Download Sample Mastery #2

Name: ________________________ Class: ___________________ Date: __________
ID: A
Sample Mastery #2
Multiple Choice
Identify the choice that best completes the statement or answers the question.
1. Determine the number of arrangements for the word BOXCAR.
a. 21
c. 720
b. 46 656
d. none of the above
2. Determine the number of arrangements for the word CANADA.
a. 240
c. 21
b. 720
d. 120
3. Determine the number of three letter arrangements using the letters of the word METAPHOR.
a. 40 320
c. 336
b. 56
d. 512
4. Determine the number of ways that a prime minister, secretary, treasurer, and publicity minister could be
chosen from an art club of 12 members.
a. 48
c. 20 736
b. 495
d. 11 880
5. Determine the number of ways you could line up 3 orange marbles, 5 blue marbles, and 1 purple marble.
a. 362 880
c. 504
b. 15
d. 24 192
6. A bag contains 3 red blocks, 2 green blocks, and 4 blue blocks. Determine how many ways that all of them can
be drawn, one at a time, without replacement.
a. 362 880
c. 15 120
b. 1260
d. none of the above
7. Determine the number of ways the 8 members of the Junior Jazz Band can stand in a line if Val must be first,
Tim sixth, and Tricia last.
a. 32 768
c. 56
b. 40 230
d. none of the above
8. Determine the number of ways that the 12 members of the boys' baseball team can be lined up if Joe, Tanner,
and Josh must all be together.
a. 1 209 600
c. 362 880
b. 604 800
d. 220
9. A person buys a ticket for a draw which has tickets made up of four digits from 0 to 9, and one winning ticket
is drawn. Assuming that all the tickets have been sold, determine the probability of the person winning the
draw.
1
1
a.
c.
3024
6561
1
1
b.
d.
10 000
5040
1
Name: ________________________
ID: A
10. Express 15 × 14 × 13 in a different manner.
a.
P(15, 13)
c.
b.
P(15, 12)
d.
15!
12!
15!
2!13!
11. The 7 members of a chess club line up for a picture. Determine the probability that Mckenzie and Johann will
be beside each other.
2
1
a.
c.
7
3
1
1
b.
d.
6
7
12. A captain and co-captain for a fencing team are chosen from a hat with the names of all 11 members in the hat.
Determine the probability that Lauren and Isabel are chosen as captain and co-captain respectively.
2
1
a.
c.
121
110
1
1
b.
d.
55
121
13. The letters of the word SIMILE are scrambled. Determine the probability that the word is spelled exactly
backwards.
1
1
a.
c.
360
6
1
b.
d. none of the above
720
14. The letters of the word CHEMISTRY are put in a hat and three letters are drawn, one at a time, without
replacement. Determine the probability that the C and R are chosen.
14
1
c.
a.
243
252
1
1
b.
d.
24
12
15. The face cards of a deck are shuffled and two cards are drawn, one at a time, without replacement. Determine
the probability that both cards are diamonds.
1
1
a.
c.
132
24
1
b.
d. none of the above
66
2
Name: ________________________
ID: A
16. Determine the number of ways that four objects can be chosen from a group of ten.
10!
a.
c. P(10, 4)
4!
b. 4C10
d. none of the above
17. The expression
a.
b.
c.
d.
14!
6!8!
is equivalent to
P(14, 6)
14C8
the number of ways that 6 out of 14 can be arranged
none of the above
ÁÊÁ 7 ˜ˆ˜
18. The expression ÁÁÁ ˜˜˜ is equivalent to
ÁÁ 2 ˜˜
Ë ¯
a. 21
7!
b.
2!
c.
42
d.
none of the above
19. From a group of seven junior and ten senior students, determine how many committees of six students can be
chosen if all the students are senior students.
a. 5040
c. 120
b. 210
d. C(6, 10)
20. From a group of seven junior and ten senior students, determine how many committees of six students can be
chosen if four are junior students.
a. 675
c. 1350
b. 32 400
d. none of the above
21. From a group of seven junior and ten senior students, determine how many committees of six students can be
chosen if at least one student is a senior student.
a. 2944
c. 1472
b. 24 738
d. 12 369
22. A bag contains four black marbles and nine orange marbles. If one marble is drawn randomly, determine the
odds of pulling out an orange marble.
