2007 State Math Contest
Wake Technical Community College
Algebra II Test
1. If 2 f ( x ) + f (1 − x ) = x 2 for all x, then find f ( 5 ) .
a. 3
b.
34
3
c.
25
16
d. 9
e. not possible to determine
2. Tom walks the same path everyday. Today he decided to run for a few minutes and it took him 15
minutes less. If Tom runs three times faster than he walks, how long did he run today?
a. 5 minutes
b. 7.5 minutes
c. 10 minutes
d. 15 minutes
e. not possible to determine
3. If a is 95 percent larger than c and b is 30 percent larger than c, then a is what percent larger than b?
a. 45%
b. 30%
c. 75%
d. 65%
e. 50%
4. Which of the following quadratic equations has as its roots the numbers obtained by increasing each root
of x 2 − 2 x − 5 = 0 by the reciprocal of the other?
a. 5 x 2 − 8 x − 16 = 0
b. 5 x 2 + 8 x − 16 = 0
c. x 2 + 8 x − 16 = 0
d. x 2 − 8 x − 16 = 0
e. 5 x 2 − 8 x + 16 = 0
5. Ted agrees to play a trivia game where a player receives 5 points for answering an easy question and 11
points for answering a hard question. What is the largest integer that cannot be Ted’s score?
a. 34
b. 38
c. 39
d. 43
e. 47
6. For all real numbers, x, f ( 2 x ) = x 2 − 2 x + 5 . Which of the following is f ( x ) ?
a. f ( x ) = 4 x 2 − x + 5
b. f ( x ) =
x2
− x+5
4
x2
d. f ( x ) = − x + 5
2
c. f ( x ) =
e. f ( x ) = 2 x 2 − 4 x + 5
1
x2 x
− +5
4 2
2007 State Math Contest
Wake Technical Community College
Algebra II Test
7. If the chances of rain are 40% and 20% for the two days of a weekend, what is the probability that it will
rain on at least one day if the two events are independent?
a. 60%
b. 80%
c. 52%
d. 56%
e. 40%
8. A customer ordered twelve Zingers. Zingers come in packages of four, three, and one. In how many
ways can the order be filled?
a. 12
b. 11
c. 10
d. 9
e. 8
9. Art invested $200 at 9% compounded annually and Bob invested $300 at 7% compounded continuously.
To the nearest year how long before these two investments will be the same?
a. never
b. 5
c. 20
d. 25
e. 35
19 x − 8
is decomposed into partial fractions (fractions reduced to lowest terms that sum to the
2 x 2 − x − 21
given fraction), what is the sum of the numerators?
10. When
a. 3 x − 4
11. Which of the following is equal to
a.
34 + 22i
20i
12. What is the value of
a. log 2 7
c. − x + 4
b. 11
b.
d. 19
e. 14
5 − 4i
?
3+i
−34 + 22i
20i
c.
11 + 17i
10
d.
−11 − 17i
10
e.
17 − 11i
10i
( log 2 3)( log 4 5)( log 6 7 )
( log 4 3)( log 6 5)( log8 7 )
b. log12 35
c. log 4
2
d. 4
e. 3
2007 State Math Contest
Wake Technical Community College
Algebra II Test
13. Officials estimated that between 300,000 and 400,000 Boston-area party-goers attended the Independence
Day celebration on the banks of the Charles River. They also estimated that the party-goers left behind 40
tons of garbage. Given that a ton equals 2,000 pounds, how many pounds of garbage did the average
party-goer leave behind?
a.
b.
c.
d.
e.
less than ½-pound each
close to one pound each
close to two pounds each
close to three pounds each
close to four pounds each
14. A printshop owner has five copiers that he uses on jobs. Each
copier is the same, each one makes 25 copies per minute. One
morning, he is faced with the jobs listed to the right. Once he
places a print job in a printer, he leaves it – that is, any print
job is done on only one printer. What is the minimum number
of minutes it will take to complete all the print jobs?
a. 100 minutes
b. 110 minutes
c. 120 minutes
d. 130 minutes
e. more than 139 minutes
15. The following snapshot came from USA Today ©.
Which of the following statements are true based on the snapshot?
(i.)
More people take pencils than any other item.
(ii.)
