Question 1

Find the partial fraction decomposition for the rational expression.
$\frac { x ^ { 2 } } { x ^ { 2 } + 6 x + 9 }$

Accepted Answer

A)  $1 + \frac { - 6 } { x + 3 } + \frac { 9 } { ( x + 3 ) ^ { 2 } }$ 
B)  $1 + \frac { 6 } { x + 3 } + \frac { - 9 } { ( x + 31 ) ^ { 2 } }$ 
C)  $\frac { 6 } { x + 3 } + \frac { - 6 } { ( x + 3 ) ^ { 2 } }$ 
D)  $\frac { - 6 } { x + 3 } + \frac { 9 } { ( x + 3 ) ^ { 2 } }$ 
A)  $1 + \frac { - 6 } { x + 3 } + \frac { 9 } { ( x + 3 ) ^ { 2 } }$ 
B)  $1 + \frac { 6 } { x + 3 } + \frac { - 9 } { ( x + 31 ) ^ { 2 } }$ 
C)  $\frac { 6 } { x + 3 } + \frac { - 6 } { ( x + 3 ) ^ { 2 } }$ 
D)  $\frac { - 6 } { x + 3 } + \frac { 9 } { ( x + 3 ) ^ { 2 } }$

Question 2

Find the values of the variables for which the statement is true, if possible.
-$\left[ \begin{array} { l l l } 4 & \mathrm {~b} & 2 \end{array} \right] + \left[ \begin{array} { l l l } \mathrm { a } & 9 & \mathrm { c } \end{array} \right] = \left[ \begin{array} { l l l } 2 & 3 & - 7 \end{array} \right]$

Accepted Answer

A)  $a = 6 ; b = 12 ; c = - 5$ 
B)  $a = \frac { 1 } { 2 } ; b = \frac { 1 } { 3 } ; c = - \frac { 7 } { 2 }$ 
C)  $a = 2 ; b = 6 ; c = 9$ 
D)  $a = - 2 ; b = - 6 ; c = - 9$ 
A)  $a = 6 ; b = 12 ; c = - 5$ 
B)  $a = \frac { 1 } { 2 } ; b = \frac { 1 } { 3 } ; c = - \frac { 7 } { 2 }$ 
C)  $a = 2 ; b = 6 ; c = 9$ 
D)  $a = - 2 ; b = - 6 ; c = - 9$

Question 3

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

Accepted Answer

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

Question 4

Find the indicated matrix.
-Let $A = \left[ \begin{array} { l l } - 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]$. Find $2 A + 3 B$.

Accepted Answer

A)  $\left[ \begin{array} { l l } - 5 & 4 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 0 & 2 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } - 3 & 4 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l } - 6 & 4 \end{array} \right]$ 
A)  $\left[ \begin{array} { l l } - 5 & 4 \end{array} \right]$ 
B)  $\left[ \begin{array} { l l } 0 & 2 \end{array} \right]$ 
C)  $\left[ \begin{array} { l l } - 3 & 4 \end{array} \right]$ 
D)  $\left[ \begin{array} { l l } - 6 & 4 \end{array} \right]$

Question 5

Graph the solution set of the system of inequalities.
-$\begin{array}{c}
3 x-2 y \geq-6 \
x-1<0
\end{array}$

Accepted Answer

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

Question 6

Find the partial fraction decomposition for the rational expression.
$\frac { 2 x ^ { 3 } - 5 x ^ { 2 } - 1 x + 7 } { ( x - 1 ) ^ { 3 } ( x + 2 ) }$

Accepted Answer

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

Question 7

Find the dimension of the matrix.
-$$\left[ \begin{array} { r r r } 
- 7 & 5 & 5 \
- 5 & 1 & - 8
\end{array} \right]$$

Accepted Answer

A)  3 x 2 
B)  $- 7,5,5 , - 5,1 , - 8$ 
C)  6 
D)  $2 \times 3$ 
A)  3 x 2 
B)  $- 7,5,5 , - 5,1 , - 8$ 
C)  6 
D)  $2 \times 3$

