Elizabeth City State University
Elizabeth City, North Carolina 27909
March 28, 2013
NC STATE REGIONAL MATHEMATICS CONTEST
COMPREHENSIVE
TEST BOOKLET
Directions: Each problem in this test is follow
ed by five suggested answers. When
you have decided which of the suggested answer is correct, shade the
corresponding circle on the answer sheet.
Sample question:
A) 6
54 – 48 =
B) 7
C) 16
D) 12
E) None of these
Sample answer:
B
C
D
E
Answer A is marked since the difference between 54 and 48 is 6.
You will have 80 minutes to work on the 40 questions in this test booklet.
Please remember to print clearly your name, test-subject, and School’s
name on the Bubble Sheet.
Comprehensive
NC State Math Contest
ECSU
March 28, 2013
1. A stock looses 60% of its value. What must the percent of increase be to recover all its lost
value?
(A) 60%
(B) 120%
(C)150%
(D) 300%
(E) None
2. Complete the identity: sin 2
2 tan 
(A)
1  tan 2 
(C) sin  (cos   sin  )
2(1  tan 2  )
(B)
cos 
(D) cos  (sin   2 cos  )
3. The prime factorization of 2,170,546,875 is 3
2,170,546,875 are there?
(A) 160
(B) 84
(C) 200
5
7 . How many factors of
(D) 240
4. The five angles in degrees of pentagon are 2 , 8
value of ?
(B) 10°
(C) 15°
(A) 5°
5, 7 , 5
5. What is the angle between the complex number
1
√
(A)
(B)
√
(C)
6. Given that sin  
(A)
9
16
(E) None
20, 6
(D) 20°
2 and its square
(E) None
5. What is the
(E) None
in exact form?
√
(D)
8
, and cos  0 , then sin 2 
9
9
16 17
(B) 
(C) 
81
17
(E) None
(D) 
8
81
(E) None
7. Tom Quig traveled 230 miles east of St. Louis. For most of the trip he averaged 60 mph, but
for one period of time he was slowed to 10 mph due to a major accident. If the total time of
travel was 8 hours, how many miles did he drive at the reduced speed?
(A) 70 mi
(B) 45 mi
(C) 60 mi
(D) 50 mi
(E) None
8. Find the ratio of the volume of a cone of height 50 m and radius 12 m to the volume of a
sphere of radius 30 m.
(A) 1/15
(B)1/12
(C)1/4
(D) 1/3
(E) None
Comprehensive
NC State Math Contest
ECSU
March 28, 2013
9. Palindromes are words or phrases that read the same in both directions, e.g. EYE,or
RACECAR . How many dates written in the format MMDDYY in the 21st century are
palindromes (such as 121112)
(A) 24
(B) 36
(C) 1
(D) 144
(E) None
10. Which of the following functions is even?


