Question 1

Find the partial fraction decomposition for the rational expression.
$\frac { 3 x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } + 12 x + 13 } { ( x + 2 ) ( x - 1 ) ^ { 4 } }$

Accepted Answer

A)  $\frac { 3 } { x + 2 } + \frac { 1 } { x - 1 } + \frac { 6 } { ( x - 1 ) ^ { 2 } } + \frac { 2 } { ( x - 1 ) ^ { 3 } } + \frac { 1 } { ( x - 1 ) ^ { 4 } }$ 
B)  $\frac { - 1 } { x + 2 } + \frac { 5 } { x - 1 } + \frac { 2 } { ( x - 1 ) ^ { 2 } } + \frac { 17 } { ( x - 1 ) ^ { 3 } } + \frac { 12 } { ( x - 1 ) ^ { 4 } }$ 
C)  $\frac { 5 } { x + 2 } + \frac { 7 } { x - 1 } + \frac { - 2 } { ( x - 1 ) ^ { 2 } } + \frac { 4 } { ( x - 1 ) ^ { 3 } } + \frac { 3 } { ( x - 1 ) ^ { 4 } }$ 
D)  $\frac { 1 } { x + 2 } + \frac { 2 } { x - 1 } + \frac { 3 } { ( x - 1 ) ^ { 2 } } + \frac { 5 } { ( x - 1 ) ^ { 3 } } + \frac { 10 } { ( x - 1 ) ^ { 4 } }$ 
A)  $\frac { 3 } { x + 2 } + \frac { 1 } { x - 1 } + \frac { 6 } { ( x - 1 ) ^ { 2 } } + \frac { 2 } { ( x - 1 ) ^ { 3 } } + \frac { 1 } { ( x - 1 ) ^ { 4 } }$ 
B)  $\frac { - 1 } { x + 2 } + \frac { 5 } { x - 1 } + \frac { 2 } { ( x - 1 ) ^ { 2 } } + \frac { 17 } { ( x - 1 ) ^ { 3 } } + \frac { 12 } { ( x - 1 ) ^ { 4 } }$ 
C)  $\frac { 5 } { x + 2 } + \frac { 7 } { x - 1 } + \frac { - 2 } { ( x - 1 ) ^ { 2 } } + \frac { 4 } { ( x - 1 ) ^ { 3 } } + \frac { 3 } { ( x - 1 ) ^ { 4 } }$ 
D)  $\frac { 1 } { x + 2 } + \frac { 2 } { x - 1 } + \frac { 3 } { ( x - 1 ) ^ { 2 } } + \frac { 5 } { ( x - 1 ) ^ { 3 } } + \frac { 10 } { ( x - 1 ) ^ { 4 } }$

Question 2

Find the value of the determinant.
-$\left| \begin{array} { l l l } 3 & 7 & 3 \ 5 & 5 & 6 \ 9 & 9 & 3 \end{array} \right|$

Accepted Answer

A)  $- 156$ 
B)  $- 600$ 
C)  960 
D)  156 
A)  $- 156$ 
B)  $- 600$ 
C)  960 
D)  156

Question 3

Find the equation of the line  y=ax + b  that passes through the points  (-1,4)  and  (-6,-6) .

Accepted Answer

A)  $y = \frac { 2 } { 5 } x + \frac { 18 } { 5 }$ 
B)   y=2x +6 
C)   y=6x +2 
D)  $y = \frac { 2 } { 5 } x + \frac { 1 } { 2 }$ 
A)  $y = \frac { 2 } { 5 } x + \frac { 18 } { 5 }$ 
B)   y=2x +6 
C)   y=6x +2 
D)  $y = \frac { 2 } { 5 } x + \frac { 1 } { 2 }$

Question 4

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary.
-$$\begin{array} { l } 
( \log 2 ) x + ( \ln 3 ) y + ( \ln 5 ) z = 4 \
( \ln 3 ) x + ( \log 8 ) y + ( \ln 10 ) z = 11 \
( \log 23 ) x + ( \ln 7 ) y + ( \ln 4 ) z = 15
\end{array}$$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ ( 9.25474,1.35236 , - 0.16880 ) \}$ 
C)  $\{ ( 0.30103,9.0309,1.38629 ) \}$ 
D)  $\{ ( 4.12321,3.91519,1.27440 ) \}$ 
A)  $\varnothing$ 
B)  $\{ ( 9.25474,1.35236 , - 0.16880 ) \}$ 
C)  $\{ ( 0.30103,9.0309,1.38629 ) \}$ 
D)  $\{ ( 4.12321,3.91519,1.27440 ) \}$