Question 8

Solve the system by substitution.
-$$\begin{array} { r } 
x + 2 y = - 16 \
2 x + 2 y = - 22
\end{array}$$

Accepted Answer

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

Question 9

Solve the system using a graphing calculator capable of performing row operations. Give solutions with values correct to
the nearest thousandth.
-$$\begin{array} { l } 
0.6 x + 4.9 y - \sqrt { 3 } z = 7 \
\sqrt { 5 } x - 18 y + 10 z = - 1 \
3 x + \sqrt { 2 } y - 4.9 z = \frac { 1 } { 4 }
\end{array}$$

Accepted Answer

A)  $\{ ( 2.084 , - 1.741,3.529 ) \}$ 
B)  $\{ ( - 7.733 , - 5.025,3.491 ) \}$ 
C)  $\{ ( 3.431,1.930,2.606 ) \}$ 
D)  $\{ ( 8.110,0.749,2.619 ) \}$ 
A)  $\{ ( 2.084 , - 1.741,3.529 ) \}$ 
B)  $\{ ( - 7.733 , - 5.025,3.491 ) \}$ 
C)  $\{ ( 3.431,1.930,2.606 ) \}$ 
D)  $\{ ( 8.110,0.749,2.619 ) \}$

Question 10

Solve the equation for x.
-$$\left| \begin{array} { l l } 
3 & x \
x & 4
\end{array} \right| = - 4$$

Accepted Answer

A)  $\{ 16 \}$ 
B)  $\{ - 4,4 \}$ 
C)  $\left\{ \frac { 4 } { 3 } \right\}$ 
D)  $\varnothing$ 
A)  $\{ 16 \}$ 
B)  $\{ - 4,4 \}$ 
C)  $\left\{ \frac { 4 } { 3 } \right\}$ 
D)  $\varnothing$

Question 11

A bakery sells three types of cakes, each requiring the amount of ingredients shown. $$\text { Cake I Cake II Cake III }$$ To fill its orders for these cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 55 eggs. How many cakes of each type were made?

Accepted Answer

A)  15 Cake I; 8 Cake II; 7 Cake III 
B)  13 Cake I; 8 Cake II; 7 Cake III 
C)  7 Cake I; 8 Cake II; 13 Cake III 
D)  37 Cake I; 8 Cake II; 12 Cake III 
A)  15 Cake I; 8 Cake II; 7 Cake III 
B)  13 Cake I; 8 Cake II; 7 Cake III 
C)  7 Cake I; 8 Cake II; 13 Cake III 
D)  37 Cake I; 8 Cake II; 12 Cake III

Question 12

$$\text { Let } A = \left[ \begin{array} { r r } 
3 & - 2 \
2 & 1
\end{array} \right] \text { and } B = \left[ \begin{array} { r r } 
- 1 & 2 \
6 & - 1
\end{array} \right]$$ Does the matrix A + B have an inverse?

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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.
-$ 4 x-5 y-7 z=-1 $
$ -7 x+2 y-8 z=1 $
$ 20 x-25 y-35 z=-5 $

Accepted Answer

A)  $ \{(-5 z+2,4 z-25, z)\} $ 
B)  $ \{(-1,1,-5)\} $ 
C)  $ \varnothing $ 
D)  $ \{(5,5,-1)\} $ 
A)  $ \{(-5 z+2,4 z-25, z)\} $ 
B)  $ \{(-1,1,-5)\} $ 
C)  $ \varnothing $ 
D)  $ \{(5,5,-1)\} $

Question 15

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { l } 
9 x + 6 y = 12 \
6 x - 2 y = - 4
\end{array}$$

Accepted Answer

A)  Cramer's rule does not apply since $D = 0$; $\varnothing$ 
B)  $\{ ( - 1,3 ) \}$ 
C)  $\{ ( 0,3 ) \}$ 
D)  $\{ ( 0,2 ) \}$ 
A)  Cramer's rule does not apply since $D = 0$; $\varnothing$ 
B)  $\{ ( - 1,3 ) \}$ 
C)  $\{ ( 0,3 ) \}$ 
D)  $\{ ( 0,2 ) \}$

