MGF 1106 Liberal Arts Mathematics
Final Review
1. Given U = { x x is a counting number less than 11} , A = {2, 4, 6, 8, 10} , B = {1, 3, 5, 7} , and
C = {1, 4, 7, 10} , which of the following is not true?
[a] 0∈ A
[b] 8∉ B
[c] 4∈( B U C )
[d] ∅ ⊆ U
2. Given U = { x x is a counting number less than 11} , A = {2, 4, 6, 8, 10} , B = {1, 3, 5, 7} , and
C = {1, 4, 7, 10} , find the set:
( A U C )′ I B
[a] {1, 3, 5, 7}
[b] {3, 5}
[c] {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
[d] {1, 2, 4, 6, 7, 8, 10}
3. Given U = { x x is a counting number less than 11} , A = { x x is an even number between 3 and 9} ,
and B = {2, 4, 6} determine if they are equal, equivalent, both, or neither.
[a] equal
[b] neither
[c] both
[d] equivalent
4. Let U = {lion, tiger, panther, leopard, cheetah} , A = {lion, tiger} , B = {lion, panther} , and
C = {panther, leopard, cheetah} . Determine A I ∅ .
[a] {panther, leopard, cheetah}
[b] ∅
[c] {lion, tiger}
[d] {∅}
1
5. In a poll of 560 voters, 53% said they prefer Candidate A. What is the margin of error for this
survey?
[a] ± 0.04%
[b] ± 23.7%
[c] ± 7.28%
[d] ± 4.0%
6. Find the median of the group of data items: 96, 96, 94, 48, 77, 96
[a] 94
[b] 95
[c] 96
[d] 48
7. Find the range of the group of data items: 9, 3, 1, 5, 4, 2
[a] 7
[b] 8
[c] 5
[d] 4
8. The scores on a test are normally distributed with a mean of 103 and a standard deviation of 15.
Using the 68-95-99.7 Rule, find the percentage of scores that are below 133.
[a] 95%
[b] 47.5%
[c] 97.5%
[d] 84%
9. A set of data items is normally distributed with a mean of 25 and a standard deviation of 4. What
data point corresponds to a z-score of –1.2?
[a] 20.2
[b] 29.8
[c] 21
[d] 19.8
2
10. What is the next number in the list of numbers? 1, 2, 4, 7, 11, 16, . . .
[a] 17
[b] 21
[c] 22
[d] 32
11. Write the contrapositive of the given statement: “If I study hard, I will get a good grade on the test.”
[a] If I do not study hard, I will get a good grade on the test.
[b] If I do not study hard, I will not get a good grade on the test.
[c] If I get a good grade on the test, then I studied hard.
[d] If I do not get a good grade on the test, then I did not study hard.
12. Write the inverse of the statement: “If you like plays, then you will like Macbeth.”
[a] If you do not like plays, then you will like Macbeth.
[b] If you do not like plays, then you will not like Macbeth.
[c] If you like Macbeth, then you will like plays.
[d] If you do not like Macbeth, then you do not like plays.
13. What can be deduced from these premises? “If he has enough votes, then he will win the election. If
he wins the election, then he will move into the White House…”
[a] If he moves into the White House, then he had enough votes.
[b] He will move into the White House.
[c] If he has enough votes, then he will move into the White House.
[d] If he wins the election, then he had enough votes.
14. A license plate consists of 2 letters followed by 4 digits. If the two letters must be different and the
first digit cannot be 0, how many license plates are possible?
[a] 6,084,000
[b] 6,760,000
[c] 1,965,600
[d] 5,850,000
3
15. The following Venn diagram describes the cars on a used car lot. How many cars on the lot are not
GM cars?
CARS
[a] 11
[b] 30
GM
[c] 37
25
[d] 41
4
Red
7
30
16. If set A has 32 subsets, the n ( A ) = ?
[a] 3
[b] 32
[c] 4
[d] 5
17. Let U = {lion, tiger, panther, leopard, cheetah} , A = {lion, tiger} , B = {lion, panther} , and
C = {panther, leopard, cheetah} . Determine A I C ′ .