About 38% of all workers take something from the office.
a.
b.
c.
d.
only (i) is true
only (ii) is true
both (i) and (ii) are true
neither (i) nor (ii) is true
3
Job
A
B
C
D
E
F
G
H
Number of Copies
1000
1500
1500
750
2500
3000
2500
2250
2007 State Math Contest
Wake Technical Community College
Algebra II Test
16. How many yards are in a mile?
a. more than 5000
b. between 3000 and 5000
c. between 1000 and 3000
d. less than 1000
17. You are preparing to tile the floor of a rectangular room that is 15 feet 9 inches by 19 feet in size. The
tiles you plan to use are square, measuring 12 inches on each side, and are sold in boxes of 25. How many
boxes of tiles must you order to complete the job?
a. 10
b. 11
c. 12
d. 13
18. The recipe shown to the right is used to make Angel Biscuits. In
preparing for a large party, a hostess wants to make as many
biscuits as she can but is limited because she has only 8 cups of
buttermilk. She figures how much of the other ingredients she
needs (below) based on using 8 cups of buttermilk. Which
ingredient is shown as the incorrect amount?
a.
6 cups of vegetable shortening
b.
2 cups of warm water
c.
20 cups of flour
d.
Both b and c are incorrect
e.
a, b, and c are incorrect
e. more than 13
Angel Biscuits
5 cups flour
3/4 cup vegetable shortening
1 teaspoon soda
1 teaspoon baking powder
1 teaspoon salt
3 tablespoons sugar
1 package yeast
1/2 cup warm water
2 cups buttermilk
19. A point ( x, y ) is called integral if both x and y are integers. How many points on the graph of
1 1 1
+ =
x y 6
are integral points?
a. 20
b. 18
c. 17
d. 15
e. 5
20. If x < y and x < 0 , which of the following is never greater than the others?
a. x + y
b. x − y
c. x + y
b. 1
e. x − y
d. 3
e. 4
x = x4 −1 ?
21. How many positive real numbers x are solutions to
a. 0
d. − x − y
c. 2
4
2007 State Math Contest
Wake Technical Community College
Algebra II Test
⎧c 2 − a 2 − b 2 = 101
22. There are positive integers a, b, and c that satisfy the system of equations: ⎨
. What is
=
ab
72
⎩
the value of a + b + c ?
a. 47
b. 37
c. 29
d. 39
e. 43
23. If the product of three numbers in geometric progression is 216 and their sum is 19, find the largest of the
three numbers.
a. 12
b. 9
c. 8
d. 6
e.
32
3
24. The amount of caffeine in your blood after reaching the maximum caffeine blood level of 120 mg
decreases at a rate of 13% per hour. How much caffeine will be in your blood stream after 5 hours?
a. 57.91 mg
b. 59.65 mg
25. Find the smallest positive value of
a. 2 6
b.
6
c. 62.64 mg
d. 59.81 mg
e. 59.77 mg
x2 + y2
where x and y are positive numbers and xy = 3.
x− y
c. 2 3
5
d. 2 2
e. 3 6
2007 State Math Contest
Wake Technical Community College
Algebra II Test
SHORT ANSWER
Place the answer in the appropriate space.
66. What is the least value of y that satisfies the inequality 4 + x + 5 + y ≤ 80 ?
67. The complex numbers 1 + i and 1 + 2i are both roots of the equation x5 − 6 x 4 + Ax3 + Bx 2 + Cx + D = 0 ,
where A, B, C, and D are integers. What is the value of A + B + C + D ?
68. What is the remainder when 7348 + 25605 is divided by 8?
69. What positive integer n satisfies the equation log ( 225!) − log ( 223!) = 1 + log ( n !)
70. A bowl contains 50 colored balls: 13 green, 10 red, 9 blue, 8 yellow, 6 black, and 4 white. If you are
blindfolded, what is the smallest number of balls you must pick to guarantee that you have at least 7 balls
of the same color?
6
2007 State Math Contest
Wake Technical Community College
Algebra II Test
1. b
2. b
3. e
4. a
5. c
6. b
7. c
8. b
9. d
10. e
11. a
12. e
13. a
14. d
15. d
16. c
17. d
18. a
19. c
20. e
21. c
22. e
23. b
24. d
25. a
66. – 85
67. 1
68. 2
69. 7
70. 35
7
2007 State Math Contest
Wake Technical Community College
Algebra II Test
1. 2 f ( 5 ) + f ( −4 ) = 25 and 2 f ( −4 ) + f ( 5 ) = 16 . So f ( 5 ) =
34
.
3
2. Let x be Tom’s walking speed and let d be the distance that Tom ran, then
d
is the time that Tom ran.
3x
d d
d
−
= 15 minutes so
= 7.5 minutes.
x 3x
3x
3. 1.95c = a and 1.3c = b . Hence
1.95
b = a and a is 50% larger than b.
