accepted Answer

A)  Every element of the matrix is zero. 
B)  The matrix has ones on the main diagonal, and all other elements are zeros. 
C)  The matrix has zeros on the main diagonal, and there are no restrictions on the other elements. 
D)  The matrix has zeros on the main diagonal, and all other elements are ones. 
A)  Every element of the matrix is zero. 
B)  The matrix has ones on the main diagonal, and all other elements are zeros. 
C)  The matrix has zeros on the main diagonal, and there are no restrictions on the other elements. 
D)  The matrix has zeros on the main diagonal, and all other elements are ones.

Question 10

Solve the system.
-$$\begin{array} { l } 
x - y + z = 1 \
x + y + z = 3 \
x + y - z = - 3
\end{array}$$

Accepted Answer

A)  $\{ ( 3 , - 1,1 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( - 1,1,3 ) \}$ 
D)  $\{ ( - 1,3,1 ) \}$ 
A)  $\{ ( 3 , - 1,1 ) \}$ 
B)  $\varnothing$ 
C)  $\{ ( - 1,1,3 ) \}$ 
D)  $\{ ( - 1,3,1 ) \}$

Question 11

Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
-Let $M$ be a $6 \times 6$ matrix. Which of the following statements are true?
(i) $\mathrm { I } _ { 6 } \mathrm { M } = \mathrm { M }$
(ii) $\mathrm { I } _ { 6 } + \mathrm { M } = \mathrm { M }$
(iii) $\mathrm { MI } _ { 6 } = \mathrm { I } _ { 6 }$
(iv) $\mathrm { MI } _ { 6 } = \mathrm { M }$

Accepted Answer

A)  (i), (iii), (iv) 
B)  (ii), (iii) 
C)  (i), (ii), (iii), (iv) 
D)  (i), (iv) 
A)  (i), (iii), (iv) 
B)  (ii), (iii) 
C)  (i), (ii), (iii), (iv) 
D)  (i), (iv)

Question 12

The following graph shows the populations of the metropolitan areas of City X and City Y over six years.  
-Express the solution of the system as an ordered pair.

Accepted Answer

A)  approximately (3.9, 0.92) 
B)  approximately (1.2, 1.03) 
C)  approximately (4.6, 0.87) 
D)  approximately (3.1, 0.95) 
A)  approximately (3.9, 0.92) 
B)  approximately (1.2, 1.03) 
C)  approximately (4.6, 0.87) 
D)  approximately (3.1, 0.95)

Question 13

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{aligned}
x ^ { 2 } - 3 y ^ { 2 } & = 1 \
4 x ^ { 2 } + 3 y ^ { 2 } & = 19
\end{aligned}$$

Accepted Answer

A)  $\{ ( 2,1 ) , ( - 2 , - 1 ) \}$ 
B)  $\{ ( - 1,2 ) , ( 1 , - 2 ) \}$ 
C)  $\{ ( 2,1 ) , ( 2 , - 1 ) , ( - 2,1 ) , ( - 2 , - 1 ) \}$ 
D)  $\{ ( 1,2 ) , ( - 1,2 ) , ( 1 , - 2 ) , ( - 1 , - 2 ) \}$ 
A)  $\{ ( 2,1 ) , ( - 2 , - 1 ) \}$ 
B)  $\{ ( - 1,2 ) , ( 1 , - 2 ) \}$ 
C)  $\{ ( 2,1 ) , ( 2 , - 1 ) , ( - 2,1 ) , ( - 2 , - 1 ) \}$ 
D)  $\{ ( 1,2 ) , ( - 1,2 ) , ( 1 , - 2 ) , ( - 1 , - 2 ) \}$

