Elizabeth City State University
Elizabeth City, North Carolina 27909
MARCH 29, 2012
NC STATE REGIONAL MATHEMATICS CONTEST
COMPREHENSIVE
TEST BOOKLET
Directions: Each problem in this test is followed by four to 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
B) 7
54 – 48 =
C) 16
D) 12
E) None of these
Sample answer:
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.
1
1. Find the quotient:
A)
4
5
3
3
+
i
B)
-
i
C) )
-
i
D)
-
i
2. A ball is dropped from a cliff that is 224 feet high. The distance s (in feet) that it falls in t seconds
is given by the formula s = 16t2. How many seconds (to tenths) will it take for the ball to hit the
ground?
A) 3136 sec
B) 15 sec
C) 3.7 sec
D) 14.4 sec
3. The number of non-text books read by college students ranges from 8 to 62. Using B as the
variable, write an absolute value inequality that corresponds to this range.
A) |B - 8| ≤ 54
B) |B - 35| ≤ 27
C) |B - 54| ≤ 8
D) |B - 27| ≤ 35
4. Find the center and radius of the circle:
x2 + y2 + 12x - 2y + 28 = 0
A) center: ( -1, 6); radius: 9
B) center: ( 6, -1); radius: 9
C) center: ( 1, -6); radius: 3
D) center: ( -6, 1); radius: 3
5. The volume V of a given mass of gas varies directly as the temperature T and inversely as the
pressure P. A measuring device is calibrated to give V = 420 in3 when T = 560o and P = 20 lb/in2.
What is the volume on this device when the temperature is 330o and the pressure is 25 lb/in2?
A) V = 188 in3
B) V = 198 in3
C) V = 208 in3
D) V = 13.2 in3.
6. Body-mass-index (BMI) varies directly as one’s weight and inversely as the square of
one’s height. A person with weight 167 pounds and height 69 inches has BMI of 24.66.
What is the BMI of a person who weighs 130 pounds and is 64 inches tall?
A) 21.6
B) 22.3
C) 22.7
D) 21.9
7. The formula N = 3x2 + 2x + 1 represents the number of households N, in thousands, in a certain
city that have a computer x years after 1990. According to the formula, in what year were there 57
thousand households with computers in this city?
A) 1992
B) 1995
C) 1994
D) 1993
8. A rain gutter is made from sheets of aluminum that are 28 inches wide. The edges are turned up to
form right angles. Determine the depth of the gutter that will allow a cross-sectional area of 96
square inches.
A) 6 in. or 8 in.
B) 6 in. or 22 in.
C) 8 in. or 12 in.
D) 8 in. or 20 in.
9. Solve:
A)
log3(x + 6) + log3(x - 6) - log3x = 2
{-3}
B) {12, -3}
C) {12}
D) Φ
10. Solve:
e2x + ex – 6 = 0
A) 2
B) -3
C) ln 2
D) ln 3
2
E) None of these
11. ABCD is a rectangle with area 48 in2 and the length of a diagonal is 10 in. What is the perimeter
of the rectangle?
A) 8 in
B) 12 in
C) 28 in
D) 36 in
E) None of these
12. A circle is inscribed in an equilateral triangle. The radius of the circle is 6 cm. Find the area of
the triangle.
A) 18 3
B) 36 3
C) 72 3
D) 108 3
E) None of these
13. In the figure below, DE is parallel to BC . BAC  x ,  BCA  x  30  and BDE  4 x . Find
the measure of ABC .
A) 30o
B) 45o
C) 60o
D) 120o
E) None of these
14. In the right triangle ABC , CF is perpendicular to AB . Find the length of CF .
A) 13 / 13
B) 2 13 / 13
C) 6 13 / 13
D) 12 13 / 13
E) None of these
15. ABC is an equilateral triangle with AB  6 . If P is the point of intersection of the three angle
bisectors, what is the length of AP ?
A) 3
B) 2 3
C) 3 3
D) 4 3
E) None of these
16. What is the area of ABC with the vertices A(1,3) , B (1,  5) , and C ( 7 ,  8) ?
A) 2
B) 20
C) 24
D) 36
E) None of these
17. A right-angled isosceles triangle is inscribed in a circle of diameter 10. What is the perimeter of
the triangle?
B) 5  5 2
C) 20  20 2
A) 20  10 2
E) None of these
D) 10  10 2
3
18. The sum of the length of the three sides of a right triangle is 26. The sum of the squares of the
lengths of the three sides is 288. Determine the area of the triangle.
A) 13
B) 15
C) 18
D) 26
E) None of these
19. A triangle of sides 20, 20, and 24 is inscribed in a circle. What is the radius of the circle?
A) 12.5
B) 13.5
C) 11.5
D) 15
E) None of these
20. An equilateral triangle is inscribed inside a circle. If a side of the triangle is 12, what is the
circumference of the circle?
A) 3
B) 4 3
C) 8 3
D) 12 3
E) None of these
21. Given that: cos  2 / 3 , and tan  0 . Find sin  .
5
5
A) 
B) 5
C) 
2
3
D) 
3
2
E) None of these
22. In the following triangle, a = 3 in, and A  60  . Find the lengths b and c.
A) b = 1.5 in, c = 3 3 in
C) b = 3 in, c = 2 3 in
23. Complete the identity:
sin 2 x  1
sin x  1
A) sin x
B) sin x  1
24. Complete the identity:
cos(   )
sin  cos 
A) cos   sin 
C) cos   sin 
25. Complete the identity:
sin( 3 )
sin 
A) 3cos2θ – sin2 θ
B) b = 3 in, c = 2 3 in
D) b = 2 3 7 in, c = 3 in
C) cos 2 x
D) sin x  1
B) cot  1
D) cot   tan 
B) 2 sin cos
4
E) None of these
E) None of these
C) cos( 2 )  sin  cos 
D) 2 cos( 2 )
E) None of these
26. Complete the identity:
sin 2

