1

Math-132

Final Exam Review (All Sections)

On this exam, which will be cumulative, questions may come from any of the following topic areas:

- Finding the equation of a straight line given two points

- Finding the equation of a straight line given information about the independent and dependent

variables in a word problem

- Finding the equation of a straight line that is to be parallel to another line

- Solving systems of two equations with two or three unknowns

- Solving systems of two or three equations set up based on a word problem

- Finding the break-even point in a word problem

- Finding the market price

- Finding and using the supply and demand functions

- Using the Gauss-Jordan method to solve systems of two or three equations

- Adding, subtracting, and multiplying matrices

- Finding the inverse of 2 x2 or 3 x 3 matrices using elementary row operations

- Using graphical method to solve a system of inequalities

- Setting up and solving a system of inequalities based on information from a word problem

- Using the geometric method to maximize or minimize an objective function given a set of

inequalities as constraints

- Using the Simplex method to maximize an objective function given a set of inequalities as

constraints

- Using the Simplex method to minimize an objective function given a set of inequalities as

constraints

- Using the Simplex method in conjunction with a word problem.

- Union and intersection of sets

- Complement of sets

- Constructing and interpreting Venn Diagrams

- Applying the additive rule to counts in sets

- Applying the additive rule to probabilities of events

- Applying the additive rule to word problems

- The multiplicative principle

- Tree diagrams

- Simple probability calculations

- Mutually exclusive and independent random outcomes

- Probabilities of mutually exclusive and independent outcomes

- Calculating probability given the odds

- Calculating the odds given probability

- Calculating the expected value

- Calculating the expected value for the information in a word problem

- Conditional probability of dependent events

- Conditional probability of independent events

2

- Probability calculation using the Baye’s theorem

- Permutation of outcomes

- Combination of outcomes

- The binomial probability

Section 1.1

#1) Find the equation of a line with slope

that passes through the point (3, 1), and write the equation

in slope-intercept form.

[answer:

]

#2) Find the equation of the line that connects the points ( ) ( ), and write the

equation in slope-intercept form.

[answer:

]

#3) Find the equation of the line that passes through the points ( ) ( ), and write the equation in slope-intercept form. [ ] Section 1.2 #4) Determine the solution of each of the following systems and put your solution in point form ( ).

a) {

[answer= ( )]

b) {

[answer= (

)]

#5) A retiree needs $10,000 per year in supplementary income. He has $150,000 to invest and can invest

in AA bonds at 10% annual interest or in Savings and Loans certificates of 5% interest per year. How

much money should be invested in each so that he realizes exactly $10,000 in extra income per year?

[answer= ]

3

Section 1.3 #6) A manufacturer produces gameday pennants at a cost of $0.85 per item and sells them for $0.95 per

item. The daily operational overhead is $350.

a) What is the daily cost function?

[ ( ) ]

b) What is the daily revenue function?

[ ( ) ]

c) How many pennants should the manufacturer sell per day in order to break-even?

[ ]

#7) The supply and demand equations for sugar have been estimated to be given by the equations:

Where is the price in dollars per pound and and are in millions of pounds.

a) Find the market price.

[ ]

b) What quantity of supply is required at this market price?

[ ]

#8) It costs a cellphone parts manufacturer $12 per item to produce a certain electronic component. There is a fixed cost of $594 per month for the manufacturer to produce the component. The company sells the components for $18 a piece.

a) What is the total cost function per month for producing the electronic component? [ ( ) ]

b) What is the total revenue function? [ ( ) ]

c) After selling how many of the electronic components per month will the manufacturer break-even? [ ]

Section 2.1 #9) Using the method of elimination, solve each of the following systems and put your solution in the form ( )

a) {

[answer= ( )

4

b) {

[answer= ]

c) {

[answer= (

)]

Section 2.2 & 2.3 #10) The following augmented matrix is already in Row Echelon Form (REF). Find the solution of the system and give the answer in the form ( ).

(

| )

[answer= ( )]

#11) Given the following augmented matrix, use row operations (i.e., the Gaussian elimination) to put the matrix into Row Echelon Form (REF). Then, determine the solution to the system and put your answer in the form ( ).

(

| )

[answer= ( )]

#12) Given the following augmented matrix, use row operations (i.e., the Gaussian elimination) to put the matrix into Row Echelon Form (REF). Then, determine the solution to the system and put your answer in the form ( ).

