Download Final Exam Review WS

Final Exam Review WS
Discrete Math
Name_________________
Which reasoning process is shown in the following example?
1. All books by Stephen King have made the best seller list. Carrie is a novel by Stephen King.
Therefore, Carrie was on the best-seller list.
2. All books by Stephen King have made the best-seller list. There fore, it is highly probably
that the novel King is currently working on will make the best-seller list.
3. Estimate the number of seconds in a day.
4. The following table relates an adult’s body weight, in pounds, to his or her dosage of a
certain medication in milligrams.
Weight 100 125 150 175 200 225
Dosage 30
37
44
a. Use inductive reasoning to fill in the missing portions of the table.
b. What would be the dosage of a person who weighs 375 pounds?
5. A car rents for $290 per week plus $0.10 per mile. Find the rental cost for a three-week trip
of 600 miles.
6. A rental car company that rents cars for local-only use charges $30 plus $.20 for each mile
the rental car is driven. If a customer gives the rental attendant $100 for a charge of $42, how
many miles did the customer drive?
7. A coin is tossed 5 times. How many ways can it come up heads 4 times and tails once?
The bar graph below represents various colors of cars sold. Use the graph to answer the
questions.
8. Estimate how many more red cars were sold than
tan cars.
9. Which color sold under 20,000 cars?
10. A publishing company sold 46,265,592 books in 2005. Round the number of books sold to
the nearest ten million.
11. A couch sells for $820. Instead of paying the total amount at the time of purchase, the
same couch can be bought by paying $400 down and $60 a month for 12 months. How much is
saved by paying the total amount at the time of purchase?
12. Write the Hindu-Arabic numeral 965 in expanded form.
13. Express (5 x 105) + (1 x 104) + (8 x 103) + (6 x 102) + (7 x 101) + (8 x 1) as a Hindu-Arabic
numeral.
14. If the Babylonian numeral V stands for one and the Babylonian numeral < stands for ten,
then write the Babylonian numeral VVV <VV VVV as a Hindu-Arabic numeral.
Convert the numeral to a numeral in base ten.
15. 231four
16. 54012six
Convert the base ten numeral to a numeral in the given base.
17. 89 to base seven
18. 295 to base six
Perform the indicated operation in the indicated base.
32 four
342 five
19.
20.
12 four
343 five
21.
34 six
 5six
22.
453seven
 6seven
23. Determine whether the sentence is a statement.
Let your sister sit there.
24. Form the negation of the statement.
Warsaw is not the capital of Italy.
Let q and r represent the following statements and express the following statement symbolically.
q: One is a guitar player
r: The commute to work is not long.
25. One is not a guitar player.
26. The commute to work is long.
Given that p, q and r each represents a simple statement, write the indicated compound
statement in its symbolic form or the symbolic statement in words.
27.
p: Tosca is an opera
q: Carmen is an opera.
Tosca is an opera and Carmen is not an opera.
28.
p: The car has been repaired.
q: The kids are home.
r: We will visit Aunt Tillie
r   p  q
Construct a truth table for the statement.
29.  r  q    r  q 
30.
 p  q   p  q
31. Determine if the set is the empty set.
 x x is a living U.S. president born after 1700
Determine whether the statement is true or false.
32. 15 1, 2,3,...,10
33. 11,3,5,7,9
34. Express the set using set-builder notation. A = {200, 201, 202, ..., 2000}
35. Write  or  in the blank so that the resulting statement is true.
{4, 34, 39} ____ {15, 34, 39, 49}
Use the Venn diagram to list the elements of the set in roster form.
36. List the elements of B, A, and U
37. A B
Let U = {q, r, s, t, u, v, w, x, y, z}
A = {q, s, u, w, y}
B = {q, s, y, z}
List the elements in the set.
38. A B '
C = {v, w, x, y, z}
39.
A
B '
40. Find the prime factorization of 1162.
Find the greatest common divisor of the numbers.
41. 126 and 231
42. 414 and 966
43. A store owner wishes to stack books into equal piles, each pile containing only one title.
there are 18 books of one title and 42 books of another title in a shipment. What is the largest
number of books that can be stacked in each pile?
Find the least common multiple of the numbers.
44. 36 and 54
45. 245 and 420
Use the order of operations to find the value of the expression.
46. -2 · 9 + (-4) · 9
49.
7  1
 
