Math 10
Final Exam Review
Name
1. State whether each statement is true or false.
a. The set of days in a month is a well defined set.
b.
(
}e0
c.
4.5e{x|A;e A^and3<x<7}
d. {2,4,6,8,...} n {3,6,9,,12,...} ={ }
2. Express the set in roster form.
{jc|jcejVand2<Jc<5}
3. Express the set in set builder form.
{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
For problems #14-20, use the following information to fill in the Venn diagram and answer the questions.
t/={a,b, c, d, e,f}
A ={a, b, d, e}
B={a, c, e,f}
C={a, c, d}
4. Find AHB.
5. Find
6. Find
7. Find
Af]B'.
AUiBHC').
niAClB').
8. Find A-B.
9. Find the distinct number of subsets of U.
10. Find the distinct number of proper subsets of U.
11. Let set A = {Sue, Jo} and set B = {Bob, Tom, Moe}, find BxA
12. Use Verm diagrams to determine whether v4 U (5 fl C)'=
Show your work.
U 5') U
13. A recent survey of 510 people revealed the following information.
175 people like the color green
227 people like the color red
285 people like the color blue
100 people like the colors green and red
96 people like the colors green and blue
87 people like the colors red and blue
59 people like all 3 colors
Construct a Verm diagram and then determine
a) the number of people who only like the color green.
b) the number of people who only like 1 color.
c) the number of people who like the colors red and blue, but not green.
d) the number of people who like the colors green or blue, but not red.
e) the number of people who only like exactly 2 colors.
14. Show that the set has a cardinality of
H'4'9'16'25'-^
U C ) for all sets A, B, and C.
15. Determine whether each number is rational (R) or irrational (I),
a) 0.532
c) 0.646879347687937
d) V7
e) V36
16. State the property that each statement illustrates.
a) a(2 + b) = ia-2) + {a-b)
b) a + (2 + *)-(a + 2) + *
17. Without using a truth table, determine the truth value of the given compound statement.
Easter is in December and Christmas is in May, or Christmas is not in May and Halloween is in
October.
p: Easter is in December (F)
q: Christmas is in May (F)
r: Halloween is in October (T)
18. Construct a true table for the following statement.
Your dinner includes soup, or salad, or a baked potato and salad.
r:
P
9
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
19. Write an equivalent statement using De Morgan's Laws, the fact that (/? -> 9) o ( ~ p v q-), or the fact
that ~ (p -> I?) o (pA ~ q).
It is not true that both 2-^3 and 1 + 1 = 2.
p.- 2 < 3
q: 1 + 1 = 2
20. Write an equivalent statement using De Morgan's Laws, the fact that (p ^ ^) <=> (~ p v .7), or the fact
that ~ (p ^ g-) » (pA ~ q).
It is not true that if you don't eat your meat, then you will have some pudding.
p: you eat your meat
q: you will have some pudding
21. Write the converse, inverse and contrapositive of the following statement.
If you do, then I do.
P'
a. Converse:
b. Inverse:
c. Contrapositive:
22. Determine whether the two statements, (~ pv ~ 9) ^ r and ~{p^q)->r are equivalent using truth
tables.
P
9
r
T
T
T
F
F
T
T
F
T
F
T
F
T
F
T
T
T
F
F
F
F
1
T \
F
F j
{p/\q)->r
23. Write an argument in symbolic form. Then use a truth table to determine whether the argument is valid
or invalid.
If the water is boiling, then the pan is ready. The water is not boiling and the pan is ready. Therefore
the pan is ready.
q:
T
T
F
F
24. Determine whether an argument is valid or invalid by comparing its symbolic form with one of the
standard symbolic forms. State the name of the standard form you are using in the comparison.
Fallacy of t h e C o n v e r s e
P->
Q
Fallacy of the I n v e r s e
L a w of D e t a c h m e n t
P -> Q
p ^ q
g
l £
p
.-. p
.-. ~£?
. q
L a w of C o n t r a p o s i t i o n
p-^q
L a w of S y l l o g i s m
P
Q
P ' ^
Q-^r
..