(

| )

[answer= ( )]

5

Section 3.2

#13) Find the product of the following matrices:

a) [ ] [ ]

[ ]

b) [ ] [

]

[ [ ] ]

c) [

] [

]

[ [

]]

d) [

] [

]

[ [

]]

Section 3.3

#14) Using elementary row operations (not the determinant method) to find the inverse of the

following matrices:

a) [

]

[ [

] ]

6

b) [

]

[ [

]]

c) #8) [

]

[ [

]]

Sections 4.1 & 4.2

#15) The given figure illustrates the graph of the set of feasible points of a linear system of inequalities.

Find the minimum and maximum values of the objective function .

[ ]

[ ]

7

#16) Using the geometrical method, find a graphical solution to each of the following linear systems of

inequalities. Then, find the coordinates of the corner points for each solution region.

a)

{

[ ( ) ( ) ( ) ( ) ( ) ]

#17) Use the geometric method to maximize the objective function subject to the

constraints: {

[ ]

#18) Using the geometric approach, maximize the objective function , subject to the

constraints:

{

[ ]

#19) Using the geometric approach, minimize the objective function , subject to the

constraints:

{

[ ]

8

Section 4.3

#20) A financial consultant wishes to invest up to a total of $25,000 in two types of securities, one that

yields 10% per year and another that yields 8% per year. Furthermore, she believes that the amount

invested in the first security should be at most one third of the amount invested in the second security.

What investment program should the consultant pursue in order to maximize income?

[

]

#21) A factory manufactures two kinds of ice skates: racing skates and figure skates. The racing skates

require 6 work-hours in the fabrication department whereas the figure skates require 4 work-hours

there. Additionally, the racing skates require 1 work-hour in the finishing department whereas the figure

skates require 2 work-hours there. The fabrication department has available at most 120 work-hours per

day, and the finishing department has no more than 40 work-hours per day available. If the profit on

each racing skate is $11 and the profit on each figure skate is $12, how many of each should be

manufactured each day to maximize profit?

[

]

Section 5.2

#22) For each of the following tableaus, determine the first pivot element, perform all the pivot

operations for the entire pivot column, and classify the next tableau that results according to the

resulting solution type as “solved,” “no solution,” or “ready for another set of pivot operations.”

a) (

| )

[ ]

b)

[ ]

9

c)

[ ]

#23) Use the Simplex Method to solve the maximum problem:

Maximize:

Subject to the constraints:

[ ]

#24) Use the Simplex Method to solve the maximum problem:

Maximize:

Subject to the constraints:

[ ]

Section 5.3 #25) Write the dual problem of the following minimum problem. Do not solve the dual problem.

Minimize

Subject to the constraints:

Subject to the constraints:

10

#26) Write the dual problem of the following minimum problem. Do not solve the dual problem.

Minimize

Subject to the constraints:

Subject to the constraints:

#27) The following tableau was obtained using the simplex method for optimizing a Minimum problem.

Determine the first pivot element, perform pivot operations to obtain the final tableau. Then, using the

final tableau, state the solution of the Minimum problem in terms of variables x1, x2, and the value of

the objective function, C.

a) What is the final tableau? Write it down.

b) What is the solution to the minimum problem?

[ ]

11

#28) The following tableau was obtained using the simplex method for optimizing a Minimum problem.

Determine the pivot element, perform pivot operations on the pivot element to obtain the final tableau.

Then, using the final tableau, state the solution of the Minimum problem in terms of variables x1, x2,

and the value of the objective function, C.

a) What is the final tableau? Write it down.

b) What is the solution to the minimum problem?

[ ]

Sections 7.1

#29) If { } { } what is:

a)

b)

[ ]

#30) If { } { } { } Find ̅ ̅

[ { } ]

#31) If { } { } { }

and { } Find

[ { } ]

#32) If { } { } { } Find ̅̅ ̅̅ ̅̅ ̅

[ { } ]

#33) If { } { } { } Find ̅̅ ̅̅ ̅̅ ̅

[ { } ]

12

#34) A class of 28 students were surveyed and asked if they ever had dogs or cats for pets at home. 8

students said they only had a dog; 6 students said they only had a cat; 10 students said they had a dog

and a cat; and 4 students said they never had a dog or a cat. Create a Venn Diagram that captures this

information.