Question 5

Find the value of the determinant.
$\left| \begin{array} { c c } 
- 5 & - 5 \
b & a
\end{array} \right|$

Accepted Answer

A)   5a - 5b 
B)   25 - ab 
C)  ba - 25 
D)   -5a+5b 
A)   5a - 5b 
B)   25 - ab 
C)  ba - 25 
D)   -5a+5b

Question 6

Two different gasohol mixtures are available. One contains 5% alcohol and the other 12% alcohol. How much of each should be mixed to obtain 1500 gal of gasohol containing 10% alcohol?

Accepted Answer

A)  429 gal 5%; 1071 gal 12% 
B)  625 gal 5%; 875 gal 12% 
C)  75 gal 5%; 180 gal 12% 
D)  1071 gal 5%; 1071 gal 12% 
A)  429 gal 5%; 1071 gal 12% 
B)  625 gal 5%; 875 gal 12% 
C)  75 gal 5%; 180 gal 12% 
D)  1071 gal 5%; 1071 gal 12%

Question 7

Find the partial fraction decomposition for the rational expression.
-$$\frac { 13 x - 48 } { x ^ { 2 } - 7 x + 12 }$$

Accepted Answer

A)  $\frac { 4 } { x - 3 } + \frac { 9 } { x - 4 }$ 
B)  $\frac { 9 } { x - 3 } + \frac { 4 } { x - 4 }$ 
C)  $\frac { 1 } { x - 3 } + \frac { 1 } { x - 4 }$ 
D)  $\frac { 9 } { x + 3 } + \frac { 4 } { x + 4 }$ 
A)  $\frac { 4 } { x - 3 } + \frac { 9 } { x - 4 }$ 
B)  $\frac { 9 } { x - 3 } + \frac { 4 } { x - 4 }$ 
C)  $\frac { 1 } { x - 3 } + \frac { 1 } { x - 4 }$ 
D)  $\frac { 9 } { x + 3 } + \frac { 4 } { x + 4 }$

Question 8

Solve the system for x and y using Cramer's rule. Assume a and b are nonzero constants.
$\begin{array} { l } 
a x + y = \frac { 1 } { a } \
b x + y = \frac { 1 } { b }
\end{array}$

Accepted Answer

A)  $\left\{ \left\{ - \frac { 1 } { a b } , \frac { a + b } { a b } \right) \right\}$ 
B)  $\left\{ \left( - \frac { 1 } { a b } , \frac { b - a } { a b } \right] \right\}$ 
C)  $\left\{ \left( \frac { 1 } { a b } , \frac { a - b } { a b } \right) \right\}$ 
D)  $\left\{ \left( \frac { 1 } { a b } , a + b \right) \right\}$ 
A)  $\left\{ \left\{ - \frac { 1 } { a b } , \frac { a + b } { a b } \right) \right\}$ 
B)  $\left\{ \left( - \frac { 1 } { a b } , \frac { b - a } { a b } \right] \right\}$ 
C)  $\left\{ \left( \frac { 1 } { a b } , \frac { a - b } { a b } \right) \right\}$ 
D)  $\left\{ \left( \frac { 1 } { a b } , a + b \right) \right\}$

Question 9

Solve the system by elimination.
$\begin{array} { l } 
\frac { 6 x } { 5 } + \frac { y } { 5 } = - 2 \
\frac { x } { 5 } + \frac { y } { 5 } = 0
\end{array}$

Accepted Answer

A)  $\{ ( - 15,15 ) \}$ 
B)  $\langle ( - 2,2 ) \rangle$ 
C)  $\left\{ \left( - \frac { 10 } { 7 } , \frac { 10 } { 7 } \right) \right\}$ 
D)  $\{ ( - 1,1 ) \}$ 
A)  $\{ ( - 15,15 ) \}$ 
B)  $\langle ( - 2,2 ) \rangle$ 
C)  $\left\{ \left( - \frac { 10 } { 7 } , \frac { 10 } { 7 } \right) \right\}$ 
D)  $\{ ( - 1,1 ) \}$

Question 10

Decide whether or not the matrices are inverses of each other.
-$$\left[ \begin{array} { l l } 
10 & 1 \
- 1 & 0
\end{array} \right] \text { and } \left[ \begin{array} { r r } 
0 & 1 \
- 1 & 10
\end{array} \right]$$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 11

Does a square matrix with a row of all zeros have an inverse?