4
9
a.
c.
9
13
b. 9:4
d. 9:13
23. Determine the odds of drawing two aces if two cards are removed from a regular deck.
a. 1:25
c. 1:220
b. 3:1349
d. 1:26
3
Name: ________________________
ID: A
24. Three cards are drawn randomly from a hat containing cards with the twenty-six letters of the alphabet on
them. Determine the probability of selecting A and B.
3
3
a.
c.
325
650
1
1
b.
d.
650
13
25. Determine the probability of choosing the jack and king of clubs out of a regular deck of cards when two cards
are randomly chosen.
1
1
a.
c.
2652
26
1
1
b.
d.
2704
1326
26. Determine the probability of choosing two hearts out of a regular deck of cards when two cards are randomly
chosen.
1
1
a.
c.
17
16
1
1
b.
d.
26
4
27. There is a 15% of winning a prize in a particular contest. Determine the odds against winning a prize.
a. 17:20
c. 17:3
b. 3:20
d. 14:15
28. The starting line up of a co-ed volleyball team must be made up of 3 males and 3 females. If the team has 9
females and 8 males, determine the probability that Emma, Mary, and Brittany are selected for the line up.
1
1
a.
c.
84
4704
3
1
b.
d.
17
1568
29. A bag contains six blue marbles, seven red marbles, and four green marbles. If four marbles are drawn
randomly, determine the probability that three are green.
2
13
a.
c.
17
595
832
13
b.
d.
83 521
2380
4
Name: ________________________
ID: A
30. Determine which of the following problems must be determined using a combination.
a. Determine the probability of rolling doubles when a pair of dice are rolled.
b. Determine the probability that Rhys, Lorne, Cheryl, and Carl are chosen when a group of
six is chosen from nineteen people.
c. Determine the number of ways that a captain and co-captain can be chosen from a football
team of 35 members.
d. Determine the probability of drawing an ace and a four from a deck of cards in that order.
31. Which of the following is not an example of a discrete random variable?
a. the number of heads observed for ten coin tosses
b. the mass of an apple randomly chosen from a supermarket bin
c. the number of hearts in a randomly chosen hand of seven playing cards
d. the sum resulting from the roll of two six-sided dice
32. A tetrahedron has four equal triangular faces. The faces of a tetrahedral die are labelled with the numbers one,
three, five, and seven. What is the expected value of the random variable representing the number observed on a
single roll of this die?
a. 3
c. 4
b. 5
d. 3.5
33. What is the probability of a sum of 5 resulting from the roll of two six-sided dice?
5
1
c.
a.
36
36
1
1
b.
d.
9
18
34. When we compare the possible values for a discrete random variable to its expected value, the following
statement must be true.
a. The possible values must all be whole numbers but the expected value may be a rational
number.
b. The expected value will not be a whole number.
c. The possible values and the expected value must all be whole numbers.
d. The expected value must be one of the possible values.
35. The probability of drawing a red card from a deck of 52 playing cards is 0.5. Which of the following
statements is not necessarily true?
a. The expected value for the number of red cards in a 7 card hand will be 3.5.
b. The expected value for the number of black cards in an 8 card hand will be 4.
c. If you successively draw and replace a card 100 times, you will see 50 red cards.
d. Over many repeated drawings you would expect the ratio of red cards drawn to total
number of drawings to be close to 1:2.
36. What is the probability of rolling less than 3 on a single roll of a six-sided die?
1
1
c.
a.
2
3
1
2
b.
d.
6
3
5
Name: ________________________
ID: A
37. A game is played by spinning a wheel that is divided into four sectors, each with a different point value. The
central angle and point value for each sector is shown in the chart below.
Central Angle
144°
108°
72°
36°
Point Value
20
30
40
50
How many total points would you expect to get for 100 spins of the wheel?
a. 2500
c. 3000
b. 3500
d. 2000
38. Which of the following is an example of a discrete random variable?
a. the cost of a U.S. dollar in Canadian currency on a given day
b. the mass of a green pepper selected at random from a bin
c. the number of red cars observed in a student parking lot
d. the time needed by a student to complete a physics lab
39. The following table shows the probability distribution for the possible sums that result from rolling two 6-sided
dice.
