Elizabeth City State University
Elizabeth City, North Carolina27909
2014
STATE REGIONAL MATHEMATICS CONTEST
COMPREHENSIVE
TEST BOOKLET
Directions:
Each problem in this test is followed 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.
Comprehensive
NC State Math Contest
ECSU
April1, 2014
1. Suppose four people weigh on average 120 lbs., and a fifth person weighing 130 lbs. joins them, the
average weight of all five people is:
A) 125
B) 128
C) 126
D) 122
E) None of these
2. The sum 2014
A) 8000 18003
C) 6000 18007
2015 2016
2014
2014
⋯
20014 is equal to
B) 9000 18001
D) 6000 18003
2014
2014
E) None of these
3. In the following circle, AB bisects CD at point E. If CD = 8 and AE = 2, then the radius is:
A
C
D
E
B
A) 3
B) 4
C) 5
D) 6
E) None of these
4. sin tan
A)
B) √
1
C)
D)
E) None of these
5. Assume that the radius of the earth is 3960 miles. A coast guard plane 5 miles above the earth spots
a boat miles away. Which formula below represents ?
3960
B)
3965 3960
A)
√3965
C)
D)
√3965 3960
√3965 √3960
E) None of these
6. Which of the following is the largest?
B) 8
A) 8
D) 4
C)2
2
E) 16
Comprehensive
NC State Math Contest
ECSU
April1, 2014
7. In the figure below, the radius of the circle is 6 cm, what is the area of the triangle?
A) 18√3 cm2
B)36√3 cm2
C)72√3 cm2
D)108√3 cm2
E) 216√3 cm2
8. A weight attached to a spring is pulled down 3 inches below the equilibrium position. Assuming that
the frequency of the system is 6/π cycles per second, determine a trigonometric model that gives the
position of the weight at time t seconds.
A)
3 cos 12
B)
3 cos 12
C)
3 πcos 6
D)
3 cos 6/ )
E) None of these
9. The Joshua Family took a summer trip by road and had 2 breaks. The first trip segment was covered
in 3 hours, the second 80 mile segment was also covered in 3 hours, and the third 130 miles long
segment took 2 hours. Suppose the overall average speed of the trip was 40 mph, how longer was the
second segment than the first one.
A) 45 miles
B) 50 miles
C) 60 miles
D) 30 miles E) None of these
10. Of the seven points (9,17), (6,11), (3,5), (7,12), (3.5,6), (5,10), (5,9), five lie on a straight line.
Which two do NOT lie on that line?
A) (5,10) and (7,12)
B) (3,5) and (5,9)
C) (9,17) and (7,12)
D) (6,11) and (3,5)
E) (3.5,6) and (5,9)
11. A cube is painted blue on all faces. If the cube is cut into 216 smaller cubes of equal sizes, how
many smaller cubes have only one blue face?
A) 6
B) 12
C) 48
D) 96
E)216
12. Complete the identity:
tan
A) 2
B) tan
1 tan
tan
C) 0
1
sec
D) 1
3
E) None of these
Comprehensive
NC State Math Contest
ECSU
April1, 2014
13. An examination consists of three parts. In part I, a student must answer 3 of 4 questions. In part II, a
student must answer 6 of 8 questions and in part III, a student must answer all 5 questions. How
many choices of questions does the student have?
A) 120
B) 288
C) 112
D) 124
E) None of these
14. Given that 6
3
3
3
A) 4096
6
3
6
6
3
3
B) 6440
6
3
6
3
6 . What is
when
3
3 .
C) 8064
D) 5044
E) None of these
15. The interior angle of a regular polygon is 144 degrees. If the length of a side is 12 units, what is the
perimeter?
A) 80
B) 100
C) 120
D) 140
E) 160
16. The expression
A) cot
sin
cos
B) cot
sin
cos
C) tan
D) tan
E) None of these
17. You have 600 feet of fencing to enclose a rectangular plot of land that borders on a river. If you do
not fence the side along the river, what is the largest area that can be enclosed in square feet?
A) 40,000
B) 90,000
C) 45,000
D) 50,000
E) None of these
|
18. Given the function
A) 3
B) 2
1|
| |
|
C)1
?
1|, what is the slope of the line segment at
D) 2
E) None of these
19. In the figure below, the chord AB of length 6 subtends an angle of 90° at the center of the circle.
Find the area of the shaded region.
A) 12
C) 36
6
18
B) 9
2 /2
D) 18 36
4
E) None of these
Comprehensive
NC State Math Contest
20. Simplify
A) tan 2
cos
cos 3
sin
sin 3
C) cot 2
B) 2 tan 2
ECSU
April1, 2014
D) 2 cot 2
E) None of these
21. Let R be a rectangle having one side of length 2 and an area of 12. Let T be a second rectangle that
has a diagonal of length 15 and is similar to rectangle R. The area of T is
B) 27√10/4
C) 9/2
D) 36
E) None of these
A) 9√10
22. Given below is the graph of the function
2 be zero?