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 12

Solve the system.
$\begin{array} { c } 
x - y + 5 z = 27 \
2 x + z = 5 \
x + 4 y + z = - 3
\end{array}$

Accepted Answer

A)  $\{ ( 0 , - 2,5 ) \}$ 
B)  $\{ ( 5 , - 2,0 ) \}$ 
C)  $\{ ( 5,0 , - 2 ) \}$ 
D)  $\varnothing$ 
A)  $\{ ( 0 , - 2,5 ) \}$ 
B)  $\{ ( 5 , - 2,0 ) \}$ 
C)  $\{ ( 5,0 , - 2 ) \}$ 
D)  $\varnothing$

Question 13

Find the inverse, if it exists, for the matrix.
-$\left[ \begin{array} { l l } 3 & 0 \ 3 & 4 \end{array} \right]$

Accepted Answer

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

Question 14

Write the augmented matrix for the system. Do not solve the system.
$\begin{array} { l } 
2 x + 3 y = 24 \
4 x + 6 y = 48
\end{array}$

Accepted Answer

A)  $\left[ \begin{array} { l l | l } 
24 & 3 & 2 \
48 & 4 & 6
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l | l } 
2 & 4 & 24 \
3 & 6 & 48
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l | l } 
2 & 3 & 48 \
6 & 4 & 24
\end{array} \right]$ 
D)  $\left[ \begin{array} { l l | l } 
2 & 3 & 24 \
4 & 6 & 48
\end{array} \right]$ 
A)  $\left[ \begin{array} { l l | l } 
24 & 3 & 2 \
48 & 4 & 6
\end{array} \right]$ 
B)  $\left[ \begin{array} { l l | l } 
2 & 4 & 24 \
3 & 6 & 48
\end{array} \right]$ 
C)  $\left[ \begin{array} { l l | l } 
2 & 3 & 48 \
6 & 4 & 24
\end{array} \right]$ 
D)  $\left[ \begin{array} { l l | l } 
2 & 3 & 24 \
4 & 6 & 48
\end{array} \right]$

Question 15

Choose the one alternative that best completes the statement or answers the question.
Graph the inequality.
$x \geq-2$

Accepted Answer

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

Question 16

A television manufacturing company has two different factories. The profit function for Factory I when x televisions are manufactured and sold is as follows: $P ( x ) = - 0.07 x ^ { 2 } + 25 x - 250$ The profit function for Factory II when x televisions are manufactured and sold is as follows: $P ( x ) = - 0.075 x ^ { 2 } + 35 x - 1500$ For what number of televisions is the profit the same at both factories, and what is the profit?

Accepted Answer

A)  134 televisions; $1843 
B)  233 televisions; $2583.33 
C)  125 televisions; $1781.25 
D)  0 televisions; $0 
A)  134 televisions; $1843 
B)  233 televisions; $2583.33 
C)  125 televisions; $1781.25 
D)  0 televisions; $0

Question 17

Solve the system.
$\begin{array} { l } 
\frac { 3 } { x } + \frac { 4 } { y } = \frac { 35 } { 8 } \
\frac { 6 } { x } - \frac { 3 } { y } = - \frac { 9 } { 4 }
\end{array}$

Accepted Answer

A)  $\left\{ \left( \frac { 8 } { 15 } , 1 \right) \right\}$ 
B)  $\{ ( - 8 , - 1 ) \}$ 
C)  $\{ ( 8,1 ) \}$ 
D)  $\{ ( 1,8 ) \}$ 
A)  $\left\{ \left( \frac { 8 } { 15 } , 1 \right) \right\}$ 
B)  $\{ ( - 8 , - 1 ) \}$ 
C)  $\{ ( 8,1 ) \}$ 
D)  $\{ ( 1,8 ) \}$