Question 16

A nonlinear system is given, along with the graphs of both equations in the system. Determine if the points of
intersection specified on the graph are solutions of the system by substituting directly into both equations.
-$\begin{array}{l}
x^{2}+y^{2}=52 \
2 y+3 x=0
\end{array}$

Accepted Answer

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

Question 17

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

Accepted Answer

A)  1 
B)  0 
C)  84 
D)   -12 
A)  1 
B)  0 
C)  84 
D)   -12

Question 18

Suppose that you are solving a system of 4 linear equations in 4 variables by the Gauss-Jordan method. If you use the transformation $10 R _ { 2 } + R _ { 3 }$ , which row or rows of the augmented matrix, if any, will change?

Accepted Answer

A)  Rows 2 and 3 
B)  Row 3 
C)  Row 2 
D)  None 
A)  Rows 2 and 3 
B)  Row 3 
C)  Row 2 
D)  None

Question 19

Choose the one alternative that best completes the statement or answers the question.
Graph the inequality.
$x \leq - 3$

Accepted Answer

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

Question 20

The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective
function.
-objective function $= 2 x + 4 y ;$ maximum

Accepted Answer

A)  56 
B)  60 
C)  38 
D)  46 
A)  56 
B)  60 
C)  38 
D)  46

Find the partial fraction decomposition for the rational expression. $\frac { x ^ { 2 } } { x ^ { 2 } + 6 x + 9 }$

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

Find the indicated matrix. -Let $A = \left[ \begin{array} { l l } - 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]$ . Find $2 A + 3 B$ .

Graph the solution set of the system of inequalities. - 3x-2y\geq-6 x-1<0 $Graph the solution set of the system of inequalities. - \begin{array}{c} 3 x-2 y \geq-6 \\ x-1<0 \end{array}$

Find the partial fraction decomposition for the rational expression. $\frac { 2 x ^ { 3 } - 5 x ^ { 2 } - 1 x + 7 } { ( x - 1 ) ^ { 3 } ( x + 2 ) }$

Find the dimension of the matrix. - $\left[ \begin{array} { r r r } - 7 & 5 & 5 \\- 5 & 1 & - 8\end{array} \right]$

Solve the system by substitution. - x+2y=-16 2x+2y=-22

Solve the system using a graphing calculator capable of performing row operations. Give solutions with values correct to the nearest thousandth. - 0.6x+4.9y-z=7 x-18y+10z=-1 3x+y-4.9z=

Solve the equation for x. - $\left| \begin{array} { l l } 3 & x \\x & 4\end{array} \right| = - 4$

A bakery sells three types of cakes, each requiring the amount of ingredients shown. $\text { Cake I Cake II Cake III }$ To fill its orders for these cakes, the bakery used 72 cups of flour, 48 cups of sugar, and 55 eggs. How many cakes of each type were made?

$\text { Let } A = \left[ \begin{array} { r r } 3 & - 2 \\2 & 1\end{array} \right] \text { and } B = \left[ \begin{array} { r r } - 1 & 2 \\6 & - 1\end{array} \right]$ Does the matrix A + B have an inverse?

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

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. - $4 x-5 y-7 z=-1$ $-7 x+2 y-8 z=1$ $20 x-25 y-35 z=-5$

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - 9x+6y=12 6x-2y=-4

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

Suppose that you are solving a system of 4 linear equations in 4 variables by the Gauss-Jordan method. If you use the transformation $10 R _ { 2 } + R _ { 3 }$ , which row or rows of the augmented matrix, if any, will change?

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. $x \leq - 3$ $Choose the one alternative that best completes the statement or answers the question. Graph the inequality. x \leq - 3$

The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective function. -objective function $= 2 x + 4 y ;$ maximum

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. x2x2+6x+9\frac { x ^ { 2 } } { x ^ { 2 } + 6 x + 9 }x2+6x+9x2​

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

Find the indicated matrix. -Let A=[−32]A = \left[ \begin{array} { l l } - 3 & 2 \end{array} \right]A=[−3​2​] and B=[10]B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]B=[1​0​] . Find 2A+3B2 A + 3 B2A+3B .

