Question 1

Provide an appropriate response.
-Fill in the blank to complete the statement. In order for a matrix to have a determinant, it must be matrix.

Accepted Answer

A)  an augmented 
B)  a square 
C)  a rectangular 
D)  a minor 
A)  an augmented 
B)  a square 
C)  a rectangular 
D)  a minor B

Question 2

Find the matrix product when possible.
-$$\left[ \begin{array} { r r } 
0 & - 3 \
4 & 3
\end{array} \right] \left[ \begin{array} { l l } 
- 2 & 0 \
- 1 & 1
\end{array} \right]$$

Accepted Answer

A) 
$$\left[ \begin{array} { r r } 
3 & - 3 \
- 11 & 3
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r } 
- 3 & 3 \
- 5 & - 11
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r } 
- 8 & - 6 \
4 & 6
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r } 
0 & 6 \
- 4 & 3
\end{array} \right]$$ 
A) 
$$\left[ \begin{array} { r r } 
3 & - 3 \
- 11 & 3
\end{array} \right]$$ 
B) 
$$\left[ \begin{array} { r r } 
- 3 & 3 \
- 5 & - 11
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r } 
- 8 & - 6 \
4 & 6
\end{array} \right]$$ 
D) 
$$\left[ \begin{array} { r r } 
0 & 6 \
- 4 & 3
\end{array} \right]$$ A

Question 3

Give all solutions of the nonlinear system of equations, including those with nonreal complex components.
-$$\begin{array} { l } 
x ^ { 2 } + y ^ { 2 } = 85 \
x + y = - 13
\end{array}$$

Accepted Answer

A)  $\{ ( 6,7 ) , ( 7,6 ) \}$ 
B)  $\{ ( - 6 , - 7 ) , ( - 7 , - 6 ) \}$ 
C)  $\{ ( - 6,7 ) , ( - 7,6 ) \}$ 
D)  $\{ ( 6 , - 7 ) , ( 7 , - 6 ) \}$ 
A)  $\{ ( 6,7 ) , ( 7,6 ) \}$ 
B)  $\{ ( - 6 , - 7 ) , ( - 7 , - 6 ) \}$ 
C)  $\{ ( - 6,7 ) , ( - 7,6 ) \}$ 
D)  $\{ ( 6 , - 7 ) , ( 7 , - 6 ) \}$ B

Question 4

Solve the linear programming problem.
-Zach is planning to invest up to $\$ 40,000$ in corporate and municipal bonds. The least he will invest in corporate bonds is $\$ 9000$ and he does not want to invest more than $\$ 29,000$ in corporate bonds. He also does not want to invest more than $\$ 29,731$ in municipal bonds. The interest is $8 \%$ on corporate bonds and $6.5 \%$ on municipal bonds. This is simple interest for one year. What is the maximum income?

Accepted Answer

A)  $\$ 32,035$ 
B)  $\$ 3035$ 
C)  $\$ 14,035$ 
D)  $\$ 19,605$ 
A)  $\$ 32,035$ 
B)  $\$ 3035$ 
C)  $\$ 14,035$ 
D)  $\$ 19,605$

Question 5

Graph the solution set of the system of inequalities.
-$\begin{array}{r}
2 x+3 y \geq 6 \
x-y \geq 3 \
y \leq 2
\end{array}$

Accepted Answer

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

Question 6

Find the partial fraction decomposition for the rational expression.
-$$\frac { 12 x ^ { 2 } - 23 x + 5 } { ( x - 3 ) \left( x ^ { 2 } + 2 \right) }$$