X
1
P(X)
0
X
7
P(X)
6
8
36
5
2
1
3
36
2
9
36
4
4
36
3
10
36
3
5
36
4
11
36
2
6
36
5
12
36
1
36
36
What is the probability that the sum rolled is even and less than 9?
1
5
a.
c.
9
36
1
7
b.
d.
2
18
6
Name: ________________________
ID: A
40. What is the probability of drawing exactly 2 red cards in a hand of 3 cards drawn from a deck of 52 cards?
1
3
a.
c.
8
8
1
2
b.
d.
2
3
41. A game involves tossing three coins. If two or more of the coins are heads, the player wins a prize. Each play
costs $0.25. What value of prize will make this a fair game?
a. $0.45
c. $0.40
b. $0.38
d. $0.50
42. The following table is a valid probability distribution for a random variable X. What must be the value for P(2)
to complete the table?
X
0
1
2
3
a.
b.
P(X)
0.15
0.2
0.4
0.3
0.15
c.
d.
0.2
0.25
43. A random variable X is defined as the number of heads observed when a coin is tossed 4 times. The probability
distribution for this random variable is shown below.
X
P(X)
0
1
1
4
2
6
3
4
4
1
16
16
16
16
16
Which of the following statements is not true?
a. The probability of no heads is the same as the probability of 4 heads.
b. The probability of not tossing 2 heads is greater than the probability of tossing 2 heads.
c. The most likely outcome is 2 heads.
6
d. The expected value is .
16
44. What is the expected value for a single roll of a 6-sided die?
a. 4
c. 3.5
1
b. 3
d.
6
7
Name: ________________________
ID: A
45. A student is preparing a probability distribution as shown below.
X
0
1
2
3
4
P(X)
0.3
0.3
0.3
0.3
A value is needed for P(3) to complete the table. Which statement below is true?
a. The required value for P(3) is –0.2.
b. The required value for P(3) is 0.3.
c. The required value for P(3) is 0.2.
d. There is no possible value for P(3) that can make this a valid probability distribution.
ÊÁ ˆ˜
ÁÁ n ˜˜
46. According to Pascal’s Identity, the single expression of the form ÁÁÁÁ ˜˜˜˜ , which is equivalent to
ÁÁ ˜˜
Ër¯
ÊÁ
ˆ
ÊÁ ˆ˜
ÁÁ 121 ˜˜˜
ÁÁ 12 ˜˜
Á
˜
ÁÁ ˜˜
˜˜
a. ÁÁÁ
c.