[

]

#35) Out of 65 students, 28 are taking English Composition and 35 are taking Chemistry. 11 students are

taking both classes.

a) Draw the Venn Diagram for this information. [ answer: Do you see how this is done? Hint: Draw

the Venn Diagram and place the intersection number, first]

b) How many are in neither class? [answer: 13]

c) How many are taking at least one of the classes? [answer: 52]

d) What is the probability that a student is taking English but not Chemistry? [answer: 0.262]

e) What is the probability that a student is taking both classes? [answer: 0.169]

f) What is the probability that a student is taking English, given that the student is taking

chemistry? [answer: 0.314]

English

Chemistry

13

17 11

Cats

24

Dogs

6 10

8

U

4

13

#36) In a group of 58 students, 24 are taking algebra, 12 are taking biology, 19 are taking chemistry, 7

are taking algebra and biology, 11 are taking algebra and chemistry, 6 are taking biology and chemistry,

and 4 are taking all three courses.

a) Draw the Venn Diagram for this information. [answer: Shown below. Do you see how this is

done? Hint: Place the intersection number for all three courses first and work your way

outward.]

b) How many students are not taking any of these courses? [answer: 23 ]

c) What is the probability that a student is not taking any of these courses? [answer: 0.397 ]

d) What is the probability that a student is taking algebra and biology but not chemistry?

[answer: 0.0517 ]

e) What is the probability that a student is taking only biology? [answer: 0.0517]

f) What is the probability that a student is taking exactly two of these courses? [answer:

]

Section 7.2

#37) Find ( ), given that ( ) and ( ) and ( )

#38) Find ( ), given that ( ) and ( ) and ( )

#39) Find ( ), given that ( ) and ( ) and ( )

A

C

B U

10

3

4

7 2

6

3

23

14

#40) Motors, Inc. manufactured 250 cars with a GPS system, 205 with satellite system, and 70 with

both these options. How many cars were manufactured if every car has at least one of these options?

[ ]

#41) In a survey of 600 business travelers, it was found that of two daily newspapers, New York times,

and Wall Street Journal, 360 read New York Times, 112 read New York Times and Wall Street Journal,

and 56 read only Wall Street Journal.

a) How many read New York Times or Wall Street Journal?

[ ]

b) How many did not read either New York Times or Wall Street Journal?

[ ]

Section 7.3

#42) A man has 13 shirts and 5 ties. How many different shirts and tie arrangements can he wear?

[ ]

#43) A restaurant offers 3 different salads, 6 different main courses, 13 different desserts, and 7

different drinks. How many different launches are possible?

[ ]

#44) How many different ways can 4 people be seated in a row of 4 seats?

[ ]

#45) How many 4-letter code words are possible using the first 5 letters of the alphabet with no letters

repeated? [ ] How many codes are possible when letters are allowed to repeat?

[ ]

Section 7.4

#46) Three letters are picked from the alphabet (repetitions are allowed, assume order is important).

Find the number of outcomes in the sample space.

[ ]

#47) Alice is going through a pile of applications for admission to the Bachelor’s program, the Master’s

program, and the PhD program in business. She is determined to select one candidate for each program

before she leaves for the day. In how many ways she may select one candidate for each program, given

the applicant’s gender? List the sample space.

[ ]

{ }

15

#48) Assume your favorite football team has 2 games left to finish the season. The outcome of each

game can be win, lose, or tie. What is the total number of possible outcomes? List the sample space.

[ ]

{ }

#49) Each customer entering a department store will either buy or not buy some merchandise. A

researcher shadows three customers to monitor their behavior in terms of whether or not each

individual buys or does not buy something. How many outcomes are possible? List the sample space.

[ ]

{ }

Section 7.5

#50) If events E and F belong to the same sample space, and ( ) , ( ) ,

( )=0.40 find ( )

[ ]

#51) If events E and F belong to the same sample space, and ( ) , ( ) ,

( )=0.80 find ( )

[ ]

#52) Anne is taking courses in both mathematics and English. She estimates her probability of passing

mathematics at 0.3 and English at 0.4, and she estimates her probability of passing at least one of

them at 0.53. What is her probability of passing both courses?

[ ]

#53) A financial consultant estimates that there is an 11% chance a mutual fund will outperform the

market during any given year. She also estimates that there is a 10% chance that the mutual fund will

outperform the market for the next two years. What is the probability that the mutual fund will

outperform the market in at least one of the next two years?

