Question 1

Use a determinant to find an equation of the line passing through the points. ​
(0,0), (3,5)
​

Accepted Answer

A) 3x - 5y = 0 
B) 5x - 3y = 5 
C) 5x - 3y = 0 
D) 5x + 3y = 3 
E) 5x + 3y = 0 
A) 3x - 5y = 0 
B) 5x - 3y = 5 
C) 5x - 3y = 0 
D) 5x + 3y = 3 
E) 5x + 3y = 0

Question 2

Use Cramer's Rule to solve (if possible)the system of equations.? $$\left\{ \begin{array} { r } 
- 7 x + 11 y = 20 \
3 x - 9 y = 0
\end{array} \right.$$ ?

Accepted Answer

A) (-6,-2) 
B) (6,-2) 
C) (-6,2) 
D) (6,2) 
E) (-2,-6) 
A) (-6,-2) 
B) (6,-2) 
C) (-6,2) 
D) (6,2) 
E) (-2,-6)

Question 3

Use Cramer's Rule to solve the following system of linear equations: $$\left\{ \begin{array} { r } 
4 x~~~~~ - 8 z = 2 \
- 4 y + 12 z = 3 \
8 x~~~ + 20 z = 0
\end{array} \right.$$

Accepted Answer

A)  $x = \frac { 1 } { 2 } ; y = - \frac { 5 } { 4 } ; z = - \frac { 1 } { 2 }$ 
B)  $x = \frac { 5 } { 18 } ; y = - \frac { 13 } { 12 } ; z = - \frac { 1 } { 9 }$ 
C)  $x = \frac { 1 } { 18 } ; y = - \frac { 13 } { 12 } ; z = - \frac { 1 } { 18 }$ 
D)  $x = \frac { 5 } { 18 } ; y = - \frac { 11 } { 12 } ; z = - \frac { 1 } { 2 }$. 
E)  $x = \frac { 1 } { 2 } ; y = - \frac { 5 } { 4 } ; z = - \frac { 1 } { 2 }$. 
A)  $x = \frac { 1 } { 2 } ; y = - \frac { 5 } { 4 } ; z = - \frac { 1 } { 2 }$ 
B)  $x = \frac { 5 } { 18 } ; y = - \frac { 13 } { 12 } ; z = - \frac { 1 } { 9 }$ 
C)  $x = \frac { 1 } { 18 } ; y = - \frac { 13 } { 12 } ; z = - \frac { 1 } { 18 }$ 
D)  $x = \frac { 5 } { 18 } ; y = - \frac { 11 } { 12 } ; z = - \frac { 1 } { 2 }$. 
E)  $x = \frac { 1 } { 2 } ; y = - \frac { 5 } { 4 } ; z = - \frac { 1 } { 2 }$.

Question 4

Write a cryptogram for the message "TWO IF BY LAND" using the matrix $$\left[ \begin{array} { c c c } 
0 & - 2 & 2 \
2 & 3 & - 3 \
3 & - 2 & 0
\end{array} \right]$$ .Show all your work.

Accepted Answer

@#IMG-DLM& 
 @#IMG-DLM& 
 @#IMG-DLM& 
 @#IMG-DLM&

Question 5

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix,by assigning a number to each letter in the alphabet (with 0 assigned to a blank space,0 = _,1 = A,2 = B and so on. )? $$\left[ \begin{array} { c c c } 
0 & 5 & 1 \
- 1 & 1 & 1 \
1 & - 5 & 0
\end{array} \right]$$ . ?