Accepted Answer

A)  $\frac { - ( 8 x + 1 ) } { x ^ { 2 } + 2 } + \frac { 4 } { x - 3 }$ 
B)  $\frac { 8 x + 1 } { x ^ { 2 } + 2 } + \frac { 4 } { x - 3 }$ 
C)  $\frac { 8 x + 4 } { x ^ { 2 } + 2 } + \frac { 1 } { x - 3 }$ 
D)  $\frac { 8 x - 1 } { x ^ { 2 } + 2 } + \frac { - 4 } { x - 3 }$ 
A)  $\frac { - ( 8 x + 1 ) } { x ^ { 2 } + 2 } + \frac { 4 } { x - 3 }$ 
B)  $\frac { 8 x + 1 } { x ^ { 2 } + 2 } + \frac { 4 } { x - 3 }$ 
C)  $\frac { 8 x + 4 } { x ^ { 2 } + 2 } + \frac { 1 } { x - 3 }$ 
D)  $\frac { 8 x - 1 } { x ^ { 2 } + 2 } + \frac { - 4 } { x - 3 }$

Question 7

Graph the solution set of the system of inequalities.
-$$\begin{array} { l } 
\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } \leq 1 \
\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } \leq 1
\end{array}$$

Accepted Answer

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

Question 8

Graph the solution set of the system of inequalities.
-$$\begin{array} { c } 
2 x + y \leq 4 \
y - 1 \leq 0
\end{array}$$

Accepted Answer

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

Question 9

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

Question 10

Solve the problem.
-A contractor mixes concrete from bags of pre-mix for small jobs. How many bags with $6 \%$ cement should he mix with 5 bags of $18 \%$ cement to produce a mix containing $12 \%$ cement?

Accepted Answer

A)  10 bags 
B)  5 bags 
C)  15 bags 
D)  7 bags 
A)  10 bags 
B)  5 bags 
C)  15 bags 
D)  7 bags

Question 11

Solve the equation for x.
-$$\left| \begin{array} { r r } 
3 & x \
2 & - 1
\end{array} \right| = 5$$

Accepted Answer

A)  $\left\{ \frac { 5 } { 3 } \right\}$ 
B)  $\left\{ \frac { 5 } { 2 } \right\}$ 
C)  $\{ - 4 \}$ 
D)  $\{ 4 \}$ 
A)  $\left\{ \frac { 5 } { 3 } \right\}$ 
B)  $\left\{ \frac { 5 } { 2 } \right\}$ 
C)  $\{ - 4 \}$ 
D)  $\{ 4 \}$

Question 12

Find the value of the determinant.
-$$\left| \begin{array} { l l l } 
2 & 3 & 4 \
1 & 1 & 2 \
3 & 3 & 5
\end{array} \right|$$

Accepted Answer

A)  $- 35$ 
B)  1 
C)  $- 1$ 
D)  79 
A)  $- 35$ 
B)  1 
C)  $- 1$ 
D)  79

Question 13

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$( x - 4 ) ^ { 2 } + ( y + 3 ) ^ { 2 } \leq 4$$

Accepted Answer

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

Question 14

Find the inverse, if it exists, for the matrix.
-$$\left[ \begin{array} { l l l } 
1 & 3 & 2 \
1 & 3 & 3 \
2 & 7 & 8
\end{array} \right]$$

Accepted Answer

A)  The inverse does not exist. 
B) 
$$\left[ \begin{array} { c c c } 
1 & \frac { 1 } { 3 } & \frac { 1 } { 2 } \
1 & \frac { 1 } { 3 } & \frac { 1 } { 3 } \
\frac { 1 } { 2 } & \frac { 1 } { 7 } & \frac { 1 } { 8 }
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r } 
- 3 & 10 & - 3 \
2 & - 4 & 1 \
- 1 & 1 & 0
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { l l l } 
- 1 & - 3 & - 2 \
- 1 & - 3 & - 3 \
- 2 & - 7 & - 8
\end{array} \right]$$ 
A)  The inverse does not exist. 
B) 
$$\left[ \begin{array} { c c c } 
1 & \frac { 1 } { 3 } & \frac { 1 } { 2 } \
1 & \frac { 1 } { 3 } & \frac { 1 } { 3 } \
\frac { 1 } { 2 } & \frac { 1 } { 7 } & \frac { 1 } { 8 }
\end{array} \right]$$ 
C) 
$$\left[ \begin{array} { r r r } 
- 3 & 10 & - 3 \
2 & - 4 & 1 \
- 1 & 1 & 0
\end{array} \right]$$ 
D)

$$\left[ \begin{array} { l l l } 
- 1 & - 3 & - 2 \
- 1 & - 3 & - 3 \
- 2 & - 7 & - 8
\end{array} \right]$$

Question 15

The following graph shows the populations of the metropolitan areas of City X and City Y over six years.  
-At the time when the populations of the two metropolitan areas were equal, what was the approximate population of each area?