Graph the solution set of the system of inequalities. - 3x-2y\geq-6 x-1<0

Find the partial fraction decomposition for the rational expression. 2x3−5x2−1x+7(x−1)3(x+2)\frac { 2 x ^ { 3 } - 5 x ^ { 2 } - 1 x + 7 } { ( x - 1 ) ^ { 3 } ( x + 2 ) }(x−1)3(x+2)2x3−5x2−1x+7​

Find the dimension of the matrix. - [−755−51−8]\left[ \begin{array} { r r r } - 7 & 5 & 5 \\- 5 & 1 & - 8\end{array} \right][−7−5​51​5−8​]

Solve the system by substitution. - x+2y=-16 2x+2y=-22

Solve the system using a graphing calculator capable of performing row operations. Give solutions with values correct to the nearest thousandth. - 0.6x+4.9y-z=7 x-18y+10z=-1 3x+y-4.9z=

Solve the equation for x. - ∣3xx4∣=−4\left| \begin{array} { l l } 3 & x \\x & 4\end{array} \right| = - 4​3x​x4​​=−4

Let A=[3−221] and B=[−126−1]\text { Let } A = \left[ \begin{array} { r r } 3 & - 2 \\2 & 1\end{array} \right] \text { and } B = \left[ \begin{array} { r r } - 1 & 2 \\6 & - 1\end{array} \right] Let A=[32​−21​] and B=[−16​2−1​] Does the matrix A + B have an inverse?

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

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - 9x+6y=12 6x-2y=-4

A nonlinear system is given, along with the graphs of both equations in the system. Determine if the points of intersection specified on the graph are solutions of the system by substituting directly into both equations. - +=52 2y+3x=0

Evaluate the determinant. ∣124253124∣\left| \begin{array} { l l l } 1 & 2 & 4 \\2 & 5 & 3 \\1 & 2 & 4\end{array} \right|​121​252​434​​

Suppose that you are solving a system of 4 linear equations in 4 variables by the Gauss-Jordan method. If you use the transformation 10R2+R310 R _ { 2 } + R _ { 3 }10R2​+R3​ , which row or rows of the augmented matrix, if any, will change?

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. x≤−3x \leq - 3x≤−3

The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective function. -objective function =2x+4y;= 2 x + 4 y ;=2x+4y; maximum

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. $\frac { x ^ { 2 } } { x ^ { 2 } + 6 x + 9 }$

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

Find the indicated matrix. -Let $A = \left[ \begin{array} { l l } - 3 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]$ . Find $2 A + 3 B$ .

Graph the solution set of the system of inequalities. - 3x-2y\geq-6 x-1<0 $Graph the solution set of the system of inequalities. - \begin{array}{c} 3 x-2 y \geq-6 \\ x-1<0 \end{array}$

Find the partial fraction decomposition for the rational expression. $\frac { 2 x ^ { 3 } - 5 x ^ { 2 } - 1 x + 7 } { ( x - 1 ) ^ { 3 } ( x + 2 ) }$

Find the dimension of the matrix. - $\left[ \begin{array} { r r r } - 7 & 5 & 5 \\- 5 & 1 & - 8\end{array} \right]$

Solve the equation for x. - $\left| \begin{array} { l l } 3 & x \\x & 4\end{array} \right| = - 4$

$\text { Let } A = \left[ \begin{array} { r r } 3 & - 2 \\2 & 1\end{array} \right] \text { and } B = \left[ \begin{array} { r r } - 1 & 2 \\6 & - 1\end{array} \right]$ Does the matrix A + B have an inverse?

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

Suppose that you are solving a system of 4 linear equations in 4 variables by the Gauss-Jordan method. If you use the transformation $10 R _ { 2 } + R _ { 3 }$ , which row or rows of the augmented matrix, if any, will change?

Choose the one alternative that best completes the statement or answers the question. Graph the inequality. $x \leq - 3$ $Choose the one alternative that best completes the statement or answers the question. Graph the inequality. x \leq - 3$

The graph shows the region of feasible solutions. Find the maximum or minimum value, as specified, of the objective function. -objective function $= 2 x + 4 y ;$ maximum