Accepted Answer

A) ? $$\begin{array} { c c c c c c c c c } 
1 & - 36 & 28 & 18 & - 29 & 14 & 13 & - 20 & 18 \
- 25 & 115 & 43 & 10 & - 67 & 11 & & &
\end{array}$$ 
B) ? $$\begin{array} { c c c c c c c c c } 
13 & - 20 & 18 & - 25 & 115 & 43 & 10 & - 67 & 11 \
1 & - 36 & 28 & 18 & - 29 & 14 & & &
\end{array}$$ 
C) ? $$\begin{array}{l}
\begin{array} { l l l l l l l l l } 
10 & - 67 & 11 & 1 & - 36 & 28 & 18 & - 29 & 14
\
13 & - 20 & 18 & - 25 & 115 & 43
\end{array}
\end{array}$$? 
D) ? $$\begin{array} { c c c c c c c c c } 
- 25 & 115 & 43 & 10 & - 67 & 11 & 1 & - 36 & 28 \
18 & - 29 & 14 & 13 & - 20 & 18 & & &
\end{array}$$? 
E) ? $$\begin{array} { c c c c c c c c c } 
18 & - 29 & 14 & 13 & - 20 & 18 & - 25 & 115 & 43 \
10 & - 67 & 11 & 1 & - 36 & 28 & & &
\end{array}$$ 
A) ? $$\begin{array} { c c c c c c c c c } 
1 & - 36 & 28 & 18 & - 29 & 14 & 13 & - 20 & 18 \
- 25 & 115 & 43 & 10 & - 67 & 11 & & &
\end{array}$$ 
B) ? $$\begin{array} { c c c c c c c c c } 
13 & - 20 & 18 & - 25 & 115 & 43 & 10 & - 67 & 11 \
1 & - 36 & 28 & 18 & - 29 & 14 & & &
\end{array}$$ 
C) ? $$\begin{array}{l}
\begin{array} { l l l l l l l l l } 
10 & - 67 & 11 & 1 & - 36 & 28 & 18 & - 29 & 14
\
13 & - 20 & 18 & - 25 & 115 & 43
\end{array}
\end{array}$$? 
D) ? $$\begin{array} { c c c c c c c c c } 
- 25 & 115 & 43 & 10 & - 67 & 11 & 1 & - 36 & 28 \
18 & - 29 & 14 & 13 & - 20 & 18 & & &
\end{array}$$? 
E) ? $$\begin{array} { c c c c c c c c c } 
18 & - 29 & 14 & 13 & - 20 & 18 & - 25 & 115 & 43 \
10 & - 67 & 11 & 1 & - 36 & 28 & & &
\end{array}$$

Question 6

Use a determinant to determine whether the points $( 1,4 ) , ( - 2,2 )$ and $( 7,8 )$ are collinear.Show all work.
​

Accepted Answer

@#IMG-DLM& ;therefor

Question 7

Use Cramer's Rule to solve (if possible)the system of equations.? $$\left\{ \begin{array} { l } 
4 x - 3 y = - 30 \
6 x + 9 y = - 18
\end{array} \right.$$ ?

Accepted Answer

A) (-6,2) 
B) (6,-2) 
C) (6,2) 
D) (2,-6) 
E) (-6,-2) 
A) (-6,2) 
B) (6,-2) 
C) (6,2) 
D) (2,-6) 
E) (-6,-2)

Question 8

Find y such that the points are collinear. ​
(5,-4), (3,y), (4,-3)
​

Accepted Answer

A) y = -2 
B) y = 1 
C) y = 0 
D) y = -1 
E) y = 2 
A) y = -2 
B) y = 1 
C) y = 0 
D) y = -1 
E) y = 2

Question 9

Write a cryptogram by assigning a number to each letter in the alphabet such as 1 = A,2 = B,3 = C and so on (with 0 assigned to a blank space)for the message "TWO IF BY LAND" using the following matrix.? $$\left[ \begin{array} { c c c } 
0 & 1 & - 1 \
- 1 & - 1 & 1 \
- 1 & 1 & 0
\end{array} \right]$$ ?