1.3
1 ⎞⎛
1 ⎞
8
16
⎛
2
4. The roots of x 2 − 2 x − 5 = 0 are 1 ± 6 . ⎜ x − 1 + 6 −
⎟⎜ x − 1 − 6 −
⎟ = x − 5 x − 5 . So
1 − 6 ⎠⎝
1+ 6 ⎠
⎝
2
5 x − 8 x − 16 = 0 has the correct roots.
5. 39 cannot be written as a linear combination of 5 and 11. 43 = 5*2 + 11*3 and 47 = 5*5 + 11*2.
2
6.
x2
⎛ x⎞ ⎛ x⎞
⎛ x⎞
f ( x) = f ⎜ 2 ⎟ = ⎜ ⎟ − 2⎜ ⎟ + 5 = − x + 5
4
⎝ 2⎠ ⎝2⎠
⎝2⎠
7. The probability that it will rain at least one day is equal to 1 minus the probability of no rain. Hence
1 − (.6 )(.8 ) = .52 or 52%
8. Make a complete list beginning with 3 packages of four and ending with 12 packages of one. There are 11
possibilities.
9. Solve 200 (1.09 ) = 300e0.07 x and get 25 years approximately.
x
10.
9
5
. So the numerators sum up to 14.
+
2x − 7 x + 3
11.
( 5 − 4i )( 3 − i ) = 11 − 17i = 11i + 17 = 34 + 22i
10
10i
20i
( 3 + i )( 3 − i )
log 3 log 5 log 7
log 2 log 4 log 6 log 8
12.
=
=3
log 3 log 5 log 7 log 2
log 4 log 6 log 8
13. (40 * 2000 ) / 300000 = 0.26 pounds per person.
14. One copier gets Jobs B & C. One copier gets job F. One copier gets jobs A & H. One copier gets job G.
One copier gets jobs D & E. 3250/25 = 130 minutes.
15. Both statements are false.
8
2007 State Math Contest
Wake Technical Community College
Algebra II Test
16. There are 1760 yards in a mile.
17. 15*19 = 285 square feet or tiles. 25*12 = 300 tiles. 300 − 285 = 15 tiles. This is not enough to nicely
finish the flooring so you should get 13 boxes of tiles.
18. (8 / 2 ) * (3 / 4 ) = 3 cups of vegetable shortening.
36
. Hence x must be a solution of x − 6 = an integer divisor of 36. There are 18 possibilities.
x−6
They all lead to an integral solution except x = 0 which gives y = 0, so there are 17 integral solutions.
19. y = 6 +
20. If y > 0, then x − y = − x − y = x − y < x + y = x + y . If y < 0, then
x + y = x − y < x − y = − x − y = x + y . Thus x − y is never greater than the others.
21. Look at the graph on the interval [ 0, 2] and there are two solutions.
22. c 2 − ( a + b ) = c 2 − a 2 − 2ab − b 2 = 101 − 2*72 = −43 and c 2 − ( a + b ) = ( c + a + b )( c − a − b ) . Since a, b,
2
2
and c are positive integers and 43 is a prime number it must be that a + b + c = 43 .
23. ( a )( ra ) ( r 2 a ) = 216 so ra = 6. Hence
6
2
3
+ 6 + 6r = 19 . Solving we get either r = or r = . Thus the
r
3
2
largest of the three numbers is 9.
24. 120 (.87 ) = 59.81 mg
5
25.
x2 + y2
2 xy
. The Arithmetic-Geometric Mean Inequality says that a + b ≥ 2 ab . If a = x − y
= x− y+
x− y
x− y
2xy
x2 + y2
2 xy
9 ⎞ ⎛
3⎞
⎛
, then
= x− y+
≥ 2 2 xy = 2 6 . Graphing f ( x ) = ⎜ x 2 + 2 ⎟ / ⎜ x − ⎟ you
x− y
x− y
x− y
x ⎠ ⎝
x⎠
⎝
find that the minimum point is 2 6 = 4.8989795
and b =
66. y = – 85 satisfies the inequality for x = – 4. If y < – 85, then 5 + y > 80 which gives 4 + x < 0 , an
impossibility.
67. ( x − 1 − i )( x − 1 + i )( x − 1 − 2i )( x − 1 + 2i )( x − 2 ) = x5 − 6 x 4 + 19 x3 − 36 x 2 + 38 x − 20 . So A + B + C + D = 1 .
68. 7 2 ≡ 1mod 8 and 25 ≡ 1mod 8 . Hence the remainder is 2.
69. log
225!
= log ( 225* 224 ) = log 50400 = log10 + log 5040 = 1 + log 7!
223!
70. Worst case scenario – You pick out 6 green + 6 red + 6 blue + 6 yellow + 6 black + 4 white = 34 balls
without having 7 of the same color. Hence to guarantee 7 of the same color you must choose 35 balls.
9