Question 18

Write the word or phrase that best completes each statement or answers the question.
Which method should be used to solve the system? Explain your answer, including a description of the first step.
$$\begin{array} { r } 
- 9 x ^ { 2 } + 8 y ^ { 2 } = 36 \
x - 6 y = - 7
\end{array}$$

Accepted Answer

Substitution should be used, b

Question 19

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable.
$\begin{array} { l } 
x - 8 y + z = 2 \
3 x - y + 2 z = 6
\end{array}$

Accepted Answer

A)  $\left\{ \left( \frac { - 46 - 15 z } { 25 } , \frac { 0 + z } { 25 } , z \right) \right\}$ 
B)  $\{ ( 2,0,1 ) \}$ 
C)  $\left\{ \left( \frac { 46 - 15 z } { 23 } , \frac { 0 + z } { 23 } , z \right) \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ \left( \frac { - 46 - 15 z } { 25 } , \frac { 0 + z } { 25 } , z \right) \right\}$ 
B)  $\{ ( 2,0,1 ) \}$ 
C)  $\left\{ \left( \frac { 46 - 15 z } { 23 } , \frac { 0 + z } { 23 } , z \right) \right\}$ 
D)  $\varnothing$

Question 20

Find the values of the variables for which the statement is true, if possible.
-$\left[ \begin{array} { l } \mathrm { a } \ 7 \ 5 \end{array} \right] = \left[ \begin{array} { l } 5 \ \mathrm { x } \ 5 \end{array} \right]$

Accepted Answer

A)  $\mathrm { a } = 7 ; \mathrm { x } = 5$ 
B)  $a = 7 ; x = 7$ 
C)  $a = 5 ; x = 7$ 
D)  $a = 5 ; x = 5$ 
A)  $\mathrm { a } = 7 ; \mathrm { x } = 5$ 
B)  $a = 7 ; x = 7$ 
C)  $a = 5 ; x = 7$ 
D)  $a = 5 ; x = 5$

Find the partial fraction decomposition for the rational expression. $\frac { 3 x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } + 12 x + 13 } { ( x + 2 ) ( x - 1 ) ^ { 4 } }$

Find the value of the determinant. - $\left| \begin{array} { l l l } 3 & 7 & 3 \\ 5 & 5 & 6 \\ 9 & 9 & 3 \end{array} \right|$

Find the equation of the line y=ax + b that passes through the points (-1,4) and (-6,-6) .

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary. - (2)x+(3)y+(5)z=4 (3)x+(8)y+(10)z=11 (23)x+(7)y+(4)z=15

Find the value of the determinant. $\left| \begin{array} { c c } - 5 & - 5 \\b & a\end{array} \right|$

Two different gasohol mixtures are available. One contains 5% alcohol and the other 12% alcohol. How much of each should be mixed to obtain 1500 gal of gasohol containing 10% alcohol?

Find the partial fraction decomposition for the rational expression. - $\frac { 13 x - 48 } { x ^ { 2 } - 7 x + 12 }$

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

Solve the system by elimination. +=-2 +=0

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { l l } 10 & 1 \\- 1 & 0\end{array} \right] \text { and } \left[ \begin{array} { r r } 0 & 1 \\- 1 & 10\end{array} \right]$

Does a square matrix with a row of all zeros have an inverse?

Solve the system. x-y+5z=27 2x+z=5 x+4y+z=-3

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { l l } 3 & 0 \\ 3 & 4 \end{array} \right]$

Write the augmented matrix for the system. Do not solve the system. 2x+3y=24 4x+6y=48

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. $x \geq-2$ $Choose the one alternative that best completes the statement or answers the question. Graph the inequality. x \geq-2$

Solve the system. += -=-

Write the word or phrase that best completes each statement or answers the question. Which method should be used to solve the system? Explain your answer, including a description of the first step. -9+8=36 x-6y=-7

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable. x-8y+z=2 3x-y+2z=6

Find the values of the variables for which the statement is true, if possible. - $\left[ \begin{array} { l } \mathrm { a } \\ 7 \\ 5 \end{array} \right] = \left[ \begin{array} { l } 5 \\ \mathrm { x } \\ 5 \end{array} \right]$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomials and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Arithmetic Sequence: Common Difference and First n Terms

Filters

Exam 6: Systems and Matrices

Find the partial fraction decomposition for the rational expression. 3x4−3x3+5x2+12x+13(x+2)(x−1)4\frac { 3 x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } + 12 x + 13 } { ( x + 2 ) ( x - 1 ) ^ { 4 } }(x+2)(x−1)43x4−3x3+5x2+12x+13​

Find the value of the determinant. - ∣373556993∣\left| \begin{array} { l l l } 3 & 7 & 3 \\ 5 & 5 & 6 \\ 9 & 9 & 3 \end{array} \right|​359​759​363​​

Find the equation of the line y=ax + b that passes through the points (-1,4) and (-6,-6) .