Accepted Answer

A) ? $$\begin{array} { l l l l l l l l l } 
- 13 & - 11 & 12 & - 4 & 10 & - 10 & - 38 & 12 & 3 \
- 15 & - 3 & 9 & - 27 & 23 & 2 & & &
\end{array}$$? 
B) ? $$\begin{array} { l l l l l l l l l } 
- 4 & 10 & - 10 & - 38 & 12 & 3 & - 15 & - 3 & 9 \
- 27 & 23 & 2 & - 13 & - 11 & 12 & & &
\end{array}$$ 
C) ? $$\begin{array} { l l l l l l l l l } 
- 27 & 23 & 2 & - 13 & - 11 & 12 & - 4 & 10 & - 10 \
- 38 & 12 & 3 & - 15 & - 3 & 9 & & &
\end{array}$$ 
D)  ?$$\begin{array} { l l l l l l l l l } 
- 15 & - 3 & 9 & - 27 & 23 & 2 & - 13 & - 11 & 12 \
- 4 & 10 & - 10 & - 38 & 12 & 3 & & &
\end{array}$$? 
E) ? $$\begin{array} { c c c c c c c c c } 
- 38 & 12 & 3 & - 15 & - 3 & 9 & - 27 & 23 & 2 \
- 13 & - 11 & 12 & - 4 & 10 & - 10 & & &
\end{array}$$ 
A) ? $$\begin{array} { l l l l l l l l l } 
- 13 & - 11 & 12 & - 4 & 10 & - 10 & - 38 & 12 & 3 \
- 15 & - 3 & 9 & - 27 & 23 & 2 & & &
\end{array}$$? 
B) ? $$\begin{array} { l l l l l l l l l } 
- 4 & 10 & - 10 & - 38 & 12 & 3 & - 15 & - 3 & 9 \
- 27 & 23 & 2 & - 13 & - 11 & 12 & & &
\end{array}$$ 
C) ? $$\begin{array} { l l l l l l l l l } 
- 27 & 23 & 2 & - 13 & - 11 & 12 & - 4 & 10 & - 10 \
- 38 & 12 & 3 & - 15 & - 3 & 9 & & &
\end{array}$$ 
D)  ?$$\begin{array} { l l l l l l l l l } 
- 15 & - 3 & 9 & - 27 & 23 & 2 & - 13 & - 11 & 12 \
- 4 & 10 & - 10 & - 38 & 12 & 3 & & &
\end{array}$$? 
E) ? $$\begin{array} { c c c c c c c c c } 
- 38 & 12 & 3 & - 15 & - 3 & 9 & - 27 & 23 & 2 \
- 13 & - 11 & 12 & - 4 & 10 & - 10 & & &
\end{array}$$

Question 10

Use Cramer's Rule to solve (if possible)the system of equations.? $$\left\{ \begin{array} { r } 
4 x - y + z = - 20 \
2 x + 2 y + 3 z = 10 \
5 x - 2 y + 6 z = 2
\end{array} \right.$$ ?

Accepted Answer

A) (6,2,6) 
B) (6,-2,6) 
C) (-6,2,6) 
D) (7,-2,6) 
E) (-6,-2,6) 
A) (6,2,6) 
B) (6,-2,6) 
C) (-6,2,6) 
D) (7,-2,6) 
E) (-6,-2,6)

Question 11

Determine a positive value for y such that a triangle with vertices $P ( 0,0 ) , Q ( 17,0 )$ ,and $R ( 17 , y )$ has an area of 17 square units.

Accepted Answer

A) 8 
B) 2 
C) 9 
D) 11 
E) 1 
A) 8 
B) 2 
C) 9 
D) 11 
E) 1

Question 12

You inherited a triangular piece of property after your Uncle Izzy passed away.You want to know the size of land,so you "step it off" to estimate the square footage.From the southernmost vertex A,you travel north 300 feet then west 220 feet (for vertex C),and from the southernmost vertex A,you travel 420 feet north then 50 feet west (for vertex B).Use a graphing utility to approximate the number of square feet of land that you have inherited.