Question 14

Use a graphing calculator to find the inverse of the matrix. Give five decimal places, if necessary.
-$$\mathrm { A } = \left[ \begin{array} { c c c } 
1 / 3 & 2 / 7 & 4 / 5 \
17 / 11 & 5 & 0 \
18 / 13 & - 5 / 6 & 8 / 7
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r r } 
- 1.10557 & 0.19216 & 0.77390 \
0.34172 & 0.14061 & - 0.23920 \
1.58861 & - 0.13028 & - 0.23703
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { l l l } 
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c c c } 
3 & 3.5 & 1.25 \
0.64706 & 5 & 0 \
0.72222 & - 1.2 & 1.14286
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { r r r } 
2.65952 & 0.15830 & - 1.86167 \
- 9.04238 & - 0.33824 & 6.32966 \
3.37129 & 0.05484 & - 1.48490
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r r } 
- 1.10557 & 0.19216 & 0.77390 \
0.34172 & 0.14061 & - 0.23920 \
1.58861 & - 0.13028 & - 0.23703
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { l l l } 
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { c c c } 
3 & 3.5 & 1.25 \
0.64706 & 5 & 0 \
0.72222 & - 1.2 & 1.14286
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { r r r } 
2.65952 & 0.15830 & - 1.86167 \
- 9.04238 & - 0.33824 & 6.32966 \
3.37129 & 0.05484 & - 1.48490
\end{array} \right]$$

Question 15

Solve the problem.
-A bookstore is having a sale. All books included in the sale have a colored sticker on them to indicate the sale price. There are green stickers, red stickers, and orange stickers. Bob, Sue, and Fred each make purchases of books that are on sale. Each row of the table gives information about the numbers of book purchases and the total cost of the purchase (before taxes).

Use this information to set up a matrix equation of the form $A X = B$, which can be solved to determine the price for each type of sale book. Solve this matrix equation to find the price of a book with a red sticker.
Use the fact that for $A = \left[ \begin{array} { l l l } 1 & 2 & 2 \ 1 & 3 & 2 \ 1 & 2 & 3 \end{array} \right] , A ^ { - 1 } = \left[ \begin{array} { r r r } 5 & - 2 & - 2 \ - 1 & 1 & 0 \ - 1 & 0 & 1 \end{array} \right]$.

Accepted Answer

A)  $\$ 6.50$ 
B)  $\$ 6.79$ 
C)  $\$ 6.87$ 
D)  $\$ 6.99$ 
A)  $\$ 6.50$ 
B)  $\$ 6.79$ 
C)  $\$ 6.87$ 
D)  $\$ 6.99$

Question 16

Find the value of the determinant.
-$\left| \begin{array} { r r r } 5 & - 5 & 8 \ 0 & 9 & 9 \ 0 & 0 & - 7 \end{array} \right|$

Accepted Answer

A)  315 
B)  270 
C)  $- 315$ 
D)  $- 360$ 
A)  315 
B)  270 
C)  $- 315$ 
D)  $- 360$

Question 17

Solve the problem.
-The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the three years considered. Find a function of the form $f ( x ) = a x ^ { 2 } + b x + c$ that fits this data. Using this function, what would you predict the dog collar production of Allied Pet Products to be in year 32 ?

Accepted Answer

A)  86,200 collars 
B)  91,200 collars 
C)  71,200 collars 
D)  100,100 collars 
A)  86,200 collars 
B)  91,200 collars 
C)  71,200 collars 
D)  100,100 collars

Question 18

Solve the system.
-$$\begin{array} { l } 
\frac { 1 } { x } + \frac { 1 } { y } - \frac { 1 } { z } = - \frac { 3 } { 28 } \
\frac { 3 } { x } - \frac { 1 } { y } + \frac { 3 } { z } = \frac { 23 } { 28 } \
\frac { 1 } { x } - \frac { 7 } { y } + \frac { 6 } { z } = \frac { 67 } { 28 }
\end{array}$$