2
sin 
2
A) sin 2

2
1
C)
2  2 cos 
B) cos 2

2
1
D)
2  2 cos 
E) None of these
27. Complete the identity:
sin x (sin 5 x  sin 7 x )
A) cos x (cos 5 x  cos 7 x )
C) cos x (cos 5 x  cos 7 x ) / 2
28. Complete the identity:
sin x  sin y
cos x  cos y
x y
A) tan

 2 
x y
C) tan

 2 
B) cos x (cos 5 x  cos 7 x ) / 2
D) cos x (cos 5 x  cos 7 x )
E) None of these
x y
B) cot

 2 
D) tan x  tan y
E) None of these
29. A radio transmission tower is 120 feet tall. How long should a guy wire be if it is to be attached
20 feet from the top and has to make an angle of 60° with the ground?
A) 200/ 3 feet
B) 200 3 feet
C) 100/ 3 feet
D) 100 3 feet
E) None of these
30. A building 100 feet tall casts a 50 foot long shadow. If a person looks down from the top of the
building, what is the measure of the angle between the end of the shadow and the vertical side of
the building?(Assume the person's eyes are level with the top of the building.)
A) sin-1(2/ 5 )
B) cos-1 (2/ 5 )
C) 60°
D) 30°
E) None of these.
31. The areas of the sides of a box are 60 and 40 square units. The top has area 80 square units. What
is the height of the box?
B) 10 2
C) 15
D) 30
E) None of these
A) 12
32. Dumbbells weigh 20, 30, or 40 lbs. The total weight of a pile of dumbbells is 800 lbs. The number
of dumbbells in the pile that weighs 30 lbs CANNOT be:
A) 2
B) 3
C) 4
D) 6
E) 10
5
33. Donald Duck can eat 2 pizza slices in 3 minutes; while Goofy can eat 3 pizza slices in 2 minutes.
At these rates, how many pizza slices can they eat together in an hour?
A) 54
B) 96
C) 120
D) 130
E) None of these
34. If a basketball team wins 60% of their first 20 games, how many games of the remaining 40 games
must they win in order to have won exactly 50% of all 60 games?
A) 18
B) 22
C) 16
D) 14
E) None of these
35. Initially glass A holds 12 ounces of water and glass B holds 12 ounces of milk. Then x ounces of
water is transferred from A to B , mixed with the milk, and then x ounces of the mixture is
transferred from B to A . If A now holds 7 ounces of water, then x equals (in ounces)
60
25
57
A)
B)
C) 8
D)
E) None of these
7
4
7
36. Students registered to take a Mathematics contest exam by taking Algebra and Geometry or both.
The number of students that actually took two exams together with the no shows is 5. The number
of students that took Geometry is 11. The number of students that took only Geometry is 4 times
that number of no shows. How many students did not show up?
A) 4
B) 2
C) 1
D) 6
E) None of these
37. Three vehicles leave points A, B and C as shown below. Vehicle A leaves and travelling straight
towards the center at 55 mph. Vehicle B leaves 10 minutes later travelling at 60 mph towards the
center and C travels 20 minutes later at 70 mph also towards center. Which car will arrive at the
center first?
A
100
B
100
C
A) A
B) B
C) C
D) A & B
E) None of these
38. Suppose the price of item A that was reduced by 5% becomes double the price of an item B that
was reduced by 10%. What was the ratio of the price of item A over that of item B before the
reduction?
6
A) 2:1
B) 36:19
C) 24:15
D) 18:13
E) None of these
39. A test that has 40 multiple choice problems has the following score policy. A correctly answered
problem earns 4 points, and an incorrectly answered problem causes a loss of a point. An
unanswered problem causes neither loss nor gain. Bianca answers 28 questions correctly and
leaves the rest unanswered. Randy guesses all problems and gets 12 problems correct. What is the
difference between Bianca’s and Randy’s scores?
A) 80
B) 92
C) -30
D) 96
E) None of these
40. Joy and Jim need to print photos for their baby. They anticipate printing around 1600 photos.
Option I is to upload the photos to an online printing site where each 8 6 inches print costs 15
cents and each order having at most 200 prints ships at a cost of $2.00. Option II is to buy an ink
printer at $199.95. They can purchase a package having Ink and a 8 1 2  14 paper ream having 500
papers at $36.99. The ink can print all photos. Additional reams of paper can be purchased at unit
price of $10. Assume each paper can hold 3 photo prints, which option is cheaper and by how
much?
A) Option I by $20
B) Option II by $20
C) Option II by 7
D) Option II by $9
E) None of these
7
Answers:
1. A
2. C
3. B
4. D
5. B
6. B
7. C
8. A
9. C
10. C
11. C
12. D
13. D
14. D
15. B
16. C
17. D
18. A
19. A
20. C
21. C
22. A
23. B
24. D
25. A
26. C
27. A
28. A
29. B
30. B
31. D
32. B
33. D
34. A
35. A
36. B
37. C
38. B
39. B
40. D
8