A) (  ,  4)  (  2, 0)  (2,  )
C) (  ,  4)  (2,  )
, for what values of
will the graph of
B) (  , 4)  (0,1)  (2,  )
D) (  ,  4)  (4,  )
E) None of these
23. Given two similar triangles, the area of the larger triangle is sixteen times the area of the smaller
triangle. Find the ratio of the perimeter of the larger triangle to the perimeter of the smaller triangle.
B)4 ∶ 1
C) 16 ∶ 1
D) 32 ∶ 1
E) 8 ∶ 1
A) √8 ∶ 1
24. Find an equation for the graph.
A) y = cos 3x B) y = cos 9x
C) y = sin 9x
D) sin 3x
5
Comprehensive
NC State Math Contest
ECSU
April1, 2014
25. In Elizabethan City in April 2010, the average lowest temperature 33° F was observed at 2:00 AM
and the average highest temperature 51°F was observed at 2:00 PM. Assume this average
temperature followed a sine graph, with t = hours after 8:00 AM. Which of the following represents
the average temperature from 8:00 AM that day for 24 hours?
 
 2 
 2 
t
t
B) 42  18sin 
C) 42  9sin 
A) 42  18sin  t 
 24 
 24 
 24 
 
D) 42  9sin  t 
E) None of these
 24 
26. The sum of all positive integers between 50 and 350 whose units digit is a 1 is
A) 5880
B) 5539
C) 5488
D) 4877
E) 4566
27. The ∆
below is an isosceles triangle with
3 and
and is a point on
such that
5. If the area of ∆
what is the length
?
4. D is a point on
is half of the area of ∆
,
A
E
B
C
D
A) 3
B) 2
C) 1
D) 0.5
E) None of these
28. Find an equation for the graph.
A) y = 5 sin (x/3)
B) y = 5 sin 3x
C) y = 5 sin(x/5)
D) y = 5 sin 5x
29. Four large and 2 small pumps can fill a swimming pool in 2 hours. Two large and 6 small pumps can
also fill the same swimming pool in 2 hours. How long does it take 8 large and 8 small pumps to fill
50% of the swimming pool? (ASSUME: all the large pumps have same power and all the small
pumps have the same power).
A) 20 minutes
B) 15 minutes
C) 25 minutes D) 45 minutes
E) None of these
6
Comprehensive
NC State Math Contest
ECSU
|2
30. The sum of the solutions of the equation 4
A) 3
B) 2
C) 1
1| is
D) 6
April1, 2014
E) 7
31. The figure below is a square. What is the ratio of the shaded area to the unshaded area?
3x
x
A)1 ∶ 3
B)1 ∶ 6
32. The solutions of tan
A) /6, /2
C) 7 /6, 11 /6
sec
C) 1 ∶ 5
D)2 ∶ 9
1 are:
B) /3, 5 /3
D) 3 /4, 2 /3
E) None of these
E) None of these
33. A certain brand of razor blades comes in packages of 6, 12, and 24 blades costing $2, $3, and $4 per
package, respectively. A store sold 12 packages containing a total of 162 razor blades and took in
$35. How many packages of the 12 blades per package were sold?
A) 5
B) 6
34. Determine the value of
A) 28
B) 7
C) 8
D) 3
⋯
for which
C) 8
E) None of these
5 is
D) 30
E) 9
35. In the triangle ABC, ∠ACB = 120°, AC = 8 and BC = 4. The bisector of ∠ACB meets the side AB at
the point D. Find the length of CD.
A) 2/3
B) 4/3
C) 8/3
D) 1
E) None of these
36.
7
Comprehensive
A) cos x
NC State Math Contest
B) sec x
C) sin x
ECSU
April1, 2014
D) csc x
37. A man can commute to work by either subway or bus. If he takes the subway in the morning, he
returns on the bus in the afternoon, and if he returns in the afternoon on the subway, he took the bus
in the morning. Over a period of weekdays, the man took the bus to work in the morning 9 times,
returned home by bus in the afternoon 13 times, and commuted by subway a total (either morning or
afternoon) of 18 times. The value of is
A) 23
B) 22
C) 21
D) 20
E) cannot be determined from the information provided
38. What are the last two digits of 7
A) 07
B) 01
39. In the triangle
, altitude
What is the length of
?
A) 1
B) 2
?
C) 49
D) 43
meets altitude
at . Suppose
C) 3
D) 1/2
E) None of these
8,
6 , and
4.
E) 5/2
40. A painter needs to cover a triangular region 60 meters by 68 meters by 72 meters. A can of paint
covers 64 square meters. How many cans will be needed?
A) 30 cans
B) 40 cans
C) 308 cans
D) 25 cans
E) None of these
8
Comprehensive
NC State Math Contest
Answers:
1. D
2. B
3. C
4. A
5. A
6. C
7. D
8. A
9. D
10. A
11. D
12. A
13. C
14. A
15. C
16. D
17. C
18. C
19. B
20. C
21. C
22. A
23. B
24. A
25. C
26. A
27. C
28. B
29. C
30. B
31. C
32. B
33. D
34. D
35. C
36. C
37. D
38. A
39. C
40. A
9
ECSU
April1, 2014