[a] {lion, tiger, panther, leopard, cheetah}
[b] {lion, tiger}
[c]
{}
[d] {panther, leopard, cheetah}
18. In a survey, 225 people were asked if they like either of two brands of soft drink, brand A or brand
B. 102 liked brand A, 97 liked brand B, and 75 liked neither brand A nor brand B. How many people
liked either brand A or brand B?
[a] 199
[b] 15
[c] 49
[d] 150
19. Write the negation of the statement: “If it is snowing, then it is cold.”
[a] If it is not cold, then it is not snowing.
[b] If it is not snowing, then it is not cold.
[c] It is snowing and it is not cold.
[d] It is not snowing and it is cold.
4
20. In a survey, 225 people were asked if they like either of two brands of soft drink, brand A or brand
B. 102 liked brand A, 97 liked brand B, and 75 liked neither brand A nor brand B. How many people
liked both brand A and brand B?
[a] 49
[b] 15
[c] 199
[d] 150
21. Let p, q, and r represent the following simple statements:
p: I am in college.
q: I like math.
r: I will major in math.
Which of the following expresses the symbolic compound statement in words? ( ~ q ∧ ~ p ) → ~ r
[a] If then I do not like math and I am in college. Therefore I will not major in math.
[b] I do not like math and I am not in college. Therefore I will not major in math.
[c] I do not like math or I am in college. Therefore I will not major in math.
[d] I do not like math or I am not in college. Therefore I will not major in math.
22. If p is true, q is false, and r is true, which of the following is false?
[a] ( p ∧ ~ q ) ∨ ~ r
[b] ( p ∧ q ) → r
[c] ( p ∧ q ) ∨ r
[d] p → ( q ∨ ~ r )
23. Which argument is valid?
[a] All people who work hard will succeed.
My electricity bill is not high.
.
∴ No people who take naps will succeed.
[b] All people who work hard will succeed.
All smart people will succeed.
.
∴ All smart people work hard.
[c] All people who work hard will succeed.
All people who enjoy their work will work hard. .
∴ All people who enjoy their work will succeed.
[d] All people who work hard will succeed.
John is successful.
.
∴ John works hard.
5
24. Two cities are 3,520 decimeters apart. Convert this distance to hectometers.
[a] 352 hm
[b] 0.3520 hm
[c] 3.52 hm
[d] 35.2 hm
25. Find the volume of the right circular cylinder with a diameter of 14cm and a height of 12cm.
[a] 1847cm3
[b] 7389cm3
[c] 616cm3
[d] 56cm3
26. Use the given right triangle to find the trigonometric function: sin A
A
5
13
14
[b]
13
5
[c]
12
12
[d]
13
[a]
52
48
C
20
B
27. If a 40-foot ladder leaning against a vertical wall makes a 75 degree angle with the horizontal
ground, how far up the wall does the ladder reach? (Round the answer to the nearest foot.)
[a] 10 ft
[b] 39 ft
[c] 149 ft
[d] 41 ft
28. The figure shows two parallel lines intersected by a transversal. Find the measure of angle A.
[a] 40°
120°
[b] 60°
[c] 120°
A
[d] 240°
6
29. You are the judge at a dog show. There are 18 entrants, and you must choose the first, second, third,
and fourth place winners (no ties are allowed). How many ways can this be done?
[a] 3060
[b] 72
[c] 104,976
[d] 73,440
30. You are dealt one card from a standard 52-card deck. Find the probability of being dealt a red card.
1
4
1
[b]
2
1
[c]
13
1
[d]
52
[a]
31. You are dealt one card from a standard 52-card deck. Find the probability of being dealt a queen or a
king.
1
169
1
[b]
13
1
[c]
26
2
[d]
13
[a]
32. You are dealt two cards from a 52-card deck, in succession, without the first card being replaced
before the second is dealt. Find the probability of getting a 5 followed by a king.
1
169
4
[b]
663
2
[c]
13
1
[d]
26
[a]
7
33. Your dad offers to play a game with you. He will pick a number between 1 and 6, and then one of
you will roll a fair die. If the die lands on the number he chose, you pay him $10. If it lands on any
other number, he pays you $2. What is your expected value in playing this game?