~ p
1P._
:.p-*r
••• Q
If I'm hungry, then I will eat. I ate. Therefore, I'm hungry.
P--
25. Determine whether two circuits are equivalent.
Disjunctive Syllogism
9
26. The local club charges a $12.50 cover charge to get in, and then $5.50 per drink after that. How many
drinks can you afford on a $51 budget?
Let
Equation:
27. Joe Gambler lost $2 more than three times as much money on Saturday than he did on Friday.
AJl together, he lost $506. How much money did he lose each day?
Let
=
Then
=
Equation:
28. A rectangular garden measures 3 yards by 5 yards. A gardener wants to place a border of bark of width
jc around the garden. He has enough bark to cover 20 square yards. How wide of a border should be
made so that the border uses all the bark and what are the dimensions of the garden and border?
Let
X
Then
3 yards
X
5 yards
X
X
And
Equation:
29. The function for the height of a ball thrown into the air is h(t) = -16t^ + 4& + 160, where h(t) is the
height of the ball after t seconds.
a. Estimate the height of the ball after 2 seconds.
b. Estimate the time it takes for the ball to be 132 feet above the ground. Round to the nearest tenth of a second.
30. The exponential graph below shows the number of world-wide shipments of smart handheld devices,
P{t), in millions, for t years between 1999 through 2004, so 0<t <5. The general formula P{t) = P^e'"
models this growth,
a. Use the graph to find the initial population, Pg.
Wiirldwide Shipments of Smart
Huiitlheld Devices
43.54
b. Use a point on the graph and the general formula
P(t) = Pfi'^Xofindthe constant grow rate, k .
Round to the third decimal place.
=
30.23
-fl)
5
f.
M
I
2(1
7.02
'99
c. Use PQ and k to find the general formula P{t) - Pfi^.
20.98
14.57
10 11
'00
'01
02
03
•04
Simrct': liucrnati<ni;il l);nii <"uip.; M S N Miinc-yt L i i l r a i :
f.SI lnc.;iKltlrli<>na! research
d. Use the formula found in part c. to predict the number of shipments in 2005.
Round to nearest tenths
e. Use the formula found in part c. to find when the number of shipments will reach 1163 million
shipments. Round to the nearest year.
31. The radioactive element, Cobalt-60, decays at an exponential rate. If there are originally 800 grams of
Cobalt-60, then the amount of Cobalt-60, P(t), remaining after t years is modeled by the formula
P(0 = 80(fe*', where/t<0, since the amount Cobalt-60 decreases as time goes on.
a. If approximately 400 grams of Cobalt-60 remain after 5.3 years, find the decay rate, k. And, state the
function that models this case. Round to the third decimal place.
b. How much Cobalt-60 will remain after 11 years? Round to the nearest gram
C.
How long will it take to have only 1 gram remain?
Round to the nearest year.
32. If P(A)=0.2, P(B)=0.3, and P(A or B)=0.45, then find P(A and B).
33. Two fair dice are rolled. Find the probability of each of the following:
a. P(the sum of the two numbers is 9).
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
b. P(the sum of the two numbers is odd).
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
c. P(the sum of the two numbers is greater than 9).
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
d. P(the sum of the two numbers is odd AND greater than 9).
e. P(the sum of the two numbers is odd OR greater than 9).
34. ONE card is drawn at random from a standard 52-card deck. Find the probability of each of the
following:
Picture cards
a. P(the card is a A).
^ ;2•4n »
b. P(the card is not a •).
c. P(the card is a number greater than 5 and less than 9).
-Hearts
rMamonds
-Spades
d. P(the card is a number greater than 5 and less than 9 OR v).
e. P(the card is a number greater than 5 and less than 9 AND y).
35. Houses are listed as 1, 2, or 3 bedroom homes, with 1 or 2 bathrooms, and have either afireplace,a
heater, or baseboards. The sample space is listed below,
a. Find P(2 bathrooms).
,S;inipic Space
b. Find P(not having either 2 bathrooms nor
baseboards).
c. Find P(2 bathrooms / there are 2 or 3 bedrooms).
d. Find P(fireplace / there are 2 bedrooms).