Accepted Answer

A)  about 0.92 million 
B)  about 0.87 million 
C)  about 0.95 million 
D)  about 1.03 million 
A)  about 0.92 million 
B)  about 0.87 million 
C)  about 0.95 million 
D)  about 1.03 million

Question 16

The following graph shows the populations of the metropolitan areas of City X and City Y over six years.  
-Use the terms increasing, decreasing, and/or constant to describe the trends for the population of the City X metropolitan area.

Accepted Answer

A)  The population of the City X metropolitan area was increasing from Year 1 to Year 5 and was constant from Year 5 to Year 6. 
B)  The population of the City X metropolitan area was decreasing from Year 1 to Year 6. 
C)  The population of the City X metropolitan area was increasing from Year 1 to Year 6. 
D)  The population of the City X metropolitan area was decreasing from Year 1 to Year 5 and was constant from Year 5 to Year 6. 
A)  The population of the City X metropolitan area was increasing from Year 1 to Year 5 and was constant from Year 5 to Year 6. 
B)  The population of the City X metropolitan area was decreasing from Year 1 to Year 6. 
C)  The population of the City X metropolitan area was increasing from Year 1 to Year 6. 
D)  The population of the City X metropolitan area was decreasing from Year 1 to Year 5 and was constant from Year 5 to Year 6.

Question 17

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants.
-$$\begin{array} { l } 
x + b y = a ^ { 2 } \
a x + y = b ^ { 2 }
\end{array}$$

Accepted Answer

A)  $\left\{ \left( - \frac { a ^ { 2 } - b ^ { 3 } } { a b } , - \frac { b ^ { 2 } - a ^ { 3 } } { a b } \right) \right\}$ 
B)  $\left\{ \left\{ \frac { a ^ { 2 } - b ^ { 3 } } { a + b } , \frac { a ^ { 2 } - b ^ { 3 } } { a + b } \right) \right\}$ 
C)  $\left\{ \left\{ \frac { \mathrm { b } ^ { 3 } } { 1 - \mathrm { ab } } , \frac { \mathrm { a } ^ { 3 } } { 1 - \mathrm { ab } } \right) \right\}$ 
D)  $\left\{ \left\{ \frac { a ^ { 2 } - b ^ { 3 } } { 1 - a b } , \frac { b ^ { 2 } - a ^ { 3 } } { 1 - a b } \right) \right\}$ 
A)  $\left\{ \left( - \frac { a ^ { 2 } - b ^ { 3 } } { a b } , - \frac { b ^ { 2 } - a ^ { 3 } } { a b } \right) \right\}$ 
B)  $\left\{ \left\{ \frac { a ^ { 2 } - b ^ { 3 } } { a + b } , \frac { a ^ { 2 } - b ^ { 3 } } { a + b } \right) \right\}$ 
C)  $\left\{ \left\{ \frac { \mathrm { b } ^ { 3 } } { 1 - \mathrm { ab } } , \frac { \mathrm { a } ^ { 3 } } { 1 - \mathrm { ab } } \right) \right\}$ 
D)  $\left\{ \left\{ \frac { a ^ { 2 } - b ^ { 3 } } { 1 - a b } , \frac { b ^ { 2 } - a ^ { 3 } } { 1 - a b } \right) \right\}$

Question 18

Graph the solution set of the system of inequalities.
-$$\begin{array} { l } 
\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } \leq 1 \
x ^ { 2 } + y ^ { 2 } \leq 36
\end{array}$$

Accepted Answer

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

Question 19

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of
the concepts involved in graphing inequalities.
-

Accepted Answer

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

Question 20

Provide an appropriate response
-If the graphs of a system of two equations are both parabolas, what are the possible 
numbers of solutions (with real coordinates) of this system?

