March 24, 2011
2011 State Math Contest
Wake Technical Community College
Algebra II Test
1. The mean October rainfall over a 30-year period is 3.28 in. How much rainfall would have to fall next
October to increase the mean by a minimum of 0.15 in?
a. 8.13 in
b. 9.99 in
c. 7.63 in
d. 8.21 in
e. 7.93 in
d. 3
e. 0
2. How many integers between 1 and 1000 have exactly three factors?
a. 15
b. 11
c. 9
8 where a, b, and c are constants and
3. Let
4 ?
a. 12
b. 8
c. 6
4
d. 2
4. Find the slope of the line through the points x = – 2 and x = 1 on the graph of
a.
b.
10. What is the value of
c.
e. 0
2
.
e. 3
d.
5. Determine the domain of the function
√
a.
∞, 2
2, ∞
b. 2, ∞
c.
2,2
2
2
2, ∞
2, 2
d.
2, ∞
e.
∞, ∞
6. A jar contains some red and some yellow jelly beans. If a child eats 1 red jelly bean, of the remaining
jelly beans would be red. If a child instead eats 5 yellow jelly beans, of the remaining jelly beans
would be red. How many yellow jelly beans are in the jar?
a. 60
b. 50
c. 48
1
d. 40
e. 36
2011 State Math Contest
Wake Technical Community College
Algebra II Test
7. Will can paint a room in 5 hours. Andy can do the same job by himself in 6 hours. They decide to work
together, but they stop to talk and sometimes get in each other’s way. Assume each boy’s productivity
decreases by the same percent. If they complete the job in 3.5 hours, what is the percent decrease in the
boy’s painting rate to the nearest percent?
a. 14%
b. 22%
c. 20%
d. 21%
e. 18%
8. The two hands on a clock point in the same direction at 12:00. Shortly after 1:00, they will again point
in the same direction. How many minutes after 1:00 does this happen?
a.
b.
c.
d.
9. Let a, b, c, and d be the four solutions of the equation 12
.
a.
b.
c.
11
e.
2
0. Determine
d.
e.
10. Each day Pam makes an open-faced sandwich using exactly one slice of bread, at most one type of
meat, and at most one type of cheese. If she chooses from 3 types of bread, 5 types of meat, and 6 types
of cheese, how many different sandwiches can she make? Note: Assume the definition of a sandwich is
that the bread has something on it.
a. 123
b. 90
c. 112
d. 135
e. 138
11. Al, Roger, Clay, Stan, and Bob are the starters on the local basketball team. Two of them are lefthanded and three are right-handed. Two are over 6 feet tall and 3 are less than 6 feet tall. Al and Clay
are of the same handedness and Stan and Bob are of different handedness. Roger and Bob are the same
height. If Clay is over 6 feet tall, then Stan is less than 6 feet tall. If Clay is less than 6 feet tall, then
Stan is over 6 feet tall. The center for the team is over 6 feet tall and is left-handed. What is the name of
the center?
a. Al
b. Bob
c. Clay
2
d. Stan
e. Roger
2011 State Math Contest
Wake Technical Community College
Algebra II Test
2
12. Determine the coefficient of the term in the expansion of
is 8.
a. 11520
b. 13440
6
13. What value of c makes
numbers x?
a. − 5
14. Let
c. 8064
5
4
a. 32
1
c. − 14
3
and
d. 3360
1
b. 2
2
for which the exponent of x
1
true for all real
d. − 3
e. 3
d. 128
e. 8
1 .
. Determine
b. 16
e. 3840
c. 6
15. Mabel throws two six-sided dice, a red one and a white one. What is the probability that the red one
beats the white one in score?
a.
b.
c.
d.
e.
16. A freight train that is 1 mile long is traveling at 60 mph. A second train that is 2.5 miles long is
traveling 45 mph. The trains are moving in opposite directions on parallel tracks. How many seconds
does it take between the time their locomotives meet and the time their cabooses fully pass each other?
a. 100 seconds
17. Let
√
b. 126 seconds
c. 115 seconds
. Determine the domain of
a. Undefined for all x
b.
1
c.
d. 108 seconds
e. 120 seconds
.
1, 0
3
0, 1
d.
∞, 1
1, ∞
e.