Accepted Answer

A)  $\{ ( 4 , - 2 , - 7 ) \}$ 
B)  $\{ ( 4 , - 2,7 ) \}$ 
C)  $\left\{ \left( \frac { 1 } { 4 } , - \frac { 1 } { 2 } , \frac { 1 } { 7 } \right) \right\}$ 
D)  $\left\{ \left( \frac { 1 } { 4 } , - \frac { 1 } { 2 } , - \frac { 1 } { 7 } \right) \right\}$ 
A)  $\{ ( 4 , - 2 , - 7 ) \}$ 
B)  $\{ ( 4 , - 2,7 ) \}$ 
C)  $\left\{ \left( \frac { 1 } { 4 } , - \frac { 1 } { 2 } , \frac { 1 } { 7 } \right) \right\}$ 
D)  $\left\{ \left( \frac { 1 } { 4 } , - \frac { 1 } { 2 } , - \frac { 1 } { 7 } \right) \right\}$

Question 19

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the
solution with y arbitrary.
-$$\begin{array} { l } 
2 x + 5 y = - 7 \
- 6 x - 15 y = 21
\end{array}$$

Accepted Answer

A)  $\{ ( - 1 , - 1 ) \}$ 
B)  $\left\{ \left( - \frac { 5 } { 2 } y - \frac { 7 } { 2 } , y \right) \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \left( - \frac { 5 } { 2 } y + \frac { 7 } { 2 } , y \right) ^ { \prime } \right.$ 
A)  $\{ ( - 1 , - 1 ) \}$ 
B)  $\left\{ \left( - \frac { 5 } { 2 } y - \frac { 7 } { 2 } , y \right) \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \left( - \frac { 5 } { 2 } y + \frac { 7 } { 2 } , y \right) ^ { \prime } \right.$

Question 20

Solve the system by using the inverse of the coefficient matrix.
-$$\begin{array} { l } 
x - 2 y + 3 z = 11 \
4 x + y - z = 4 \
2 x - y + 3 z = 10
\end{array}$$

Accepted Answer

A)  $\{ ( 2,3,5 ) \}$ 
B)  $\{ ( 2 , - 3,1 ) \}$ 
C)  $\{ ( - 2 , - 3,3 ) \}$ 
D)  $\varnothing$ 
A)  $\{ ( 2,3,5 ) \}$ 
B)  $\{ ( 2 , - 3,1 ) \}$ 
C)  $\{ ( - 2 , - 3,3 ) \}$ 
D)  $\varnothing$

Graph the solution set of the system of inequalities. -

Solve the system. - 6x+6y-z=32 x+9y-4z=-2 6x+y+z=35

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

Find the indicated matrix. -Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & 2 \\ 0 & 2 \end{array} \right]$ . Find $3 \mathrm {~A}$ .

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { r r r r } 2 & - 4 & 0 & 2 \\6 & - 8 & 0 & 4 \\6 & - 16 & 0 & 6 \\0 & 0 & - 5 & 0\end{array} \right]$

Find the partial fraction decomposition for the rational expression. - $\frac { 4 } { x ^ { 2 } \left( x ^ { 2 } + 7 \right) }$

Find the cofactor of the indicated element. - $\text { a33 }$ $\left|\begin{array}{rrr}3 & 3 & 2 \\3 & 5 & 4 \\-1 & -1 & 4\end{array}\right|$

Provide an appropriate response. -Describe the elements of a $3 \times 3$ zero matrix.

Solve the system. - x-y+z=1 x+y+z=3 x+y-z=-3

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -Express the solution of the system as an ordered pair.

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - -3 =1 4+3 =19

Use a graphing calculator to find the inverse of the matrix. Give five decimal places, if necessary. - $\mathrm { A } = \left[ \begin{array} { c c c } 1 / 3 & 2 / 7 & 4 / 5 \\17 / 11 & 5 & 0 \\18 / 13 & - 5 / 6 & 8 / 7\end{array} \right]$

Find the value of the determinant. - $\left| \begin{array} { r r r } 5 & - 5 & 8 \\ 0 & 9 & 9 \\ 0 & 0 & - 7 \end{array} \right|$

Solve the system. - +-=- -+= -+=

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the solution with y arbitrary. - 2x+5y=-7 -6x-15y=21

Solve the system by using the inverse of the coefficient matrix. - x-2y+3z=11 4x+y-z=4 2x-y+3z=10

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