1

Algebra II – Chapter 2 Test Review Standards/Goals:

A.1.e.: o A.CED.2.: I can solve systems of linear equations using various methods, including substitution,

elimination with addition and elimination with multiplication. o A.REI.6: I can solve systems of equations for both an exact answer and an approximation. o A.CED.3.: I can interpret solutions to systems of equations as being inconsistent, independent

or dependent (one solution, no solution or infinitely many solutions). A.CED.2.: I can graph equations on the coordinate axes with labels and scales. A.1.f.:

o I can write a linear equation in slope-intercept form when given two points, a point and the slope, or from the graph of an equation.

o A.CED.1.: I can understand linear equations and can apply them to solving real-life problems. o A.CED.1.: I can understand the relationship between slope & parallel & perpendicular lines. o A.CED.1/F.LE.2.: I can write linear equations in standard form, point slope & slope intercept

form. o A.CED.2.: I can graph equations on the coordinate axes with labels and scales.

D.1.c./A.CED.2.: I can solve systems of THREE linear equations using various methods, including substitution and the use of matrices.

I.1.a./N.VM.8.: I can add and subtract matrices of appropriate dimensions. I.1.a./N.VM.10.: I can understand that the zero and identity matrices play a role in matrix addition and

multiplication similar to the role of 0 and 1. I.1.b.: I can use addition, subtraction, and multiplication of matrices to solve real-world problems.

o N.VM.6.: I can use matrices to manipulate data. o N.VM.7.: I can multiply matrices by scalars to produce new matrices.

I.1.c./N.VM.10: I can calculate the determinant of 2 x 2 and 3 x 3 matrices. I.1.d./ N.VM.12.: I can find the inverse of a matrix. I.1.e.: I can solve a system of linear equations using a single matrix equation and inverses and

determinants. o A.REI.8: I can represent a system of linear equations as a single matrix equation.

I.1.f.: I can use technology to perform operations on matrices, find determinants, and find inverses. o A.CED. 2: I can create equations that involve two or more variables to represent relationships

between quantities. F.IF.4:

o I can interpret the key features of graphs and sketch key features given the known intercepts, through the use of a scatter plot.

o I can use graphs and linear models to make predictions from linear data displaying relationships between quantities.

H.2.a./A.SSE.4.: I can identify whether a sequence is arithmetic or geometric. o I can derive the formula for sequence for both its explicit and recursive definitions.

H.2.d.: I can use sequences and series to solve real-world problems. #1. Short Answer: If A is a 4 x 2 matrix, B is a 3 x 7 matrix, and C is a 2 x 3 matrix, what are the dimensions of A x C x B?

Below are some problems that you can easily do without a calculator:

#2. Determine the sum [

] [

] if it exists.

2

#3. Determine the difference: [

] [

] if it exists.

#4. Consider the following matrix to answer the questions.

[ ]

a. What are the correct dimensions of matrix A?

b. How many elements are in matrix A?

c. Suppose you wanted to multiply matrix A by another matrix B, so that you are calculating A · B. How many rows would matrix B need to have in order to do this calculation?

d. Suppose you decided to multiply matrix A by another matrix, C, which has dimensions of 3 rows and 8 columns. What would the dimensions be of this resulting matrix be by finding this product (A C)?

e. What would be the dimensions be of the resulting matrix by finding the product of (A*D), if D is a 2 x 3 matrix?

#5. Determine what matrix is equivalent to X in the equation:

[

] [

]?

#6. Determine what matrix is equivalent to ‘D’ in the equation:

[

] [

]?

#7. Let W + Z = [

]. If W = [

], determine what matrix is equivalent to Z?

#8. Determine what matrix is equivalent to [

]?

3

#9. Determine what matrix is the solution of:

[

] [

] ?

#10. The matrix [

] represents the vertices of a polygon. Determine what matrix

represents the vertices of the image of the polygon after a dilation of 4?

Below are some operations that you will need to be proficient in the use of the calculator for: #11. What is A B?

[

] [

]

#12. Find the inverse of [

] (Check to see if determinant is zero).

#13. What is the inverse of this matrix? [

] (Check to see if determinant is zero).

#14. Evaluate the determinant: |

|

#15. Evaluate the determinant: |

|

#16. What is the product of [

] [ ]?

4

#17. What is the product of [ ] [ ]

#18. Problem Solving: Four teams are going to participate in a speech competition. The number of 1st, 2nd, 3rd and 4th places finishes in each round determines the final score. The matrix shown shows the results of all 10 rounds of this competition. Teams will earn 4 points for each 1st place finish, 3 points for each 2nd place finish, 2 points for each 3rd finish, and 1 point for each 4th place finish. What is the ranking of the four teams?