Accepted Answer

A) 3,000 square feet 
B) 38,700 square feet 
C) 53,700 square feet 
D) 43,500 square feet 
E) 22,500 square feet 
A) 3,000 square feet 
B) 38,700 square feet 
C) 53,700 square feet 
D) 43,500 square feet 
E) 22,500 square feet

Question 13

Use Cramer's Rule to solve (if possible)the system of equations.? $$\left\{ \begin{array} { l } 
4 x - 2 y + z = - 3 \
2 x + 2 y + 3 z = 9 \
5 x - 2 y + 6 z = - 15
\end{array} \right.$$ ?

Accepted Answer

A) (3,-6,-3) 
B) (3,6,-3) 
C) (-3,6,-3) 
D) (-3,-6,-3) 
E) (7,-6,-3) 
A) (3,-6,-3) 
B) (3,6,-3) 
C) (-3,6,-3) 
D) (-3,-6,-3) 
E) (7,-6,-3)

Question 14

Use Cramer's Rule to solve (if possible)the system of equations.? $$\left\{ \begin{array} { r } 
- 0.4 x + 0.8 y = 1.6 \
0.2 x + 0.3 y = 2.3
\end{array} \right.$$ ?

Accepted Answer

A)  $$\left( - \frac { 34 } { 7 } , - \frac { 31 } { 7 } \right)$$ 
B)  $$\left( \frac { 34 } { 7 } , \frac { 31 } { 7 } \right)$$ 
C)  $$\left( \frac { 34 } { 7 } , - \frac { 31 } { 7 } \right)$$ 
D)  $$\left( \frac { 31 } { 7 } , \frac { 34 } { 7 } \right)$$ 
E)  $$\left( - \frac { 34 } { 7 } , \frac { 31 } { 7 } \right)$$ 
A)  $$\left( - \frac { 34 } { 7 } , - \frac { 31 } { 7 } \right)$$ 
B)  $$\left( \frac { 34 } { 7 } , \frac { 31 } { 7 } \right)$$ 
C)  $$\left( \frac { 34 } { 7 } , - \frac { 31 } { 7 } \right)$$ 
D)  $$\left( \frac { 31 } { 7 } , \frac { 34 } { 7 } \right)$$ 
E)  $$\left( - \frac { 34 } { 7 } , \frac { 31 } { 7 } \right)$$

Question 15

Use determinants to find the area of a triangle with the given vertices and confirm your answer by plotting the points in a coordinate plane and using the formula $\text { Area } \left. = \frac { 1 } { 2 } \text { (base)(height } \right)$ . ?
(5,-3), (2,-2), (7,5)
?

Accepted Answer

A) 21 
B) 13 
C) 15 
D) 19 
E) 17 
A) 21 
B) 13 
C) 15 
D) 19 
E) 17

Question 16

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix $$\left[ \begin{array} { c c c } 
0 & 5 & 1 \
- 2 & 1 & 2 \
2 & - 5 & 0
\end{array} \right]$$ .Show all your work.
​

Accepted Answer

​ @#IMG-DLM& 
 @#IMG-DLM& 
 @#IMG-DLM&

Question 17

Use a determinant and the given vertices of a triangle to find the area of the triangle.?   ?

Accepted Answer

A) 52 
B)  $\frac { 4 } { 19 }$ 
C) 25 
D)  $\frac { 2 } { 17 }$ 
E) 54 
A) 52 
B)  $\frac { 4 } { 19 }$ 
C) 25 
D)  $\frac { 2 } { 17 }$ 
E) 54

Question 18

Use a determinant and the given vertices of a triangle to find the area of the triangle.?   ?