ÁÁ ˜˜
˜˜
ÁÁ
ÁÁ ˜˜
˜
12
Ë
¯
Ë 3¯
ÊÁ ˆ˜
ÊÁ ˆ˜
ÁÁ 11 ˜˜
ÁÁ 12 ˜˜
Á
˜
b. ÁÁÁ ˜˜˜
d. ÁÁÁÁ ˜˜˜˜
ÁÁ ˜˜
ÁÁ ˜˜
Ë 5¯
Ë 4¯
ÁÊÁ ˜ˆ˜
Á8˜
47. According to Pascal’s Identity, what two expressions may be added to obtain ÁÁÁÁ ˜˜˜˜ ?
ÁÁ ˜˜
Ë6¯
ÊÁ ˆ˜ ÊÁ ˆ˜
ÊÁ ˆ˜ ÊÁ ˆ˜
ÁÁ 8 ˜˜ ÁÁ 8 ˜˜
ÁÁ 7 ˜˜ ÁÁ 7 ˜˜
Á
˜
Á
˜
a. ÁÁÁ ˜˜˜ + ÁÁÁ ˜˜˜
c. ÁÁÁÁ ˜˜˜˜ + ÁÁÁÁ ˜˜˜˜
ÁÁ ˜˜ ÁÁ ˜˜
ÁÁ ˜˜ ÁÁ ˜˜
Ë 4¯ Ë 5¯
Ë 4¯ Ë 5¯
ÊÁ ˆ˜ ÊÁ ˆ˜
ÊÁ ˆ˜ ÊÁ ˆ˜
ÁÁ 4 ˜˜ ÁÁ 4 ˜˜
ÁÁ 7 ˜˜ ÁÁ 7 ˜˜
Á
˜
Á
˜
b. ÁÁÁ ˜˜˜ + ÁÁÁ ˜˜˜
d. ÁÁÁÁ ˜˜˜˜ + ÁÁÁÁ ˜˜˜˜
ÁÁ ˜˜ ÁÁ ˜˜
ÁÁ ˜˜ ÁÁ ˜˜
Ë 3¯ Ë 3¯
Ë 5¯ Ë 6¯
7
48. How many terms are in the expansion of ÊÁË 2x − 3y ˆ˜¯ ?
a. 8
c. 5
b. 7
d. 6
5
49. In the expansion of ÊÁË x + y ˆ˜¯ , what is the coefficient of the fourth term?
a. 5
c. 4
b. 10
d. 7
8
ÊÁ ˆ˜ ÊÁ ˆ˜
ÁÁ 11 ˜˜ ÁÁ 11 ˜˜
ÁÁ ˜˜ + ÁÁ ˜˜ ?
ÁÁ ˜˜ ÁÁ ˜˜
ÁÁ ˜˜ ÁÁ ˜˜
Ë 3¯ Ë 4¯
Name: ________________________
ID: A
n
50. In the expansion of ÊÁË x + y ˆ˜¯ , which term or terms must have coefficients with a value of n?
a. the second and second from last terms
c. the middle term or middle two terms
b. none
d. the second term
ÊÁ ˆ˜ ÊÁ ˆ˜ ÊÁ ˆ˜ ÊÁ ˆ˜ ÊÁ ˆ˜ ÊÁ ˆ˜
ÁÁ 5 ˜˜ ÁÁ 5 ˜˜ ÁÁ 5 ˜˜ ÁÁ 5 ˜˜ ÁÁ 5 ˜˜ ÁÁ 5 ˜˜
51. What is the value of ÁÁÁÁ ˜˜˜˜ + ÁÁÁÁ ˜˜˜˜ + ÁÁÁÁ ˜˜˜˜ + ÁÁÁÁ ˜˜˜˜ + ÁÁÁÁ ˜˜˜˜ + ÁÁÁÁ ˜˜˜˜ ?
ÁÁ ˜˜ ÁÁ ˜˜ ÁÁ ˜˜ ÁÁ ˜˜ ÁÁ ˜˜ ÁÁ ˜˜
Ë 0¯ Ë 1¯ Ë 2¯ Ë 3¯ Ë 4¯ Ë 5¯
a. 20
c. 64
b. 30
d. 32
52. In the expansion of (a + b ) , what is the value of the exponent k in the term that contains a 5 b k ?
a. 5
c. 4
b. 56
d. 3
8
ÊÁ ˆ˜ ÊÁ ˆ˜ ÊÁ ˆ˜ ÊÁ ˆ˜
ÊÁ ˆ˜
ÁÁ n ˜˜ ÁÁ n ˜˜ ÁÁ n ˜˜ ÁÁ n ˜˜
ÁÁ n ˜˜
Á
˜
Á
˜
Á
˜
Á
˜
53. If ÁÁÁ ˜˜˜ + ÁÁÁ ˜˜˜ + ÁÁÁ ˜˜˜ + ÁÁÁ ˜˜˜ +. . .+ ÁÁÁÁ ˜˜˜˜ = 64, what is the value of n?
ÁÁ ˜˜ ÁÁ ˜˜ ÁÁ ˜˜ ÁÁ ˜˜
ÁÁ ˜˜
Ë 0¯ Ë 1¯ Ë 2¯ Ë 3¯
Ën¯
a.
b.
5
4
c.
d.
8
6
54. What is the coefficient of the third term in the expansion of (3x + 1) ?
a. 10
c. 15
b. 150
d. 270
5
ÊÁ ˆ˜
ÁÁ 5 ˜˜
5−rÊ ˆr
Á y ˜ is the general term in the expansion of which binomial power?
55. ÁÁÁÁ ˜˜˜˜ (2x)
Ë ¯
ÁÁ ˜˜
Ër¯
a.
b.
r
ÁËÊ 2x − y ˆ˜¯
ÊÁ 2x + y ˆ˜ r
Ë
¯
c.
d.