[a] $0
[b] $2
[c] −$10
[d] $1.67
34. If a single die is rolled, find the odds in favor of getting a 3.
[a] 1: 6
[b] 6 :1
[c] 1: 5
[d] 5 :1
35. A fair die is tossed one time. What is the probability of getting an even number given that the
number is greater than 3?
1
2
2
[b]
3
1
[c]
3
1
[d]
6
[a]
36. A club consists of five men and seven women. Three members are selected at random to attend a
conference. Find the probability that the selected group consists of three men.
3
35
1
[b]
220
1
[c]
22
3
[d]
5
[a]
8
37. The weights, in pounds, of a group of dogs are 34, 37, 42, 57, 64 and 82. Find the standard deviation
for this set of weights.
2
( data item − mean )
∑
Standard deviation =
[a] 16.9
n −1
[b] 18.5
[c] 24.8
[d] 343
38. Use the following scatter plot to estimate the value of the correlation coefficient.
y
[a] r = −1
12
[b] r = −0.8
10
[c] r = 0
8
••
6
[d] r = 0.7
••
•
4
•
2
1
2
3
4
5
•
6
x
39. The circle graph below shows the percentages of students living on and off campus at a large
University. If the enrollment at the University is 30,137, what is the number of students living off
campus?
68.8%
off campus
[a] 20,734
[b] 9,403
31.2%
on campus
[c] 69
[d] 18, 956
40. Find the area and circumference of a circle with diameter 20m.
[a] 1257 m 2 ; 126m
[b] 126m 2 ; 1257m
[c] 314m 2 ; 63m
[d] 63m 2 ; 314m
41. A rectangular plot of land measures 22,563 square feet. That same rectangular plot measures how
many square yards?
[a] 7521 yd 2
[b] 1880.25 yd 2
[c] 2507 yd 2
[d] 203067 yd 2
9
42. Find the surface area of a cube with a side of length 24m.
[a] 512m 2
[b] 13824m 2
[c] 72m 2
[d] 3456m 2
43. Convert 86° F to its equivalent temperature on the Celsius scale. (Round to the nearest tenth.)
[a] 97.2°C
[b] 186.8°C
[c] 30.0°C
[d] 65.6°C
44. Given S = {a, b, c, d , e} , how many subsets can be made.
[a] 32
[b] 31
[c] 24
[d] 25
45. Amy, Jean, Keith, Tom, Susan, and Dave have all been invited to a birthday party. They arrive
randomly and each person arrives at a different time. In how many ways can they arrive? In how
many ways can Jean arrive first and Keith last? Find the probability that Jean will arrive first and
Keith will arrive last.
1
12
1
[b] 720; 24;
30
1
[c] 120; 6;
20
1
[d] 720; 15;
48
[a] 120; 10;
46. Select the statement that is logically equivalent to: “Everything is not expensive or I am broke.”
[a] If everything is expensive, then I am broke.
[b] If everything is expensive, then I am not broke.
[c] Everything is not expensive and I am broke.
[d] If everything is not expensive, then I am broke.
10
47. Most of the following symbols would create a false statement. Which one of these symbols would
create a true statement? {1,2} _____ {3, 2, 2,1,1,1}
[a] ⊄
[b] =
[c] ∈
[d] ⊆
48. A class of 10 male students and 12 female students must select groups to work with to complete
their class project. How many groups are possible if the groups must consist of one male and two
female students?
[a] 660
[b] 880
[c] 1320
[d] 1540
49. Write the converse of the statement: “If you are physically fit, then you work out at the gym.”
[a] If you are physically fit, then you work out at the gym.
[b] If you work out at the gym, then you are physically fit.
[c] If you are not physically fit, then you work out at the gym.
[d] If you do not work out at the gym, then you are not physically fit.
50. A student survey estimates that the probability of passing Math or English is 0.9, the probability of
passing English is 0.8, but the probability of passing both is 0.6. What should be the estimate of the
probability of passing Math?
[a] 0.23
[b] 0.10
[c] 0.70
[d] 0.50
51. All of the following arguments have true conclusions, but one of the arguments is not valid. Select
the argument that is not valid.
[a] All mathematics is abstract. Physics is not abstract. Therefore, physics is not mathematics.