(A) f ( x)  2 cos x  
(B) f ( x )  sin 2 x
2



(D) f ( x)  csc x  
(C) f ( x)  csc x  2
2

(E) None
11. How much pure acid should be mixed with 9 gallons of a 50% acid solution in order to get an
80% acid solution?
(A) 36 gal
(B) 4.5 gal
(C) 13.5 gal
(D) 22.5 gal (E) None
12. Two identical circles intersect. The area of the common region is equal to the sum of the
areas of the two uncommon regions. If the area of the common region is 48π, determine the
circumference of each circle.
(A) 4√2
(B) 12
(C)12√2
(D) 24
(E) None
13. If the sum of the first four terms of an arithmetic progression is 32, and the sum of the first 6
terms is 60, then the first term is
(A) 3
(B) 5
(C) 17/3
(D) 10
(E) None
14. Find the sum of the frequency, amplitude and period of f ( x )  9 sin( 5x  6)  3 .
101
12
119
61
(B) 
(C)
(D)
(E) None
(A)
10
10
10
10
15. With 31.5% commitment, a company can clean up 45% of an acid rain forest. With 36.5%
commitment, it can clean up 90%. Find a linear equation that models the percent clean up
for x percent commitment.
(A)
9
238.5
(B)
4
(C)
5
45
(D)
9
238.5
(E) None
Comprehensive
NC State Math Contest
ECSU
March 28, 2013
16. In the right ∆ABC, altitude from the point C meets the hypotenuse AB at D. If AD = 6,
and DB = 24, what is the length AC?
A
D
z
(A) 4 5
B
C
(B) 5 5
(C) 6 5
(D) 10 5
(E) None
17. A student took 6 exams. If the average on the first 3 exams was 82, and the average on the
next two was 87, what score did the student need on the last exam to have an average of 85
for all six exams?
(A) 84
(B) 90
(C) 86
(D) 87
(E) None
18. Which of the following is equivalent to: csc 5 x tan x  csc 3 x tan x ?
sin x
sin x
cos x
cos x
(A)
(B)
(C) 2
(D)
2
4
cos x
cos x
sin x
sin 4 x
(E) None
19. A person's emotional quotient (EQ) is found by multiplying emotional age by 100 and
dividing by chronological age. Using this information, if a 21 yr old person has an EQ of 102
what is that person's emotional age?
(A) 24.6 years old
(B) 21.42 years old
(C) 21 years old
(D) 20.59 years old
(E) None
20. In the given figure, DE and BC are parallel. If m∠BAC = x◦, m∠BCA = (x + 30)◦, and
m∠BDE = 4x◦, find the measure of ∠ABC.
A
D
B
E
C
Comprehensive
NC State Math Contest
ECSU
March 28, 2013
(A) 120◦
(B) 60◦
(C) 45◦
(D) 15◦
(E) None
21. If an investment is compounded annually for 10 years at a rate , then the value of the
investment will triple if
1
/
(B)10
(C)3 /
(D)log 10 (E) None
(A) log 3
22. A triangle has three angles A, B and C . Compute tan C in terms of x .
(A) tan(1  x )
(B) tan 1  tan x
(C)  tan(1  x )
(D) tan 1  tan x
(E) None
. Write the answer in standard form of
23. Find the quotient
(A)
+
(B)
-
(C)
+
(D)
-
(E) None
24. In the figure shown, a circle passes through two adjacent vertices of a square and is tangent
to the opposite side of the square. If the length of a side of the square is 3, what is the area of
the circle?
(A) π
(B)
π
(C) 6
(D)
π
(E) None
25. In a class of 40 students every student takes at least one of the subjects Math, Science and
English. Of these students 27 take Math, 15 take Math and Science, 19 take Math and
English, and 17 either take only Math or else take all three. How many students take only
Math?
(A) 5
(B) 6
(C) 7
(D) 8
(E) None
26. The sum of all solutions to the equation sin 2 x  sin x on the interval [0, 2 ] .
7
(B) 
(C)
(D) 2
(E) None
(A) 5
3
27. The length of a rectangle is 10 inches more than its width. If 5 inches are taken from the
length and added to the width, the figure becomes a square with an area of 196 square inches.
What are the dimensions of the original figure?
(A) 14 in. by 14 in.
(B) 9 in. by 14 in.
Comprehensive
NC State Math Contest
ECSU
March 28, 2013
(C) 9 in. by 19 in.
(D) 4 in. by 14 in.
(E) None
28. In the figure shown, the measure of an angle formed by the bisectors of two angles in triangle
ABC is 120◦. Find the measure of angle B.
A
B
(A) 40◦
(B) 60◦
C
(C) 50◦
(D) 45◦
(E) None
29. Two cars leave an intersection at the same time. One car travels north; the other east. When
the car traveling north had gone 9 mi, the distance between the cars was 3 mi more than the
distance traveled by the car heading east. How far had the east bound car traveled?
(A) 9 mi
(B) 12 mi
(C) 18 mi
(D) 15 mi
(E) None
30. Find the center and radius of the circle: x2 + y2 + 12x - 2y + 28 = 0
(A) center: (6, -1); radius: 9
(B) center: (1, -6); radius: 3
(C) center: (-6, 1); radius: 3
(D) center: (-1, 6); radius: 9
(E) None
31. Jacob walks along a level road and then up a hill. At the top he immediately turns and walks
back to his starting point. He walks 4mph on level ground, 3mph uphill, and 6mph
downhill. If the entire walk takes 6hours, how far does he walk?
(A) 8mi
(B) 10mi
(C)12mi
(D) 20mi
(E) None
32. Ed starts with no money and receives successive payments of$1, $2, $4, $8, ….Dona starts
with $22 and receives $15 each time Ed receives a payment. After how many times will Ed
have at least as much money as Dona?
(A) never
(B) 10
(C) 6
(D) 7
(E) None
33. Beginning with 100 ounces of 50% salt solution, 50 ounces of solution is removed and 50
ounces of plain water is then added. This is then repeated for a total of ten times. The
resulting solution is approximately what percent salt?
(A) 5
(B) 1
(C) 50
(D) .05
(E) None
Comprehensive
NC State Math Contest
ECSU
March 28, 2013
x
 x
34. What is the period of the function f ( x)  tan x  sin    cos  ?
2
4
(A) 
(B) 2
(C) 4
(D) 8



(E) None

35. Suppose 0  x  1 . A possible value of sin cos 1 x  cos sin 1 x is
1
x
2x
(A) 2 1  x 2
(B)
(C)
(D)
1 x2
1 x2
1 x2
(E) None
36. Helen Weller invested $15,000 in an account that pays 11% simple interest. How much
additional money must be invested in an account that pays 14% simple interest so that the
average return on the two investments amounts to 12%?
(A) $7500
(B) $12,000
(C) $15,000
(D) $11,000 (E) None
37. A 10 10 box is filled with 1
of the box?
(A) 600
(B) 512
1
1 blocks. How many blocks touch the six inside faces
(C) 488
(D) 448
(E) None
38. In triangle ABC, AB = 260, AC = 400 and BC = 520. Point D is chosen on BC so that the
circle inscribed in triangle ABD and ADC are tangent to AD at the same point. What is the
length of BD?
A
B
(A) 130
(B) 150
D
(C) 170
C
(D) 190
(E) None
39. In a triangle ABC, BC = 24, AC = 18 and the medians to the sides BC and AC are
perpendicular. Find the length AB.
(C) 6√5
(D) 5√65
(E) None
(A) 30
(B) 15√2
40. Joe pays $ for an article and sells it to Tom for 5% profit. Tom then sells it to Bill at 10%
loss. If the difference between the amounts Tom and Bill paid is 70 cents, the value of x is
(A) 14
(B) 7/3
(C) 20/3
(D) 30/7
(E) None
Comprehensive
NC State Math Contest
Comprehensive Answers:
1. A
2. A
3. A
4. D
5. A
6. C
7. D
8. A
9. A
10. D
11. C
12. C
13. B
14. C
15. A
16. C
17. B
18. D
19. B
20. A
21. C
22. C
23. A
24. D
25. A
26. A
27. C
28. B
29. B
30. C
31. C
32. D
33. D
34. D
35. A
36. A
37. C
38. D
39. C
40. C
ECSU
March 28, 2013