Accepted Answer

0, 1, 2, 3

Provide an appropriate response. -Fill in the blank to complete the statement. In order for a matrix to have a determinant, it must be matrix.

Find the matrix product when possible. - $\left[ \begin{array} { r r } 0 & - 3 \\4 & 3\end{array} \right] \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 1\end{array} \right]$

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - +=85 x+y=-13

Graph the solution set of the system of inequalities. - 2x+3y\geq6 x-y\geq3 y\leq2 $Graph the solution set of the system of inequalities. - \begin{array}{r} 2 x+3 y \geq 6 \\ x-y \geq 3 \\ y \leq 2 \end{array}$

Find the partial fraction decomposition for the rational expression. - $\frac { 12 x ^ { 2 } - 23 x + 5 } { ( x - 3 ) \left( x ^ { 2 } + 2 \right) }$

Graph the solution set of the system of inequalities. - +\leq1 +\leq1 $Graph the solution set of the system of inequalities. - \begin{array} { l } \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } \leq 1 \\ \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } \leq 1 \end{array}$

Graph the solution set of the system of inequalities. - 2x+y\leq4 y-1\leq0 $Graph the solution set of the system of inequalities. - \begin{array} { c } 2 x + y \leq 4 \\ y - 1 \leq 0 \end{array}$

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

Solve the problem. -A contractor mixes concrete from bags of pre-mix for small jobs. How many bags with $6 \%$ cement should he mix with 5 bags of $18 \%$ cement to produce a mix containing $12 \%$ cement?

Solve the equation for x. - $\left| \begin{array} { r r } 3 & x \\2 & - 1\end{array} \right| = 5$

Find the value of the determinant. - $\left| \begin{array} { l l l } 2 & 3 & 4 \\1 & 1 & 2 \\3 & 3 & 5\end{array} \right|$

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { l l l } 1 & 3 & 2 \\1 & 3 & 3 \\2 & 7 & 8\end{array} \right]$

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -At the time when the populations of the two metropolitan areas were equal, what was the approximate population of each area?

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -Use the terms increasing, decreasing, and/or constant to describe the trends for the population of the City X metropolitan area.

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. - x+by= ax+y=

Graph the solution set of the system of inequalities. - -\leq1 +\leq36 $Graph the solution set of the system of inequalities. - \begin{array} { l } \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } \leq 1 \\ x ^ { 2 } + y ^ { 2 } \leq 36 \end{array}$

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of the concepts involved in graphing inequalities. -

Provide an appropriate response -If the graphs of a system of two equations are both parabolas, what are the possible numbers of solutions (with real coordinates) of this system?

Review of Basic Concepts

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

Filters

Exam 10: Systems and Matrices

Provide an appropriate response. -Fill in the blank to complete the statement. In order for a matrix to have a determinant, it must be matrix.

Find the matrix product when possible. - [0−343][−20−11]\left[ \begin{array} { r r } 0 & - 3 \\4 & 3\end{array} \right] \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 1\end{array} \right][04​−33​][−2−1​01​]

Give all solutions of the nonlinear system of equations, including those with nonreal complex components. - +=85 x+y=-13

Graph the solution set of the system of inequalities. - 2x+3y\geq6 x-y\geq3 y\leq2

Find the partial fraction decomposition for the rational expression. - 12x2−23x+5(x−3)(x2+2)\frac { 12 x ^ { 2 } - 23 x + 5 } { ( x - 3 ) \left( x ^ { 2 } + 2 \right) }(x−3)(x2+2)12x2−23x+5​

Graph the solution set of the system of inequalities. - +\leq1 +\leq1

Graph the solution set of the system of inequalities. - 2x+y\leq4 y-1\leq0

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

Solve the problem. -A contractor mixes concrete from bags of pre-mix for small jobs. How many bags with 6%6 \%6% cement should he mix with 5 bags of 18%18 \%18% cement to produce a mix containing 12%12 \%12% cement?