[b] All mathematics is abstract. Algebra is mathematics. Therefore, algebra is abstract.
[c] All algebra is mathematics. Geometry is not mathematics. Therefore, geometry is not algebra.
[d] All algebra is abstract. All mathematics is abstract. Therefore, all algebra is mathematics.
11
52. A college president wants to find out which courses are popular with students. The president decides
to conduct a survey of a sample of the students. What procedure would be most appropriate for
obtaining a statistically unbiased sample of the college’s students?
[a] Survey a random sample of students from the English department.
[b] Survey a random sample of students from a list of the entire student body.
[c] Have students voluntarily mail in their preferences.
[d] Randomly survey the first hundred students to register for classes.
53. What kind of correlation would you expect when comparing the economies of different nations to
the infant mortality rates of those nations?
[a] Perfect positive correlation
[c] Negative correlation
[d] Perfect negative correlation
54. Which of the following is true according to the graph?
Number of People (in thousands)
[b] Positive correlation
25
20
15
10
5
1
[a] The graph is based on a sample of approximately 62 thousand people.
3
5
2
4
Years of Education
6
Beyond High School
[b] More people had 4 years of education beyond high school than 3 years.
[c] If the sample is truly representative, then for a group of 50 people, we can expect about 32 of
them to have one year of education beyond high school.
[d] The percent of people with years of higher education greater than those shown by any
rectangular bar is equal to the percent of people with years of education less than those
shown by that bar.
55. The center of a circle is also the center of a square of side 6cm . The circle passes through the four
vertices of the square. Find the area of the region that is inside the circle and outside the square (use
π = 3.14 ).
[a] 77.04cm 2
[b] 20.52cm 2
[c] 10.88cm 2
6cm
[d] 7.74cm 2
12
56. Construct a truth table for the statement: ~ ( q → ~ p )
[a]
[c]
p
q
T
T
F
F
T
F
T
F
p
q
T
T
F
F
T
F
T
F
~ p q →~ p ~ ( q → ~ p)
F
F
T
T
T
F
F
F
F
T
T
T
~ p q →~ p ~ ( q → ~ p)
F
F
T
T
F
T
T
T
[b]
[d]
T
F
T
F
p
q
T
T
F
F
T
F
T
F
p
q
T
T
F
F
T
F
T
F
~ p q →~ p ~ ( q → ~ p)
F
F
T
T
F
F
T
T
T
T
F
F
~ p q →~ p ~ ( q → ~ p)
F
F
T
T
F
T
T
T
T
F
F
F
57. Given the following statements:
p: I am an environmentalist.
q: I recycle my aluminum cans.
r: I recycle my newspapers.
Select the symbolic representation of the statement: “If I recycle my aluminum cans or newspapers,
then I am an environmentalist.”
[a]
[b]
[c]
[d]
(q ∨ r ) ↔ p
(q ∨ r ) → p
(q ∧ r ) → p
(q ∧ r ) ↔ p
58. How many arrangements can be made using 4 letters of the word HYPERBOLAS if no letter is to be
used more than once?
[a] 302, 400
[b] 151, 200
[c] 5040
[d] 210
59. Find the cardinal number for the set:
{27,
29, 31, 33, 35}
[a] 5
[b] 4
[c] 6
[d] 27
13
60. You are dealt one card from a 52-card deck. Find the probability that you are not dealt a diamond.
1
4
2
[b]
5
4
[c]
13
3
[d]
4
[a]
61. The table below describes the smoking habits of a group of asthma sufferers:
Nonsmoker Light Smoker Heavy Smoker
Female
Male
382
382
88
73
86
61
If one person is randomly selected, find the probability that the person is a woman given that the
person is a light smoker.
[a] 0.086
[b] 0.547
[c] 0.159
[d] 0.257
62. Given the ages of teachers in the mathematics department of a certain high school, find the a. mean
b. median c. mode and d. midrange: 28, 54, 35, 58, 28, 54, 28, 50, 50, 44
[a] a. 26.9; b. 47; c. 28; d . 43
[b] a. 42.9; b. 47; c. 50; d . 43
[c] a. 31.9; b. 47; c. 28; d . 43
[d] a. 42.9; b. 47; c. 28; d . 43
63. An architect is considering bidding for the design of a new shopping mall. The cost of drawing plans
and submitting a model is $10,000. The probability of being awarded the bid is 0.07, and anticipated
profits are $100,000, resulting in a possible gain of this amount minus the $10,000 cost for plans and
a model. What is the expected value in this situation?