Accepted Answer

A) ? $\frac { 6 } { 7 }$ 
B) 4 
C) ? $\frac { 2 } { 17 }$ 
D)  $\frac { 6 } { 11 }$ 
E) 7 
A) ? $\frac { 6 } { 7 }$ 
B) 4 
C) ? $\frac { 2 } { 17 }$ 
D)  $\frac { 6 } { 11 }$ 
E) 7

Question 19

​Find a value of y such that the triangle with the given vertices has an area of 26 square units. ​
​(-6,-6),​(2,-2),​(-9,y)
​

Accepted Answer

A) ​y = -1 or y = -14 
B) ​y = -1 or y = 14 
C) ​y = -1 or y = 0 
D) ​y = 1 or y = 14 
E) ​y = 1 or y = -14 
A) ​y = -1 or y = -14 
B) ​y = -1 or y = 14 
C) ​y = -1 or y = 0 
D) ​y = 1 or y = 14 
E) ​y = 1 or y = -14

Question 20

Find a value of y such that the triangle with the given vertices has an area of 40 square units. ​
​(7,0),​(7,-5),​(-9,y)
​

Accepted Answer

A) ​y = -9 or y = -10 
B) ​​y = 7 or y = -3 
C) ​y = 9 or y = 0 
D) ​y = 8 or y = 3 
E) ​y = 10 or y = 8 
A) ​y = -9 or y = -10 
B) ​​y = 7 or y = -3 
C) ​y = 9 or y = 0 
D) ​y = 8 or y = 3 
E) ​y = 10 or y = 8

Exam 51: Applications of Matrices and Determinants

Use a determinant to find an equation of the line passing through the points. ​ (0,0), (3,5) ​

Use Cramer's Rule to solve (if possible)the system of equations.? {−7x+11y=203x−9y=0\left\{ \begin{array} { r } - 7 x + 11 y = 20 \\3 x - 9 y = 0\end{array} \right.{−7x+11y=203x−9y=0​ ?

Use Cramer's Rule to solve the following system of linear equations: {4x −8z=2−4y+12z=38x +20z=0\left\{ \begin{array} { r } 4 x~~~~~ - 8 z = 2 \\- 4 y + 12 z = 3 \\8 x~~~ + 20 z = 0\end{array} \right.⎩⎨⎧​4x −8z=2−4y+12z=38x +20z=0​

Write a cryptogram for the message "TWO IF BY LAND" using the matrix [0−2223−33−20]\left[ \begin{array} { c c c } 0 & - 2 & 2 \\2 & 3 & - 3 \\3 & - 2 & 0\end{array} \right]​023​−23−2​2−30​​ .Show all your work.

Use a determinant to determine whether the points (1,4),(−2,2)( 1,4 ) , ( - 2,2 )(1,4),(−2,2) and (7,8)( 7,8 )(7,8) are collinear.Show all work. ​

Use Cramer's Rule to solve (if possible)the system of equations.? {4x−3y=−306x+9y=−18\left\{ \begin{array} { l } 4 x - 3 y = - 30 \\6 x + 9 y = - 18\end{array} \right.{4x−3y=−306x+9y=−18​ ?

Find y such that the points are collinear. ​ (5,-4), (3,y), (4,-3) ​

Use Cramer's Rule to solve (if possible)the system of equations.? {4x−y+z=−202x+2y+3z=105x−2y+6z=2\left\{ \begin{array} { r } 4 x - y + z = - 20 \\2 x + 2 y + 3 z = 10 \\5 x - 2 y + 6 z = 2\end{array} \right.⎩⎨⎧​4x−y+z=−202x+2y+3z=105x−2y+6z=2​ ?

Determine a positive value for y such that a triangle with vertices P(0,0),Q(17,0)P ( 0,0 ) , Q ( 17,0 )P(0,0),Q(17,0) ,and R(17,y)R ( 17 , y )R(17,y) has an area of 17 square units.