Solve the equation for x. - ∣3x2−1∣=5\left| \begin{array} { r r } 3 & x \\2 & - 1\end{array} \right| = 5​32​x−1​​=5

Find the value of the determinant. - ∣234112335∣\left| \begin{array} { l l l } 2 & 3 & 4 \\1 & 1 & 2 \\3 & 3 & 5\end{array} \right|​213​313​425​​

Which method should be used to solve the system? Explain your answer, including a description of the first step. - (x−4)2+(y+3)2≤4( x - 4 ) ^ { 2 } + ( y + 3 ) ^ { 2 } \leq 4(x−4)2+(y+3)2≤4

Find the inverse, if it exists, for the matrix. - [132133278]\left[ \begin{array} { l l l } 1 & 3 & 2 \\1 & 3 & 3 \\2 & 7 & 8\end{array} \right]​112​337​238​​

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -At the time when the populations of the two metropolitan areas were equal, what was the approximate population of each area?

The following graph shows the populations of the metropolitan areas of City X and City Y over six years. -Use the terms increasing, decreasing, and/or constant to describe the trends for the population of the City X metropolitan area.

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants. - x+by= ax+y=

Graph the solution set of the system of inequalities. - -\leq1 +\leq36

Determine the inequality which matches the calculator graph. Do not use your calculator. Instead, use your knowledge of the concepts involved in graphing inequalities. -

Provide an appropriate response -If the graphs of a system of two equations are both parabolas, what are the possible numbers of solutions (with real coordinates) of this system?

Review of Basic Concepts

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

Filters

Find the matrix product when possible. - $\left[ \begin{array} { r r } 0 & - 3 \\4 & 3\end{array} \right] \left[ \begin{array} { l l } - 2 & 0 \\- 1 & 1\end{array} \right]$

Graph the solution set of the system of inequalities. - 2x+3y\geq6 x-y\geq3 y\leq2 $Graph the solution set of the system of inequalities. - \begin{array}{r} 2 x+3 y \geq 6 \\ x-y \geq 3 \\ y \leq 2 \end{array}$

Find the partial fraction decomposition for the rational expression. - $\frac { 12 x ^ { 2 } - 23 x + 5 } { ( x - 3 ) \left( x ^ { 2 } + 2 \right) }$

Graph the solution set of the system of inequalities. - +\leq1 +\leq1 $Graph the solution set of the system of inequalities. - \begin{array} { l } \frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } \leq 1 \\ \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } \leq 1 \end{array}$

Graph the solution set of the system of inequalities. - 2x+y\leq4 y-1\leq0 $Graph the solution set of the system of inequalities. - \begin{array} { c } 2 x + y \leq 4 \\ y - 1 \leq 0 \end{array}$

Solve the problem. -A contractor mixes concrete from bags of pre-mix for small jobs. How many bags with $6 \%$ cement should he mix with 5 bags of $18 \%$ cement to produce a mix containing $12 \%$ cement?

Solve the equation for x. - $\left| \begin{array} { r r } 3 & x \\2 & - 1\end{array} \right| = 5$

Find the value of the determinant. - $\left| \begin{array} { l l l } 2 & 3 & 4 \\1 & 1 & 2 \\3 & 3 & 5\end{array} \right|$

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { l l l } 1 & 3 & 2 \\1 & 3 & 3 \\2 & 7 & 8\end{array} \right]$

Graph the solution set of the system of inequalities. - -\leq1 +\leq36 $Graph the solution set of the system of inequalities. - \begin{array} { l } \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } \leq 1 \\ x ^ { 2 } + y ^ { 2 } \leq 36 \end{array}$