Question 1

Solve the equation for x.
-$P ( - 3 , - 3 ) , Q ( 2,5 ) , R ( 3 , - 3 )$

Accepted Answer

A)  0 
B)  32 
C)  48 
D)  24 
A)  0 
B)  32 
C)  48 
D)  24

Question 2

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{array} { l } 
5 x - y + z = 8 \
7 x + y + z = 6 \
12 x + 2 z = 14
\end{array}$$

Accepted Answer

A)  $\{ ( x , - x + 1,6 x - 7 ) \}$ 
B)  $\{ ( x , - x - 1 , - 6 x + 7 ) \}$ 
C)  $\{ ( x , - x + 1,6 x + 7 ) \}$ 
D)  $\{ ( x , - x + 1 , - 6 x + 7 ) \}$ 
A)  $\{ ( x , - x + 1,6 x - 7 ) \}$ 
B)  $\{ ( x , - x - 1 , - 6 x + 7 ) \}$ 
C)  $\{ ( x , - x + 1,6 x + 7 ) \}$ 
D)  $\{ ( x , - x + 1 , - 6 x + 7 ) \}$

Question 3

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth.
-A rectangular plot has area $126 \mathrm { yd } ^ { 2 }$ with a perimeter of $46 \mathrm { yd }$. What is the length of the longest side?

Accepted Answer

A)  $16 \mathrm { yd }$ 
B)  $13 \mathrm { yd }$ 
C)  $14 \mathrm { yd }$ 
D)  $12 \mathrm { yd }$ 
A)  $16 \mathrm { yd }$ 
B)  $13 \mathrm { yd }$ 
C)  $14 \mathrm { yd }$ 
D)  $12 \mathrm { yd }$

Question 4

Find the cofactor of the indicated element.
-
$\text { a } 33$
$$\begin{array} { l } 
\left| \begin{array} { r r r } 
1 & 5 & 3 \
3 & 5 & 2 \
- 1 & - 1 & 4
\end{array} \right|
\end{array}$$

Accepted Answer

A)  $- 5$ 
B)  $- 10$ 
C)  5 
D)  10 
A)  $- 5$ 
B)  $- 10$ 
C)  5 
D)  10

Question 5

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$x - y > - 4$$

Accepted Answer

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

Question 6

Find the partial fraction decomposition for the rational expression.
-$$\frac { - 2 x ^ { 2 } + 7 x + 24 } { ( x + 3 ) ^ { 2 } ( 3 x + 4 ) }$$

Accepted Answer

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

Question 7

Provide an appropriate response.
-Does a square matrix with a row of all zeros have an inverse?

Accepted Answer

A)  Yes 
B)  $\mathrm { No }$ 
A)  Yes 
B)  $\mathrm { No }$

Question 8

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$x \leq 6$$

Accepted Answer

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

Question 9

Find the partial fraction decomposition for the rational expression.
-$$\frac { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 27 } { x ^ { 2 } + x - 2 }$$

Accepted Answer

A)  $2 x - 5 + \frac { - 7 / 3 } { x + 2 } + \frac { 13 / 3 } { x - 1 }$ 
B)  $2 x - 5 + \frac { - 1 } { x + 2 } + \frac { 8 } { x - 1 }$ 
C)  $2 x - 9 + \frac { - 5 / 3 } { x + 2 } + \frac { 1 / 3 } { x - 1 }$ 
D)  $2 x - 9 + \frac { - 13 } { x + 2 } + \frac { 7 } { x - 1 }$ 
A)  $2 x - 5 + \frac { - 7 / 3 } { x + 2 } + \frac { 13 / 3 } { x - 1 }$ 
B)  $2 x - 5 + \frac { - 1 } { x + 2 } + \frac { 8 } { x - 1 }$ 
C)  $2 x - 9 + \frac { - 5 / 3 } { x + 2 } + \frac { 1 / 3 } { x - 1 }$ 
D)  $2 x - 9 + \frac { - 13 } { x + 2 } + \frac { 7 } { x - 1 }$

Question 10

Use a graphing calculator to Express solutions with approximations to the nearest thousandth.
-$$\begin{array} { l } 
x - y + z = 1 \
x + y + z = 3 \
x + y - z = - 3
\end{array}$$