[a] $6300
[b] $7000
[c] $6000
[d] −$3000
14
64. The bar graph shows the percentage of adults that use the Internet for specific tasks. Use the graph to
represent the set of tasks in which usage exceeds 20%.
E-mail 33%
[a] {e-mail, information search}
Information Search 28%
[b] { job, school}
News 24%
[c] {e-mail, information search, news}
Job 17%
[d] {e-mail, information search, news, job}
School 12%
65. Set A contains 6 elements, set B contains 13 elements, and 4 elements are common to sets A and B.
How many elements are in A U B?
[a] 15
[b] 19
[c] 14
[d] 16
66. Write a negation of the statement: “Cats are lazy or dogs aren't friendly.”
[a] Cats aren't lazy or dogs are friendly.
[b] Cats aren't lazy or dogs aren't friendly.
[c] Cats are lazy and dogs are friendly.
[d] Cats aren't lazy and dogs are friendly.
67. A club elects a president, vice-president, and secretary-treasurer. How many sets of officers are
possible if there are 11 members and any member can be elected to each position? No person can
hold more than one office.
[a] 495
[b] 990
[c] 7920
[d] 330
68. Express the set using the roster method. Remember that
{ x x ∈ and x is greater than 3}
[a] {3, 4, 5, K}
[b] {4, 6, 8, K}
[c] {4, 5, 6, K}
[d] {4, 5, 6}
15
represents the set of natural numbers:
69. Numbered disks are placed in a box and one disk is selected at random. If there are 6 red disks
numbered 1 through 6, and 4 yellow disks numbered 7 through 10, find the probability of selecting a
yellow disk, given that the number selected is less than or equal to 3 or greater than or equal to 8.
3
10
1
[b]
2
3
[c]
5
3
[d]
4
[a]
70. Write the contrapositive of the statement: “If I am in the city of Jokdrifa, then I am on the planet,
Plochus.”
[a] If I am not in the city of Jokdrifa, then I am on the planet Plochus.
[b] If I am not in the city of Jokdrifa, then I am not on the planet Plochus.
[c] If I am not on the planet Plochus, then I am not in the city of Jokdrifa.
[d] If I am not on the planet Plochus, then I am in the city of Jokdrifa.
71. Construct a truth table for the statement:
[a]
[c]
p
q
~q
p∧ ~ q
T
T
F
F
T
F
T
F
F
T
F
T
F
T
F
T
p
q
~q
p∧ ~ q
T
T
F
F
T
F
T
F
F
T
F
T
F
T
F
F
p∧ ~ q
[b]
[d]
p
q
~q
p∧ ~ q
T
T
F
F
T
F
T
F
F
T
T
T
F
T
F
F
p
q
~q
p∧ ~ q
T
T
F
F
T
F
T
F
F
T
F
T
T
T
F
F
72. Identify the pattern and determine the next number: 1, 4, 9, 16, …
[a] 25
[b] 32
[c] 50
[d] 36
16
73. Find the measure of angle B for the triangle with angle A = 22o and angle C = 38o.
A
[a] 150°
C
[b] 30°
[c] 90°
B
[d] 120°
74. Set A contains 11 letters and 6 numbers, set B contains 9 letters and 7 numbers. Four letters and 3
numbers are common to sets A and B. How many elements are in A or B?
[a] 24
[b] 26
[c] 40
[d] 33
75. Write the converse of the statement: “If you are not with me, then you are against me.”
[a] If you are not against me, then you are with me.
[b] If you are with me, then you are not against me.
[c] If you are not with me, then you are not against me.
[d] If you are against me, then you are not with me.
76. Find the probability of correctly answering the first 5 questions on a multiple choice test if random
guesses are made and each question has 4 possible answers.
5
4
4
[b]
5
1
[c]
625
1
[d]
1024
[a]
77. An IRS auditor randomly selects 3 tax returns from 44 returns of which 12 contain errors. What is
the probability that she selects none of those containing errors?