NOTE: COLUMNS: The 1st column represents 1st place, the 2nd column represents the 2nd place, the 3rd column represents the 3rd place, and the 4th column represents the 4th place. ROWS: The 1st row represents Team 1, the 2nd row represents Team 2, the 3rd row represents Team 3, and the 4th row represents Team 4.

[

]

#19. Problem Solving: A used CD and DVD store sells DVDs for $5.00 each, CD’s for $3.00, and VHS tapes for $1.00 each. On a recent weekend, they sold 47 DVD’s, 38 CD’s, and 12 VHS tapes. Write a set of matrices that could show how you would compute the store’s total income for that day.

5

#20. CD/DVD Shop: At a cd and dvd bookstore, cd’s sell for $10 each and dvd’s sell for $15 each. You purchase 40 items and spend $450. How many cd’s did you buy? #21. Fruit Market: A fruit market is selling oranges in a 5 lb bag for $6 and a 10 lb bag for $10. You spend $68 and buy a total of 8 bags of oranges. Using a matrix, how many 5 lb bags and 10 lb bags of oranges did you buy? How many total pounds of oranges did you buy? #22. Numbers: The sum of three numbers is 21. The second number is two more than twice the first number. The second number is three times the third number. What are the numbers? #23. Geometry: One angle of a right triangle measures 90 degrees. The measure of the second angle is 5 times the measures of the third. What are the measures of these angles?

6

#24. Thrift Shop: You walk into Goodwill with $20 in your pocket. The tags for t-shirts say that each shirt cost $1.00. The tags for the long-sleeve t-shirts say that each shirt cost $2.00. You spend exactly the amount in your pocket(s) (assume no sales tax) and walk out with 14 items of clothing. How many t-shirts did you buy? How many long-sleeve t-shirts did you buy?

#25. What is the solution of the system?

{

I can identify when a system of equations has NO SOLUTION or when it has INFINITELY MANY

SOLUTIONS. Consider the following:

#26. { –

–

#27. {

7

#28. Write a matrix equation to express the following:

{

#29. What is the solution of the matrix equation below?

[

] [ ]

#30. What would be a system of equations that would represent the following matrix?

|

| ||

#31. Consider this matrix: |

|

a. What is in matrix M?

b. How many rows AND columns does the matrix have?

c. How many elements does the matrix have? What are the x & y intercepts of #32. 4x + 7y = 28? #33. 3x + y = 6

8

Consider an equation with a slope of -1/2 and passes through the point (-12, 10). #34. Write the equation of the line in point-slope form. #35. Write the equation of the line in slope-intercept form. #36. Write the equation of the line in standard form. #37. Write the equation of the line that would be parallel to this line in both standard and slope-intercept forms. #38. Write the equation of the line that would be perpendicular to this line in both standard and slope-intercept forms. #39. What would the slope be of a line parallel to y = ¼ x + 9 look like? #40. What would the slope of a line perpendicular to y = 5x – 10 look like?

9

#41. Find the 200th term of this arithmetic sequence: 9, 13, 17, 21, 25,… Show your work and the formula needed to derive your answer. #42. What is the 150th term in the arithmetic sequence beginning with 4, 16, …? Show your work and the formula needed to derive your answer. #43. Consider the boxplot below. Answer the following questions. The following boxplot represents the test scores on a mathematics exam at a very large university. The class had 500 students enrolled in it.

a. Approximately at what value is the 25th percentile? 50th percentile?; 75th percentile?

b. Approximately between what two numbers is the middle 50%?

c. Approximately between what two numbers is the lowest 75% of scores?

d. Approximately between what two numbers is the highest 75% of scores?

e. In this particular boxplot, would the mean be greater than, less than or equal to the value of the

median?

f. What actual number of students enrolled in the class makes up the middle 50%?

g. What actual number of students enrolled in the class makes up the highest 75%?

10

#44. The table and scatterplot below show the relationship between student enrollment (in thousands) and total number of property crimes (burglary and theft) in 2006 for eight colleges and universities in a certain U.S. state.

Enrollment (in 1000s) (x)

No. of Property Crimes (y)

16 201

2 6

9 42

10 141

14 138

26 601

21 230

19 294

Predictor COEF

Constant -112.58

X 21.83

a. Write the equation of the least-squares regression line in the format of: ̂ . Define any variables used. State whether the correlation is positive or negative.

b. Use your ‘line of best fit/LSRL’ to predict the number of property crimes when there are 20,000 students enrolled.

c. Use your equation from part #1 to predict the number of property crimes (in thousands) for an enrollment of 15,000 students.

#45. A quality control engineer finds that the mean life expectancy of its best-selling model of refrigerators is 11 years and the standard deviation is 2 years. The lifespans are approximately normally-distributed.

a. What percent of refrigerators models will last from 9 to 13 years?

b. What percent of refrigerators modes will last from 5 years to 17 years?

c. What percent of refrigerators models will last for over 13 years? For less than 13 years?

11

12

d.