Accepted Answer

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

Question 11

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { c } 
x + 3 y = - 12 \
4 x + 4 y = - 16
\end{array}$$

Accepted Answer

A)  $\{ ( 0 , - 4 ) \}$ 
B)  $\{ ( 4,0 ) \}$ 
C)  $\{ ( 1 , - 5 ) \}$ 
D)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
A)  $\{ ( 0 , - 4 ) \}$ 
B)  $\{ ( 4,0 ) \}$ 
C)  $\{ ( 1 , - 5 ) \}$ 
D)  Cramer's rule does not apply since $D = 0 ; \varnothing$

Question 12

Which method should be used to solve the system? Explain your answer, including a description of the first step.
-$$\begin{aligned}
2 x ^ { 2 } - 4 y ^ { 2 } & = 4 \
9 x + 9 y & = - 1
\end{aligned}$$

Accepted Answer

Substitution should be used, b

Question 13

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

Accepted Answer

A)  $\left\{ \left( a ^ { 2 } + a b + b ^ { 2 } , - b - a \right) \right\}$ 
B)  $\left\{ \left( a ^ { 2 } + 2 a b + b ^ { 2 } , b - a \right) \right\}$ 
C)  $\left\{ \left( a ^ { 2 } + 2 a b + b ^ { 2 } , - b - a \right) \right\}$ 
D)  $\left\{ \left( a ^ { 2 } + a b + b ^ { 2 } , a + b \right) \right\}$ 
A)  $\left\{ \left( a ^ { 2 } + a b + b ^ { 2 } , - b - a \right) \right\}$ 
B)  $\left\{ \left( a ^ { 2 } + 2 a b + b ^ { 2 } , b - a \right) \right\}$ 
C)  $\left\{ \left( a ^ { 2 } + 2 a b + b ^ { 2 } , - b - a \right) \right\}$ 
D)  $\left\{ \left( a ^ { 2 } + a b + b ^ { 2 } , a + b \right) \right\}$

Question 14

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

Accepted Answer

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

Question 15

If the system has infinitely many solutions, write the solution set with x arbitrary.
-$$\begin{aligned}
4 x + y + z & = 6 \
x - y + 2 z & = 12 \
4 x + y + z & = 21
\end{aligned}$$

Accepted Answer

A)  $\{ ( 4,5 , - 3 ) \}$ 
B)  $\{ ( 4 , - 3,5 ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( 5 , - 3,4 ) \}$ 
A)  $\{ ( 4,5 , - 3 ) \}$ 
B)  $\{ ( 4 , - 3,5 ) \}$ 
C)  $\varnothing$ 
D)  $\{ ( 5 , - 3,4 ) \}$

Question 16

Provide an appropriate response.
-Indicate whether the following statement is If two of the rows of a 3 × 3 matrix are 
identical, but its other row is different, then the determinant of the matrix equals 0.

Accepted Answer

A) True 
 B)False

Question 17

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

Question 18

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

Accepted Answer

A)  0 
B)  $- 12$ 
C)  1 
D)  72 
A)  0 
B)  $- 12$ 
C)  1 
D)  72

Question 19

The sizes of two matrices are given. Find the size of the product AB and the size of the product BA, if the given product
can be calculated.
-A is $2 \times 1$; B is $1 \times 1$.

Accepted Answer

A)  $\mathrm { AB }$ cannot be calculated; $\mathrm { BA }$ is $1 \times 2$. 
B)  $\mathrm { AB }$ is $1 \times 2$; $\mathrm { BA }$ is $1 \times 1$. 
C)  $\mathrm { AB }$ is $2 \times 1$; BA cannot be calculated. 
D)  $\mathrm { AB }$ is $2 \times 2$; $\mathrm { BA }$ is $1 \times 1$. 
A)  $\mathrm { AB }$ cannot be calculated; $\mathrm { BA }$ is $1 \times 2$. 
B)  $\mathrm { AB }$ is $1 \times 2$; $\mathrm { BA }$ is $1 \times 1$. 
C)  $\mathrm { AB }$ is $2 \times 1$; BA cannot be calculated. 
D)  $\mathrm { AB }$ is $2 \times 2$; $\mathrm { BA }$ is $1 \times 1$.