[a] 0.3847
[b] 0.0203
[c] 0.3745
[d] 0.0166
17
78. Hanna owns 23 acres of land of which she rents to a farmer for $3853 per acre per year. Her
property taxes are $920 per acre per year. How much profit does she make on the land each year?
[a] $67, 459
[b] $87, 699
[c] $109,779
[d] $89,539
79. Which diagram below shows the set A I ( B U C ′ ) ?
[a]
[b]
[c]
[d]
80. Given that AB CD , which of the following statements is true for the figure shown?
[a] y = 37°
x
[b] x = y
C
[c] w = 53°
[d] v = w
B
y
v
w
D
A
z
E
18
53°
81. Select the statement that is not logically equivalent to “If you wear shoes, then you must wear
socks.”
[a] You do not wear shoes or you must wear socks.
[b] If you must not wear socks, then you do not wear shoes.
[c] You wear shoes and you don't wear socks.
[d] It is not true that you wear socks and you do not wear shoes.
82. If the probability of Sam winning an election is
5
, what are the odds against Sam winning?
12
[a] 5:12
[b] 12:5
[c] 5:7
[d] 7:5
83. Given a standard deck of 52 cards, what is the probability of drawing exactly two aces and exactly
two kings if five cards are selected at random and without replacement?
C2 × 4 C2 × 44 C1
52 C5
[a]
4
[b]
8
[c]
4
[d]
5 3
C4 × 44 C1
52 C5
P2 × 4 P2 × 44 P1
52 P5
P × 5 P2
52 P5
84. According to the American Medical Association, in 1997, 22% of physicians in the U. S. were
female, 18% were under 35 years old, and 6% were female under 35 years old. If one of these
physicians is randomly selected, find the probability that he or she is a female or under 35 years old.
[a] 0.04
[b] 0.34
[c] 0.46
[d] 0.06
85. Which one of the following is true?
19
[a] If the same number is added to each item in a data set, the standard deviation does not change.
[b] If each number in a data set is multiplied by 4, the standard deviation is doubled.
[c] It is possible for a set of scores to have a negative standard deviation.
[d] Data sets with different means cannot have the same standard deviation.
86. Suppose a circular pizza with a diameter of 6 inches costs $3.25. Assuming that the cost depends
upon the size (area) of the pizza, find the cost of a similar pizza whose diameter is one foot.
[a] $6.50
[b] $19.50
[c] $9.75
[d] $13.00
87. The Chamber of Commerce needs to estimate the average income of the citizens of a city. To do so,
the Chamber decides to conduct a survey. Which of the following procedures would be most
appropriate for selecting a statistically unbiased sample?
[a] Survey a random sample of people at a certain shopping mall in the city.
[b] Randomly select geographic regions of the city and then survey a random sample of the
people within the regions.
[c] Select the largest employer in the city and survey a random sample of its employees.
[d] Survey all of the residents of the largest subdivision in the city.
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Additional Final Exam Review Questions
MGF 1106
Topic
Page
Exercises
Sets
(p. 102-104)
3-6, 8, 9, 11, 14-25, 29-39 odds, 45, 47, 55,
61
Logic
(p. 9)
(p. 190-192)
6-11
1-3, 7-14, 17, 19, 21, 24, 27-45 odds, 51, 53,
57, 59, 65, 67
Counting/Probability
(p. 673-767)
1, 5, 7, 16, 17, 19, 21, 26, 27-39 odds, 44,
48, 50, 52, 55, 57, 59, 71, 74-76, 81, 86, 90,
91, 101, 107
Statistics
(p. 742-745)
11, 13, 16, 19, 25, 31, 35, 37, 42, 45, 46, 50,
55, 60, 63-69, 70, 71
Measurement/Geometry (p. 534-535)
(p. 601-605)
3, 5-10, 12, 33-36, 47, 49, 51, 55
5, 9, 11-13, 15, 17, 19, 22, 24, 26, 31, 36,
38, 42, 43, 45, 46, 50, 51, 52, 55, 57, 59, 63
In addition to these problems, you will want to review your in-class exams.
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