[ ]

#54) Three letters, with repetition allowed, are selected from the alphabet. What is the probability that

none is repeated?

[ ]

#55) Determine the probability of E if the odds in favor of E are 3 to 1.

[ ( )

]

16

#56) Determine the probability of E if the odds in favor of E are 1 to 1.

[ ( )

]

#57) Determine the odds for and against the event E if ( ) .

[ ]

#58) Determine the odds for and against the event F if ( )

[ ]

#59) Suppose events A and B are independent, with ( ) and ( ) .

a) Find the odds for A.

[ ]

b) Find the odds for ̅ (i.e., the odds against A)

[ ]

#60) Events A and B are mutually exclusive, with ( ) and ( ) .

a) Find the odds for A

[ ]

b) Find the odds for ̅ (i.e., the odds against A)

[ ]

Section 7.6

#61)The information about attendance at a football game in a certain city is given in the table below.

How many fans are expected to attend each game?

Extremely Cold Cold Moderate Warm

Attendance 40,000 50,000 70,000 90,000

Probability 0.09 0.41 0.41 0.09

[ ]

#62) In a lottery, 1000 tickets are sold at $0.32 each. There are 3 cash prizes: one for $130, one for $80,

and one for $10. Alice buys 7 tickets.

A) Considering the expected value of each ticket, what would have been a fair price for a ticket?

[ ]

B) In total, how much extra did Alice pay?

[ ]

17

#63) A fair coin is tossed 3 times, and a player wins $15 if 3 tails occur, wins $7 if 2 tails occur, and

losses $15 if no tails occur. If tail occurs, no one wins. What is the expected value of the game? Round

your answer to two decimal places.

[ ]

Section 8.1

#64) If E and F are events with P(E)=0.2 P(F)=0.3 ( )

a) Find P(E|F)

[ ]

b) Find P(F|E)

[ ]

c) Find ( )

[ ]

#65) Given the following tree diagram probabilities, find the probability of obtaining D in any final

outcome.

[ ]

#66) If E and F are events with ( ) and P(E|F)=0.5 what is P(F)=?

[ ( ) ]

#67) A jar contains 6 white marbles, 2 yellow marbles, 2 red marbles, and 5 blue marbles. Two marbles

are picked at random one after another and without replacement.

a) Draw the tree diagram for this experiment.

[ ]

b) What is the probability that both are blue?

[ ]

A

B

D

C

D

0.11

C

0.89

0.87

0.13

0.28

0.72

18

c) What is the probability that exactly one is blue?

[ ]

d) What is the probability that at least one is blue?

[ ]

#68) If P(E)=0.71 P(F)=0.79 ( ) find ( )

[ ( ) ]

#69) If P(E)=0.20 P(F)=0.65 ( ) P(E|F)=?

[ ( | ) ]

#70) Assume you have applied to two graduate schools: school A and school B. Suppose it is very

competitive to get into school A and only 8% of all applicants are accepted, while the probability of

getting accepted by school B is 0.25.

a) What is the probability of your application being accepted by both schools? [ ]

b) What is the probability of receiving at least one letter of acceptance? [ ]

c) What is the probability that both schools decline your application? [ ]

#71) If you are fishing in South Puget Sound in the state of Washington in October, there is a 30%

chance that you’ll catch Salmon, and a 5% chance that you’ll catch Flat Fish. You are there to catch two

fish before heading back home. Assuming that catching the fish is mutually exclusive:

a) What is the probability that you’ll catch either a Salmon or a Flat Fish?

[ ]

b) What is the probability that you won’t be able to catch either kind of fish?

[ ]

#72) A survey of magazine subscribers showed that 63.2% rented a car during the past 12 months for

19

business reasons, 36% rented a car during the past 12 months for personal reasons, and 23% rented a

car during the past 12 months for both business and personal reasons.

a) What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons? [ ]

b) What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons? [ ]

#73) The following table of data is the result of a survey of 574 individuals at a shopping mall:

Likes the deodorant Does not like the deodorant No opinion

Group I 167 65 24

Group II 100 85 13

Group III 48 63 9

a) what is the probability that a customer likes the deodorant given he/she is from group I?