Question 20

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set.
-$$\begin{array} { r } 
x + y + z = 9 \
2 x - 3 y + 4 z = 7
\end{array}$$

Accepted Answer

A)  Cramer's rule does not apply since $D = 0 ; \left\{ \left\{ \frac { z + 34 } { 5 } , \frac { 2 z + 11 } { 5 } , z \right) \right\}$ 
B)  Cramer's rule does not apply since $\mathrm { D } = 0 ; \left\{ \left[ \frac { - 7 \mathrm { z } + 34 } { 5 } , \frac { 2 \mathrm { z } - 11 } { 5 } , \mathrm { z } \right) \right\}$ 
C)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
D)  Cramer's rule does not apply since $D = 0 ; \left\{ \left( \frac { - 7 z + 34 } { 5 } , \frac { 2 z + 11 } { 5 } , z \right) \right\}$ 
A)  Cramer's rule does not apply since $D = 0 ; \left\{ \left\{ \frac { z + 34 } { 5 } , \frac { 2 z + 11 } { 5 } , z \right) \right\}$ 
B)  Cramer's rule does not apply since $\mathrm { D } = 0 ; \left\{ \left[ \frac { - 7 \mathrm { z } + 34 } { 5 } , \frac { 2 \mathrm { z } - 11 } { 5 } , \mathrm { z } \right) \right\}$ 
C)  Cramer's rule does not apply since $D = 0 ; \varnothing$ 
D)  Cramer's rule does not apply since $D = 0 ; \left\{ \left( \frac { - 7 z + 34 } { 5 } , \frac { 2 z + 11 } { 5 } , z \right) \right\}$

Solve the equation for x. - $P ( - 3 , - 3 ) , Q ( 2,5 ) , R ( 3 , - 3 )$

If the system has infinitely many solutions, write the solution set with x arbitrary. - 5x-y+z=8 7x+y+z=6 12x+2z=14

Use a graphing calculator to solve the nonlinear system. Give x- and y-coordinates to the nearest hundredth. -A rectangular plot has area $126 \mathrm { yd } ^ { 2 }$ with a perimeter of $46 \mathrm { yd }$ . What is the length of the longest side?

Find the cofactor of the indicated element. - $\text { a } 33$ 1 5 3 3 5 2 -1 -1 4

Which method should be used to solve the system? Explain your answer, including a description of the first step. - $x - y > - 4$

Find the partial fraction decomposition for the rational expression. - $\frac { - 2 x ^ { 2 } + 7 x + 24 } { ( x + 3 ) ^ { 2 } ( 3 x + 4 ) }$

Provide an appropriate response. -Does a square matrix with a row of all zeros have an inverse?

Which method should be used to solve the system? Explain your answer, including a description of the first step. - $x \leq 6$ $Which method should be used to solve the system? Explain your answer, including a description of the first step. - x \leq 6$

Find the partial fraction decomposition for the rational expression. - $\frac { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 27 } { x ^ { 2 } + x - 2 }$

Use a graphing calculator to Express solutions with approximations to the nearest thousandth. - x-y+z=1 x+y+z=3 x+y-z=-3

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x+3y=-12 4x+4y=-16

Which method should be used to solve the system? Explain your answer, including a description of the first step. - 2-4 =4 9x+9y =-1

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

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

If the system has infinitely many solutions, write the solution set with x arbitrary. - 4x+y+z =6 x-y+2z =12 4x+y+z =21

Provide an appropriate response. -Indicate whether the following statement is If two of the rows of a 3 × 3 matrix are identical, but its other row is different, then the determinant of the matrix equals 0.

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

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

The sizes of two matrices are given. Find the size of the product AB and the size of the product BA, if the given product can be calculated. -A is $2 \times 1$ ; B is $1 \times 1$ .

Use Cramer's rule to solve the system of equations. If D = 0, use another method to determine the solution set. - x+y+z=9 2x-3y+4z=7

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

Solve the equation for x. - P(−3,−3),Q(2,5),R(3,−3)P ( - 3 , - 3 ) , Q ( 2,5 ) , R ( 3 , - 3 )P(−3,−3),Q(2,5),R(3,−3)