5
ÁËÊ 2x − y ˆ˜¯
5
ÁËÊ 2x + y ˆ˜¯
56. Pascal’s Triangle can be arranged into the following format.
n=0
n=1
n=2
n=3
n=4
r=0
1
1
1
1
1
r=1
r=2
r=3
r=4
1
2
3
4
1
3
6
1
4
1
Which of the following statements is not true?
a. In any row for which n is prime, all values in the row other than the first and last must be
divisible by n.
b. The value in any cell is the sum of the cell directly above and the cell above and one left.
c. The sum of the nth row must be 2 n .
d. Values in each row must alternate between even and odd numbers.
9
Name: ________________________
ID: A
6
57. For the expansion of ÊÁË 3x + y ˆ˜¯ , which statement is not true?
a. The degree of each term in the expansion is 6.
b. There are seven terms in the expansion.
c. The coefficient of the term containing y 6 is one.
d. The coefficient of the second term is equal to the coefficient of the fifth term.
ÁÊÁ
58. What is the value of the constant term in the expansion of ÁÁÁÁ x 2 +
ÁË
a. 8
c. 32
b. 4
d. 16
6
2 ˜ˆ˜˜
˜˜ ?
x ˜˜¯
59. How many different paths will spell the word CASH using the diagram below?
C
A A
S S S
H H H H
a.
b.
10
6
c.
d.
4
8
60. Which of the following is a correct expression of Pascal’s Identity?
ÁÊÁ
˜ˆ ÁÊ
˜ˆ ÁÊ ˜ˆ
ÁÊÁ
˜ˆ ÁÊ
˜ˆ ÁÊ
˜ˆ
ÁÁ n − 2 ˜˜˜ ÁÁÁ n − 1 ˜˜˜ ÁÁÁ n ˜˜˜
ÁÁ n + 1 ˜˜˜ ÁÁÁ n + 1 ˜˜˜ ÁÁÁ n + 2 ˜˜˜
Á
˜
Á
˜
Á
˜
Á
˜
Á
˜
Á
˜˜
a. ÁÁ
c. ÁÁ
˜˜ + ÁÁ
˜˜ = ÁÁ ˜˜
˜˜ + ÁÁ
˜˜ = ÁÁ
˜˜
ÁÁ
˜˜ ÁÁ
˜˜ ÁÁ ˜˜
ÁÁ
˜˜ ÁÁ
˜˜ ÁÁ
˜
Ë r−2¯ Ë r−1¯ Ë r ¯
Ë r−1¯ Ë r ¯ Ë r+1¯
ÊÁ ˆ˜ ÊÁ
ˆ Ê
ˆ
ÊÁ
ˆ Ê
ˆ Ê ˆ
ÁÁ n ˜˜ ÁÁ n ˜˜˜ ÁÁÁ n + 2 ˜˜˜
ÁÁ n − 1 ˜˜˜ ÁÁÁ n − 1 ˜˜˜ ÁÁÁ n ˜˜˜
Á
˜
Á
˜
Á
˜
Á
˜
Á
˜˜ = ÁÁ ˜˜
b. ÁÁÁ ˜˜˜ + ÁÁÁ
d. ÁÁÁ
˜˜˜ = ÁÁÁ
˜˜˜
˜˜˜ + ÁÁÁ
˜˜ ÁÁ ˜˜
ÁÁ ˜˜ ÁÁ
˜˜ ÁÁ
˜˜
ÁÁ
˜˜ ÁÁ
˜˜ ÁÁ ˜˜
r
r
+
1
r
+
1
r
−
1
r
Ë ¯ Ë
¯ Ë
¯
Ë
¯ Ë
¯ Ër¯
10
ID: A
Sample Mastery #2
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
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25.
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27.
28.
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30.
31.
32.
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34.
35.
36.
37.
38.
C
D
C
D
C
B
D
B
B
C
A
C
A
D
D
D
B
A
B
A
D
B
C
A
D
A
C
A
C
B
B
C
B
A
C
C
C
C
1
ID: A
39.
40.
41.
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43.
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46.
47.
48.
49.
50.
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52.
53.
54.
55.
56.
57.
58.
59.
60.
D
C
D
D
D
C
D
D
D
A
B
A
D
D
D
D
D
D
D
D
D
D
2