[ ]

b) What is the probability that a customer is from group II and does not like the deodorant?

[ ]

c) What is the probability that a customer has no opinion, if he/she is from group III?

[ ]

#74) The two-way table below shows the favorite leisure activities for 50 adults (30 women and 20

men).

Dance Sports TV Total

Women 16 6 8 30

Men 2 10 8 20

Total 18 16 16 50

a) Calculate the probability of an individual whose favorite leisure activity is sports, given that the individual is a woman. [ ]

b) (4 points) Calculate the probability of an individual whose favorite leisure activity is dance, given that the individual is a man. [ ]

20

c) What is the probability that someone’s favorite activity is watching TV? [ ]

d) What is the probability of a woman who likes watching TV?

[ ]

Section 8.2

#75) Let E and F be independent events, P(E)=0.11 P(F)=0.35. What is ( )

[ ]

#76) If E and F are independent, and if P(E)=0.3 and ( ) , find P(F)=?

[ ( ) ]

#77) If E and F are independent and P(E)=0.40, ( ) , find P(F)=?

[ ]

Section 8.3

#78) Events and are mutually exclusive and form a complete partition of a sample space with

( ) , ( ) . If E is an event with ( | ) ( | ) , compute ( )

[ ( ) ]

#79) Events , , and are mutually exclusive and form a complete partition of a sample space

with ( ) , ( ) ( ) . If E is an event with ( | )

( | ) , and ( | ) compute ( )

[ ( ) ]

#80) Events and are mutually exclusive and form a complete partition of a sample space with

( ) , and ( ) . If E is an event in with ( | ) and

( | ) , compute ( | ) and ( | )

[ ( | ) ( | ) ]

#81) Events , , and are mutually exclusive and form a complete partition of a sample space

with ( ) , ( ) ( ) . If E is an event in and ( | ) ,

( | ) and ( | ) . Compute ( | ) ( | ) and ( | )

[ ( | ) ( | ) ( | ) ]

21

Section 8.4

#82) Nine horses are competing in a race. Assume you have purchased a Trifecta ticket (In Trifecta, the

player selects three horses as first, second, and third place winners, and to win, those three horses must

finish the race in the precise order the player has selected). How many possibilities of a Trifecta exist?

[ ]

#83) From 1200 lottery tickets that are sold, three are to be selected for first, second, and third prizes.

How many possible outcomes are there?

[ ]

#84) Five girls and six boys are sitting in a row. How many ways can they sit if all girls are sitting together

and all boys are sitting together in the same row?

[ ]

#85) Ten individuals are candidates for positions of president, vice president of an organization. How

many possibilities exist for a pair of president and vice president?

[ ]

Section 8.5

#86) Three individuals are to be selected from a pool of nine English professors to form a panel of judges

in a poetry contest. How many different groups of three may be selected at random from the entire

pool?

[ ]

#87) Eleven people are waiting to get on a bus. When the bus arrives, the driver tells the awaiting

customers that there are only four seats available on the bus and no one is allowed to stand up when

the bus is moving. In how many different ways can any four customers get on the bus?

[ ]

#88) From a stack of 12 cards of 7 spades and 5 hearts, 3 cards are randomly selected. Find the

probability that there are 2 spades and 1 heart.

[ ]

Section 8.6

#89) A multiple choice test has 65 questions and each question has 6 possible answer choices, only 1 of

which is correct. If a student guesses on every question, how many questions should he expect to get

wrong?

[ ]

22

#90) A baseball player has a 0.25 batting average. If he is at bat 4 times, then:

a) What is the probability that the player will have at least 2 hits?

[ ]

b) What is the probability of at least 1 hit?

[ ]

c) What is the expected number of hits at bat?

[ ]

#91) Approximately 23% of the North American unexpected deaths are due to heart attacks. What is

the probability that exactly 4 of the next 10 unexpected deaths reported will be due to heart attacks?

[ ]

#92) For tax evasion cases that actually reached prosecution in 2006, the IRS reported a conviction rate

of 91.5%. Suppose 20 tax evasion cases prosecuted by the IRS in 2006 are randomly selected.

a) What is the probability that all 20 resulted in conviction?

[ ]

b) What is the probability that exactly 19 resulted in conviction?

[ ]

c) What is the probability that at least 18 resulted in conviction?

[ ]

d) What is the expected number of convictions from this sample?

[ ]