17  2 
Simplify.
51. 175
47. (2 + 1)3 - (3 - 1)3
 3  1 
48.      
 10   21 
1 1 6
50.      
5 6 9
52.
2  2 98  4 8
53. 4 175  4 20  4 125  28
54. Write the first six terms of the arithmetic sequence with the first term a 1 = 6 and d = 5.
55. Find a8 for the arithmetic sequence when a1 = -10 and d = -3
56. Write a formula for the general term of the arithmetic sequence. Then use the formula to
find a20.
-1, 1, 3, 5, 7, ...
57. Write the first six terms of the geometric sequence where a 1 = 6 and r = 4.
58. Find a6 for the geometric sequence when a1 = 7 and r = 5
59. Find a6 for the geometric sequence when a1 = -4 and r = -2
60. As part of a fundraiser, a local charity has sold 6 raffle tickets, with a goal of selling 100
tickets. What percentage of the goal has been sold?
61. A dress regularly sells for $130. The sale price is $84. Find the percent decrease of the
sale price from the regular price.
62. The principal P is borrowed at a simple interest rate r for a period of time t. Find the simple
interest owed for the use of the money. P = $5000, r = 6%, t = 3 years.
63. The principal represents an amount of money deposited in a savings account subject to
compound interest at the given rate. Find how much money will be in the account after the
given number of years, and how much interest was earned. Principal: $8000, rate: 4%,
compounded: quarterly, time: 2 years.
64. A mother invests $2000 in a bank account at the time of her daughter's birth. The interest
is compounded quarterly at a rate of 8%. What will be the value of the daughter's account on
her twentieth birthday, assuming no other deposits or withdrawls are made during this period?
65. Find the value of an annuity that has a periodic deposit of $100 at the end of each year, a
rate 5% compounded annually, and a time of 9 years.
66. The price of a home is $260,000. The bank requires a 20% down payment and one point at
the time of closing. The cost of the home is financed with a 15-year fixed-rate mortgage at 7%.
a. Find the required down payment.
b. Find the amount of the mortgage.
c. How much must be paid for the one point at closing?
d. Find the total cost of interest over 15 years, to the nearest dollar.
67. A restaurant offers 7 entrees and 6 desserts. In how many ways can a person order a twocourse meal?
68. License plates in a particular state display 2 letters followed by 2 numbers. How many
different license plates can be manufactured?
69. There are 7 performers who are to present their acts at a variety show. One of them insists
on being the first act of the evening. If this request is granted, how many different ways are
there to schedule the apperances?
70. In a contest in which 8 contestants are entered, in how many ways can the 5 distinct prizes
be awarded?
71. A record club offers a choice of 7 records from a list of 45. In how many ways can a
member make a selection?
72. From 10 names on a ballot, a committee of 4 will be elected to attend a political national
convention. How many different committees are possible?
73. A die is rolled. The set of equally likely outcomes is {1, 2, 3, 4, 5, 6}. Find the probability of
getting a 7.
74. You are dealt one card from a standard 52-card deck. Find the probability of being dealt a
picture card.
75. A single die is rolled twice. Find the probability of getting two numbers whose sum is
greater than 10.
76. A group consists of 6 men and 5 women. Three people are selected to attend a
conference. In how many ways can 3 people be selected from this group of 11? In how many
ways can 3 men be selected from the 6 men? Find the probability that the selected group will
consist of all men.
The Theater Society members are voting for the kind of play they will perform next semester: a
comedy (C), a drama (D), or a musical (M). Their votes are summarized in the following
preference table.
Number of votes
10
8
4
2
First Choice
C
M M D
Second Choice
D
C
D M
Third Choice
M
D
C
C
77. Which type of play is selected using the plurality method?
78. Which type of play is selected using the Borda count method?
79. Which type of play is selected using the plurality-with-elimination method?
80. Which type of play is selected using the pairwise comparison method?
Use the following table to solve 81 - 85.
Number of votes
180 100 40 30
First Choice
A
B
D
C
Second Choice
B
D
B
B
Third Choice
C
A
C
A
Fourth Choice
D
C
A
D
81. Using the plurality method, which candidate wins the election?
82. Using the Borda count method, which candidate wins the election?
83. Using the plurality-with-elimination method, which candidate wins the election?
84. Using the pairwise comparison method, which candidate wins the election?
85. Which candidate has a majority of first-place votes? Based on your answers to 81 - 84,
which voting system violates the majority criterion in this election?
A country is composed of four states, A, B, C, and D. The population of each state is given in
the following table. Congress will have 200 seats, divided among the four states according to
their respective populations.
State
A
B
C
D
Population
3320
10,060
15,020
19,600
86. Use Hamilton's method to apportion the congressional seats.
87. Use Jefferson's method to apportion the congressional seats.
88. Use Adam's method to apportion the congressional seats.
89. Use Webster's method to apportion the congressional seats.
90. Draw a graph that models the connecting relationships in the floor plan. Use vertices to
represent the rooms and the outside, and edges to represent the connecting doors.
B
A
C
G
E
D
F
91. Determine whether the graph has an Euler path, an Euler circuit or neither. If the graph has
an Euler path or circuit find one.
A
B
C
E
D
F
92. Determine if the graph must have Hamilton circuits. If the graph must have Hamilton
circuits determine the number of circuits.
A
E
B
F
H
G
C
D
Find the sum or difference.
1 2 5  2 7 3
93. 


3 2 1   1 2 5 
 0 2   5 6 
94. 


 4 1  9 1
Solve each matrix equation.
95.
2
6 8   1 2 4  X
Find the value of each variable.
9   7 w  1
x  5

97. 
t  2 8  r
1 
 4
7 1
 4 9
X 
96. 


0 8 
 3 11
 4  t
98. 
 r
2 y   2t
11


w  5  2r  12 9 
Use matrices A, B, C, and D to find each scalar product and sum, or difference, if possible.
 2 1 
 4 0
 6 1 0 8
 1 3
5 2 


A
D
B

C





 2 2
 4 3 7 11
3 6 
 2 4


 1 1
99. 3A
100. B - 2A
101. AB
102. BA
103. AC - BD
104. 4B - 3D
Evaluate the determinant. For the 2X2 matrices find the inverse if possible.
 1 0 2
6 1 
 5 2
105. 
106. 
107.  1 0 1 


0 4 
10 4
 1 2 0 
Solve each system using Cramer's Rule.
 x  y  z  1

108.  x  y  3z  3
2 x  y  2 z  0

 x  y  z  3

109.  3y   z  4
2 x  y  2 z  5

110. Use row-reduced Echelon form to solve the system of equations.
3x  y  7

5 x  2 z  3