Use a graphing calculator and the method of matrix inverses to Give five decimal places, if necessary. - (2)x+(3)y+(5)z=4 (3)x+(8)y+(10)z=11 (23)x+(7)y+(4)z=15

Find the value of the determinant. ∣−5−5ba∣\left| \begin{array} { c c } - 5 & - 5 \\b & a\end{array} \right|​−5b​−5a​​

Two different gasohol mixtures are available. One contains 5% alcohol and the other 12% alcohol. How much of each should be mixed to obtain 1500 gal of gasohol containing 10% alcohol?

Find the partial fraction decomposition for the rational expression. - 13x−48x2−7x+12\frac { 13 x - 48 } { x ^ { 2 } - 7 x + 12 }x2−7x+1213x−48​

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

Solve the system by elimination. +=-2 +=0

Decide whether or not the matrices are inverses of each other. - [101−10] and [01−110]\left[ \begin{array} { l l } 10 & 1 \\- 1 & 0\end{array} \right] \text { and } \left[ \begin{array} { r r } 0 & 1 \\- 1 & 10\end{array} \right][10−1​10​] and [0−1​110​]

Does a square matrix with a row of all zeros have an inverse?

Solve the system. x-y+5z=27 2x+z=5 x+4y+z=-3

Find the inverse, if it exists, for the matrix. - [3034]\left[ \begin{array} { l l } 3 & 0 \\ 3 & 4 \end{array} \right][33​04​]

Write the augmented matrix for the system. Do not solve the system. 2x+3y=24 4x+6y=48

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. x≥−2x \geq-2x≥−2

Solve the system. += -=-

Write the word or phrase that best completes each statement or answers the question. Which method should be used to solve the system? Explain your answer, including a description of the first step. -9+8=36 x-6y=-7

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, let the last variable be the arbitrary variable. x-8y+z=2 3x-y+2z=6

Find the values of the variables for which the statement is true, if possible. - [a75]=[5x5]\left[ \begin{array} { l } \mathrm { a } \\ 7 \\ 5 \end{array} \right] = \left[ \begin{array} { l } 5 \\ \mathrm { x } \\ 5 \end{array} \right]​a75​​=​5x5​​

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomials and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Arithmetic Sequence: Common Difference and First n Terms

Filters

Find the partial fraction decomposition for the rational expression. $\frac { 3 x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } + 12 x + 13 } { ( x + 2 ) ( x - 1 ) ^ { 4 } }$

Find the value of the determinant. - $\left| \begin{array} { l l l } 3 & 7 & 3 \\ 5 & 5 & 6 \\ 9 & 9 & 3 \end{array} \right|$

Find the value of the determinant. $\left| \begin{array} { c c } - 5 & - 5 \\b & a\end{array} \right|$

Find the partial fraction decomposition for the rational expression. - $\frac { 13 x - 48 } { x ^ { 2 } - 7 x + 12 }$

Decide whether or not the matrices are inverses of each other. - $\left[ \begin{array} { l l } 10 & 1 \\- 1 & 0\end{array} \right] \text { and } \left[ \begin{array} { r r } 0 & 1 \\- 1 & 10\end{array} \right]$

Find the inverse, if it exists, for the matrix. - $\left[ \begin{array} { l l } 3 & 0 \\ 3 & 4 \end{array} \right]$

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. $x \geq-2$ $Choose the one alternative that best completes the statement or answers the question. Graph the inequality. x \geq-2$

Find the values of the variables for which the statement is true, if possible. - $\left[ \begin{array} { l } \mathrm { a } \\ 7 \\ 5 \end{array} \right] = \left[ \begin{array} { l } 5 \\ \mathrm { x } \\ 5 \end{array} \right]$