Question 1

Use a determinant to find y such that $$( 6 , - 15 ) , ( 12 , y )$$ ,and $( 15 , - 6 )$ are collinear.

Accepted Answer

A)  $$y = - 33$$ 
B)  $y = 15$ 
C)  $y = 3$ 
D)  $y = - 6$ 
E)  $y = - 9$ 
A)  $$y = - 33$$ 
B)  $y = 15$ 
C)  $y = 3$ 
D)  $y = - 6$ 
E)  $y = - 9$

Question 2

Use a determinant and the given vertices of a triangle to find the area of the triangle. ?
(-2,4), (2,5), (6,-4)
?

Accepted Answer

A)  $\frac { 6 } { 19 }$ 
B) 20 
C)  $\frac { 5 } { 17 }$ 
D) -20 
E) 40 
A)  $\frac { 6 } { 19 }$ 
B) 20 
C)  $\frac { 5 } { 17 }$ 
D) -20 
E) 40

Question 3

​Find a value of y such that the triangle with the given vertices has an area of 4 square units. ​
​(-1,8),​(0,4),​(-1,y)
​

Accepted Answer

A) ​​y = 16 or y = -8 
B) ​​y = 16 or y = 0 
C) ​y = -8 or y = 0 
D) ​y = -16 or y = 0 
E) ​y = 8 or y = 0 
A) ​​y = 16 or y = -8 
B) ​​y = 16 or y = 0 
C) ​y = -8 or y = 0 
D) ​y = -16 or y = 0 
E) ​y = 8 or y = 0

Question 4

Find the uncoded 1 × 3 row matrices for the message "MERRY CHRISTMAS";then encode the message using the encoding matrix $$\left[ \begin{array} { c c c } 
0 & 1 & 1 \
- 2 & 1 & 2 \
2 & - 1 & 0
\end{array} \right]$$ .Show all your work.

Accepted Answer

The answer of Find the uncoded 1 × 3 row...

Question 5

Use Cramer's rule to find the solution of the system,if possible.? $$\left\{ \begin{aligned}
5 x - y & = 12 \
x + y & = 0
\end{aligned} \right.$$ ?

Accepted Answer

A) x = 2,y = -2 
B) x = 5,y = -12 
C) x = 5,y = -5 
D) The system is inconsistent. 
E) The equations are dependent. 
A) x = 2,y = -2 
B) x = 5,y = -12 
C) x = 5,y = -5 
D) The system is inconsistent. 
E) The equations are dependent.

Question 6

Use Cramer's Rule to solve (if possible)the system of equations.? $$\left\{ \begin{array} { r } 
6 x - 5 y = 23 \
- 13 x + 3 y = - 89
\end{array} \right.$$ ?

Accepted Answer

A) (-8,-5) 
B) (8,-5) 
C) (-8,5) 
D) (8,5) 
E) (5,8) 
A) (-8,-5) 
B) (8,-5) 
C) (-8,5) 
D) (8,5) 
E) (5,8)

Question 7

A large region of forest has been infested with gypsy moths.The region is roughly triangular,as shown in the figure on the next page.From the northernmost vertex A of the region,the distances to the other vertices are x = 25 miles south and 10 miles east (for vertex B),and 20 miles south and 28 miles east (for vertex C).Use a graphing utility to approximate the number of square miles in this region.​   ​

Accepted Answer

A) 250 mi2 
B) 280 mi2 
C) 260 mi2 
D) 290 mi2 
E) 270 mi2 
A) 250 mi2 
B) 280 mi2 
C) 260 mi2 
D) 290 mi2 
E) 270 mi2

Question 8

A hair product company sells three types of hair products for $30,$20 and $10 per unit.In one year,the total revenue for the three products was $820,000 which corresponded to the sale of 42,000 units.The company sold half as many units of the $30 products as units of the $20 product.Use Cramer's Rule to solve a system of linear equations to find how many units of each product were sold. ​

Accepted Answer

A) 10,000 units at $30,20,000 units at $20,12,000 units at $10. 
B) 10,000 units at $30,20,000 units at $20,18,000 units at $10. 
C) 10,000 units at $30,20,000 units at $20,20,000 units at $10. 
D) 10,000 units at $30,20,000 units at $20,14,000 units at $10. 
E) 10,000 units at $30,20,000 units at $20,16,000 units at $10. 
A) 10,000 units at $30,20,000 units at $20,12,000 units at $10. 
B) 10,000 units at $30,20,000 units at $20,18,000 units at $10. 
C) 10,000 units at $30,20,000 units at $20,20,000 units at $10. 
D) 10,000 units at $30,20,000 units at $20,14,000 units at $10. 
E) 10,000 units at $30,20,000 units at $20,16,000 units at $10.

Question 9

Use a determinant to determine whether the points (-5,1), (-8,-1)and (1,5)are collinear. ?