∞, ∞
2011 State Math Contest
Wake Technical Community College
Algebra II Test
18. Facebook membership can be modeled by the piecewise function
1.5 0
1.5
46
126
3
3
5
where F is the membership in millions and t is the number of years since the beginning of 2004.
Determine the average rate of change from the beginning of 2005 to the beginning of 2008.
a. 17.5 million per year
b. 46 million per year
d. 7.8 million per year
c. 16.5 million per year
e. 18.7 million per year
19. Two fair coins are flipped simultaneously. This process is continued until at least one of the coins
comes up heads. What is the probability that both coins show heads on this last flip?
a.
b.
20. If
c.
,
a. 18
21. If
2
2
14,
b. 18.5
2
2 and
a. 136
2
d.
6, and
4
c. 19
2
b. 115
1, determine
e.
8, determine
3 .
d. 19.5
e. 20
d. 105
e. 124
30 .
c. 120
22. The sum of five consecutive positive integers beginning at n is a perfect cube. The first such n is 23.
Which of the following is true about the next possible value of n?
a. It is divisible by 13.
b. It is divisible by 5.
d. It is prime.
c. It is divisible by 7.
e. It is divisible by 11.
4
2011 State Math Contest
Wake Technical Community College
Algebra II Test
| |
23. Find the area of the intersection of the graphs of
a. 25 sq units
24. Let
a.
b. 60 sq units
ln
1
b.
5 and
c. 75 sq units
. Determine
5.
d. 50 sq units
e. 45 sq units
ln 7 .
c.
d.
e.
25. How many different 3-letter strings can be formed from the letters of ALGEBRAIC (no letter can be
used in a given string more times than it appears in the word)?
a. 336
b. 252
c. 126
5
d. 357
e. 504
2011 State Math Contest
Wake Technical Community College
Algebra II Test
SHORT ANSWER
Place the answer in the appropriate space.
66. What is the probability that rolling four fair, six-sided dice produces an odd-numbered sum?
67. Determine the sum of all real number solutions of 3
9
9
3
3
9
12 .
68. A test consists of 100 questions for a total of 100 points. Each true-false is worth 0.5 points. Multiple
choice questions are worth 3 points each. The essay questions are worth 10 points each. How many
essay questions are there?
69. Solve for x:
log 16
log 25
log 3
log 7
log 32
log 125
70. Twelve students – 4 from Mr. Adam’s theater class, 4 from Mrs. Jones’s theater class, and 4 from Ms.
Smith’s theater class – bought a block of 12 seats at the local Performing Arts Center. How many ways
can the students sit in their block so that the students from each class sit together in a block of 4
consecutive seats?
6
2011 State Math Contest
Wake Technical Community College
Algebra II Test
1. e
2. b
3. c
4. b
5. d
6. a
7. b
8. c
9. a
10. a
11. d
12. d
13. c
14. a
15. e
16. e
17. a
18. e
19. c
20. b
21. c
22. e
23. d
24. b
25. d
66.
67. 3.5
68. 5
69.
70. 82944
7
2011 State Math Contest
Wake Technical Community College
1. 31
3.28
0.15
30 3.28
Algebra II Test
7.93 inches.
2. To have exactly three factors a number must be the square of a prime, for example 4 and 9 have exactly
three factors. Hence count the primes between 1 and 31. There are 11.
3.
4
Hence 10
4
4
4
4 8
16. Thus 4
6.
4
4
4
8
16
4
16 .
4.
5. All real numbers greater than or equal to – 2 except 2. Hence
2, 2
2, ∞ .
6. Let x be the number of red jelly beans and y be the number of yellow jelly beans. Then the two
equations are
1
1 and
5 . Solving gives 11 red jelly beans and 60
yellow jelly beans.
7. Let x be the decreased productivity hours for Will and y be the decreased productivity hours for Andy.
Then
.
or
. Hence
.
and solving gives x approximately equal to 6.42. Thus
.78 or a decrease of 22%.
8. Let x be the number of minutes past 1:00 that the hands point in the same direction. The hour hand
5
and solving gives
.
travels at of a minute per minute. Hence
9. 12
11
2
4
1 3
2 . Hence
10. She can make 90 sandwiches with a meat and a cheese: 3*5*6. She can make 15 sandwiches with just
meat: 3*5. She can make 18 sandwiches with just cheese: 3*6. That makes 123 sandwiches.