Use Cramer's Rule to solve (if possible)the system of equations.? {4x−2y+z=−32x+2y+3z=95x−2y+6z=−15\left\{ \begin{array} { l } 4 x - 2 y + z = - 3 \\2 x + 2 y + 3 z = 9 \\5 x - 2 y + 6 z = - 15\end{array} \right.⎩⎨⎧​4x−2y+z=−32x+2y+3z=95x−2y+6z=−15​ ?

Use Cramer's Rule to solve (if possible)the system of equations.? {−0.4x+0.8y=1.60.2x+0.3y=2.3\left\{ \begin{array} { r } - 0.4 x + 0.8 y = 1.6 \\0.2 x + 0.3 y = 2.3\end{array} \right.{−0.4x+0.8y=1.60.2x+0.3y=2.3​ ?

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix [051−2122−50]\left[ \begin{array} { c c c } 0 & 5 & 1 \\- 2 & 1 & 2 \\2 & - 5 & 0\end{array} \right]​0−22​51−5​120​​ .Show all your work. ​

Use a determinant and the given vertices of a triangle to find the area of the triangle.? ?

Use a determinant and the given vertices of a triangle to find the area of the triangle.? ?

​Find a value of y such that the triangle with the given vertices has an area of 26 square units. ​ ​(-6,-6),​(2,-2),​(-9,y) ​

Find a value of y such that the triangle with the given vertices has an area of 40 square units. ​ ​(7,0),​(7,-5),​(-9,y) ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

Hyperbolas

Rotation of Conics

Parametric Equations

Use a determinant to find an equation of the line passing through the points. (0,0), (3,5)

Use Cramer's Rule to solve (if possible)the system of equations.? $\left\{ \begin{array} { r } - 7 x + 11 y = 20 \\3 x - 9 y = 0\end{array} \right.$ ?

Use Cramer's Rule to solve the following system of linear equations: $\left\{ \begin{array} { r } 4 x~~~ - 8 z = 2 \\- 4 y + 12 z = 3 \\8 x~ + 20 z = 0\end{array} \right.$

Write a cryptogram for the message "TWO IF BY LAND" using the matrix $\left[ \begin{array} { c c c } 0 & - 2 & 2 \\2 & 3 & - 3 \\3 & - 2 & 0\end{array} \right]$ .Show all your work.

Use a determinant to determine whether the points $( 1,4 ) , ( - 2,2 )$ and $( 7,8 )$ are collinear.Show all work.

Use Cramer's Rule to solve (if possible)the system of equations.? $\left\{ \begin{array} { l } 4 x - 3 y = - 30 \\6 x + 9 y = - 18\end{array} \right.$ ?

Find y such that the points are collinear. (5,-4), (3,y), (4,-3)

Use Cramer's Rule to solve (if possible)the system of equations.? $\left\{ \begin{array} { r } 4 x - y + z = - 20 \\2 x + 2 y + 3 z = 10 \\5 x - 2 y + 6 z = 2\end{array} \right.$ ?

Determine a positive value for y such that a triangle with vertices $P ( 0,0 ) , Q ( 17,0 )$ ,and $R ( 17 , y )$ has an area of 17 square units.

Use Cramer's Rule to solve (if possible)the system of equations.? $\left\{ \begin{array} { l } 4 x - 2 y + z = - 3 \\2 x + 2 y + 3 z = 9 \\5 x - 2 y + 6 z = - 15\end{array} \right.$ ?

Use Cramer's Rule to solve (if possible)the system of equations.? $\left\{ \begin{array} { r } - 0.4 x + 0.8 y = 1.6 \\0.2 x + 0.3 y = 2.3\end{array} \right.$ ?

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix $\left[ \begin{array} { c c c } 0 & 5 & 1 \\- 2 & 1 & 2 \\2 & - 5 & 0\end{array} \right]$ .Show all your work.

Find a value of y such that the triangle with the given vertices has an area of 26 square units. (-6,-6),(2,-2),(-9,y)

Find a value of y such that the triangle with the given vertices has an area of 40 square units. (7,0),(7,-5),(-9,y)