Accepted Answer

A)  $$\left| \begin{array} { c c c } 
- 5 & 1 & 1 \
- 8 & - 1 & 1 \
1 & 5 & 1
\end{array} \right| = 0 \text {; therefore, the points are collinear. }$$ 
B)  $$\left| \begin{array} { c c c } 
- 5 & 1 & 1 \
- 8 & - 1 & 1 \
1 & 5 & 1
\end{array} \right| = 1 \text {; therefore, the points are not collinear }$$ 
A)  $$\left| \begin{array} { c c c } 
- 5 & 1 & 1 \
- 8 & - 1 & 1 \
1 & 5 & 1
\end{array} \right| = 0 \text {; therefore, the points are collinear. }$$ 
B)  $$\left| \begin{array} { c c c } 
- 5 & 1 & 1 \
- 8 & - 1 & 1 \
1 & 5 & 1
\end{array} \right| = 1 \text {; therefore, the points are not collinear }$$

Question 10

Find a value of y such that the triangle with the given vertices has an area of 8 square units. ​
​(5,6),​(5,8),​(-3,y)
​

Accepted Answer

A) y = 10 or y = -9 
B) y = -10 or y = -9 
C) y = 10 or y = 9 
D) y = 10 or y = 0 
E) y = -10 or y = 9 
A) y = 10 or y = -9 
B) y = -10 or y = -9 
C) y = 10 or y = 9 
D) y = 10 or y = 0 
E) y = -10 or y = 9

Question 11

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

Accepted Answer

A)  $\frac { 2 } { 13 }$ 
B)  $\frac { 6 } { 19 }$ 
C) 18 
D) 38 
E) 40 
A)  $\frac { 2 } { 13 }$ 
B)  $\frac { 6 } { 19 }$ 
C) 18 
D) 38 
E) 40

Question 12

Find y such that the points are collinear. ​
(-3,3), (-4,y), (-2,4)
​

Accepted Answer

A) y = 3 
B) y = 4 
C) y = -2 
D) y = 2 
E) y = 5 
A) y = 3 
B) y = 4 
C) y = -2 
D) y = 2 
E) y = 5

Question 13

A triangular region of farmland has become overrun with deer and will be open to the public for hunting to reduce the population.In order to know how many hunters to allow on the land at one time,you need to know the area of the region in square miles.In order to estimate the area of the region,you travel from the southernmost vertex C north 20 miles then west 24 miles (for vertex B),and from the southernmost vertex C you travel 54 miles north then 7 miles east (for vertex A).Use a graphing utility to approximate the number of square miles of land.

Accepted Answer

A) 578 square miles 
B) 51 square miles 
C) 119 square miles 
D) 429 square miles 
E) 718 square miles 
A) 578 square miles 
B) 51 square miles 
C) 119 square miles 
D) 429 square miles 
E) 718 square miles

Question 14

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

Accepted Answer

A) (-1,-1,2) 
B) (-1,-1,-1) 
C) (-1,1,-2) 
D) (-1,-1,-2) 
E) (-1,1,2) 
A) (-1,-1,2) 
B) (-1,-1,-1) 
C) (-1,1,-2) 
D) (-1,-1,-2) 
E) (-1,1,2)

Exam 51: Applications of Matrices and Determinants

Use a determinant to find y such that (6,−15),(12,y)( 6 , - 15 ) , ( 12 , y )(6,−15),(12,y) ,and (15,−6)( 15 , - 6 )(15,−6) are collinear.

Use a determinant and the given vertices of a triangle to find the area of the triangle. ? (-2,4), (2,5), (6,-4) ?

​Find a value of y such that the triangle with the given vertices has an area of 4 square units. ​ ​(-1,8),​(0,4),​(-1,y) ​

Find the uncoded 1 × 3 row matrices for the message "MERRY CHRISTMAS";then encode the message using the encoding matrix [011−2122−10]\left[ \begin{array} { c c c } 0 & 1 & 1 \\- 2 & 1 & 2 \\2 & - 1 & 0\end{array} \right]​0−22​11−1​120​​ .Show all your work.

Use Cramer's rule to find the solution of the system,if possible.? {5x−y=12x+y=0\left\{ \begin{aligned}5 x - y & = 12 \\x + y & = 0\end{aligned} \right.{5x−yx+y​=12=0​ ?

Use Cramer's Rule to solve (if possible)the system of equations.? {6x−5y=23−13x+3y=−89\left\{ \begin{array} { r } 6 x - 5 y = 23 \\- 13 x + 3 y = - 89\end{array} \right.{6x−5y=23−13x+3y=−89​ ?

Use a determinant to determine whether the points (-5,1), (-8,-1)and (1,5)are collinear. ?

Find a value of y such that the triangle with the given vertices has an area of 8 square units. ​ ​(5,6),​(5,8),​(-3,y) ​

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

Find y such that the points are collinear. ​ (-3,3), (-4,y), (-2,4) ​

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

Polar Coordinates

Polar Equations of Conics

Graphs of Polar Equations

Filters

Use a determinant to find y such that $( 6 , - 15 ) , ( 12 , y )$ ,and $( 15 , - 6 )$ are collinear.

Find a value of y such that the triangle with the given vertices has an area of 4 square units. (-1,8),(0,4),(-1,y)

Find the uncoded 1 × 3 row matrices for the message "MERRY CHRISTMAS";then encode the message using the encoding matrix $\left[ \begin{array} { c c c } 0 & 1 & 1 \\- 2 & 1 & 2 \\2 & - 1 & 0\end{array} \right]$ .Show all your work.

Use Cramer's rule to find the solution of the system,if possible.? $\left\{ \begin{aligned}5 x - y & = 12 \\x + y & = 0\end{aligned} \right.$ ?

Use Cramer's Rule to solve (if possible)the system of equations.? $\left\{ \begin{array} { r } 6 x - 5 y = 23 \\- 13 x + 3 y = - 89\end{array} \right.$ ?

Find a value of y such that the triangle with the given vertices has an area of 8 square units. (5,6),(5,8),(-3,y)

Find y such that the points are collinear. (-3,3), (-4,y), (-2,4)