36. A bag contains 7 cherry, 5 orange, and 3 lemon candies.
a. You reach in and take 3 pieces of candy at random at the same time. Find P(2 cherry and 1 lemon).
b. You reach in and take 3 pieces of candy at random one at a time without replacing the any of the
candies. Find P(l'' cherry, 2"*^ cherry, 3"^ lemon).
37. A license plate is to consist of 3 letters followed by 2 digits. Determine the number of different license
plates possible if
a. The T' letter has to be a vowel {a, e, i, o, u) and the 1"' number has to be greater than 6. Also,
repetition of letters and numbers is permitted.
'T'
^
^
^
b. The 1^' letter has to be a vowel {a, e, i, o, u) and the 1^' number has to be greater than 6. Also,
repetition of letters and numbers is NOT permitted.
'T^ ^
^
^
38. You want to arrange 6 books (math, science, fiction, history, biology, and English) on a bookshelf
a. How many different ways can all 6 books be arranged?
b. How many different ways can the 6 books be arranged if science isfirst,math is in the middle and
history is on the end?
c. How many different ways can the books be arranged if you threw out 2 books and only kept 4 of the
6 books?
39. Say there were 8 women and 9 men applying for 7 positions at a company.
a. Find the number of different combinations the company can hire the 7 people.
b. Find the number of different combinations the company can hire the 7 people, if 4 have to be women
and 3 have to be men.
40. Below is a list of the current gas prices in Santa Cruz from GasPriceWatch.com
a. Create a frequency distribution for the data.
Price
#
Station
I
Arco
705 Capitola Rd ©Tth Ave
Santa Cruz, CA 95062
Frequency
2
Price
Costco (Membership Required)
220 Sylvania Ave Harvey W Blvd
Santa Cruz, CA 95060
3
4
Valero
1319 Ocean St ©Franklin St
Santa Cruz, CA 95060
6
$3.59
$3.63
SavOn
1415 Ocean St gPryce St
Santa Cruz, CA 95060
5
$3.59
Rod & Ros Gas Mart
1024 Water St gSeabright Ave
Santa Cruz, CA 95062
$3.63
$3.63
Valero
1701 Capitola Rd
ei7th Ave
$3.63
Santa Cruz, CA 95062
7
Kwik Serv
1725 Mission St (SR-l) ePalm St
$3.63
Santa Cruz, CA 95060
8
b. Create a histogram for the data.
9
10
n
12
13
Valero
335 Mission St (CA-1) ©Highland
Santa Cruz, CA 95060
$3.65
Valero
za32 Mission St (CA-1) ©Miramar DR
Santa Cruz, CA 95060
$3.65
Rod & Ros
1024 Water St Seabright Ave
Santa Cruz, CA 95062
$3.65
US Pro
823 Mission St ©Walnut Ave
Santa CniZ, CA 95060
$3.65
Live Oak American
1200 17th Ave ©Brommer St
Santa Cmz, CA 95062
$3.65
76
904 Mission St ©Walnut Ave
Santa Cruz, CA 95060
14
15
Rotten Robbie (#3S)
1906 Mission St ©Berkshire Ave
Santa Caiz, CA 95060
18
0
$3.75
76
805 Ocean St ©Water St
Santa Cruz, CA 95060
17
$3.75
76 - Steve's
1500 Sequel Dr ©Paul Sweet Rd
Santa Cniz, CA 95065
16
$3.75
$3.79
Stiell-Gas & Car Wash
2700 Soquel Ave ©7th Ave
Santa Cruz, CA 95062
$3.79
Chevron
404 Soquel Ave Ocean St
Santa Cruz, CA 95062
$3.83
c. Create a circle graph for the data.
d. Find the mean of the data.
e. Find the median of the data,
f
Find the mode of the data.
g. Find the range of the data.
h. Find the standard deviation of the data.
Data
Data-Mean
(Data-Mean) 2
Total:
Standard Deviation:
i.
Pretend that the distribution is normal. Using the z-score, find the number of gas stations that lie
between thefirststandard deviation on the left and the third standard deviation on the right.