11. Since Al and Clay are of the same handedness and Stan and Bob are of different handedness that means
that either Bob or Stan must be the same handedness as Al and Clay making them right-handed (not the
center). Similarly since Roger and Bob are of the same height and Clay and Stan are on opposite sides
of 6 feet, Roger and Bob must be on the same side of six feet as either Stan or Clay. That means they
are under 6 feet and not the center. That leaves Stan as the center.
12. Let n be the term in the expansion for which the exponent of x is 8. Then the nth term is
10
10
10
2
2
2
10
Hence x has an exponent of 8 in the sixth term. Thus
2
210 16 3360.
6
13. Repeated long division with
method of solving.
,
1 yields a = 1; b = 3; c = − 14; and d = 14. Using a matrix is another
8
2011 State Math Contest
Wake Technical Community College
1
14.
3
3
Algebra II Test
2
32.
15. There are 36 possible rolls. Six of those have the numbers equal leaving 30 where the numbers are
unequal. On half of them the red number is greater than the white die. Hence the probability is
16. Let t be the time in hours required. Then 60
seconds.
2 1
2
17.
45
2 1
2
1
2
2 1
2
2
3.5 and t equals a thirtieth of an hour or 120
√
| |
2
1
.
1
. Regardless of the value of x this
composition has no real values.
18.7 million per year.
18.
19. The previous tosses are irrelevant. On the last toss we know one of the coins is heads so of the three
possible tosses only one of them show two heads. Hence the probability is .
20. One method for solving this problem is to find the quadratic regression for the three given points. This
yields
0.5
2
8. Hence
3
18.5.
21.
4
2
2
2 1; 6
3
the pattern is established that 2
4
3 2 1;
. Hence 30
8
4
120.
6
4
3
2
1; and so
22. Let
2. Then
2
1
1
2
5 and 5x must be a perfect cube;
the first time that happens is when x = 25 or when n = 23. The next time that happens is when x = 25*8
= 200 or when n = 198. Hence it is divisible by 11.
23. The intersection is two triangles of base 10 units and height 5 units. Hence the area is 50 square units.
7. Solving for x gives
1
24. The equation to solve is
.
25. Assume all three letters chosen are different then there are 336 different strings. If the two A’s are
chosen then there are seven ways to choose the additional letter and 3 ways to arrange the 3 letters.
Hence an additional 21 strings. This gives 357 strings.
66. There are 216 sums for the first three die. For each sum when added to the fourth die number half of
them will be odd and half will be even, Hence the probability is .
67. 3
3
9
9
9
3
9
3
3
9
3
9
3 3
9 9
3
3 3
9 9
3
3 3
9 9
3
3 3
9 9
3
9
9
9
3 . Hence
3 .
2011 State Math Contest
Wake Technical Community College
Hence 0 3 3
9 9
3
3 3
9 9
3
3 3
9 9
the solutions are x = 2; x = 0.5; and x = 1. This gives a sum of 3.5.
3 3
Algebra II Test
9 9
3 . Thus
68. Let x be the number of true-false questions; let y be the number of multiple choice questions; and let z
be the number of essay questions. Then there are two equations
100 and 0.5
3
10
100. Solve by multiplying the first equation by negative one and adding it to two times the
second equation giving 5
19
100. The only integer solution of this equation is y = 1 and z = 5.
This gives x = 94. Since 0.5 94 3 1 10 5 100. There are 5 essay questions.
69. Using the change of base formula and multiplying the first equation by
equation by
and multiplying the second
yields the following system:
log 5
log 3
log 2
log 7
4log 2
log 7
2 log 5
log 3
log 5 5log 2
log 3 2log 7
log 2 3 log 5
log 7 2 log 3
Adding the two equations together gives:
log 5 log 2
2
log 3 log 7
Hence 2x = 1 or
log 5 log 2
log 3 log 7
70. There are 12 ways to pick the student to sit in the first seat, then there are 3 ways for the next, 2 ways
for the next and 1 way for the fourth seat. For the fifth seat there are 8 choices, followed by 3, 2, and 1
again. Finally for the last four seats there are 4, 3, 2, and 1. This gives
12 4 3 2 1 8 3 2 1 4 3 2 1 82944
10