Elizabeth City State University
Elizabeth City, North Carolina 27909
2011
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
2
1. Suppose that b and c are constants and ( x  3)( x  b)  x  cx  12 . What is c?
A) 4
B) 5
C) 6
D) 7
2. Completely factor the polynomial x 3  2 x 2  6 x .
A) x( x 2  2 x  6)
B) x( x  1  7 )( x  1  7 )
D) x ( x  4)( x  2 )
E) None of these
March 31, 2011
E) None of these
C) x( x  1  7 )( x  1  7 )
3. Joy Palmer, a pharmacist, needs 60 liters of 60% alcohol solution. She has available a 70% alcohol solution
and a 30% alcohol solution. How many liters of each solution should she mix to obtain 60 liters of 60%
alcohol solution?
A) 10 liters of 70% solution and 50 liters of 30% solution
B) 20 liters of 70% solution and 40 liters of 30% solution
C) 15.5 liters of 70% solution and 44.5 liters of 30% solution
D) 45 liters of 70% solution and 15 liters of 30% solution
E) 28 liters of 70% solution and 32 liters of 30% solution.
4. Find the value of the sum: 2011 2010 2009 2008   4  3  2  1
A) 1006
B) 845700
C) 0
D) 46850
E) None of these
5. Find the domain of the function f ( x)  ( x  4) 1 / 2 (5  x)1 / 3 .
A) ( 4,  )  (5,  )
B) ( 4,  )
C) ( ,5)
E) None of these
D) ( 4,  )  (  ,5)
6. Determine the value of A so that the line whose equation is x  Ay  3 is perpendicular to the line containing
the points ( 2, 1) and ( 4,2) .
A) 2
B)  1/ 2
C) 1 / 2
D) 3
E) None of these
7. Given the following equation: 3 2 x  3 x  20  0 , solve for x.
ln 5
B)  4, 5
A)
C) 20,  1
ln 3
8. Let i   1 . Simplify i 1997  i 2011 .
A) i
B)  i
C) 0
D)
ln 5 ln 5
,
ln 4 ln 3
D) 2i
E) None of these
E) None of these
9. Find the cubic equation with integral coefficients having roots 3 and 1 2i .
B) x 3  x 2  11 x  15
C) x 3  5 x 2  11 x  15
A) x 3  x 2  11 x  15
E) None of these
D) x 3  5 x 2  11 x  15
2
Comprehensive
NC State Math Contest
ln ln ln( x ) 
is
10. The domain of the function
ln ln( x ) 
A) { x | x  0}
B) { x | x  e}
C) {x | x  e 2 }
ECSU
D) {x | 0  x  e 2 }
March 31, 2011
E) None of these
11. In the right triangle ABC with legs of length 5 and 12, the hypotenuse is of length 13. A semicircle is
inscribed in the triangle. Find the radius of the semicircle.
A) 17/3
B) 13/5
C) 10/3
D) 12/5
E) None of these
12. 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.
A) 8 : 1
B) 4 : 1
C) 16 : 1
D) 32 : 1
E) None of these
13. In the right triangle ABC, CF is perpendicular to AB. Find the length of CF.
B
F
4
C
A) 13 / 13
B) 2 13 / 13
6
C) 6 13 / 13
A
D) 12 13 / 13
14. Find the sum of all solutions of the equation x 4  2 x 2  15  0 .
D) 14
A) 2
B) 0
C) 2 5
E) None of these
E) None of these
15. Find the area of a trapezoid in which the bases are 17 m and 42 m and the legs are 15 m and 20 m.
B) 354 m2
C) 400 m2
D) 475 m2
E) None of these
A) 225 m2
16. The interior angles of a polygon are in arithmetic progression; the least angle is 120 and the common
difference is 5 . Find the greatest number of sides that this polygon can have.
C) 12
D) 9
E) None of these
A) 20
B) 16
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Comprehensive
NC State Math Contest
ECSU
March 31, 2011
17. In the figure below, two small circles are inscribed and tangent to the large circle at points A and D. Let
points B and C be the centers of the circles, where AB = BC = CD = 2. Find the area of the shaded region.
A)
B) 4.5
C) 3
D) 1.5
E) None of these
18. In a triangle ABC, AC = 12. If one of the trisectors of angle B is the median to AC and the other trisector of
angle B is the altitude to AC, find the length of the altitude.
A) 2 3
B) 3 3
C) 4 3
D) 7 5
E) None of these
19. In the circle shown, AB = 24, and the perpendicular chord CD bisects AB. How long is BD if DM is 4 times
as long as CM?
A) 4 5
B) 8 5
C) 12 5
D) 16 5
E) None of these
20. If the diameter of a cylindrical can is increased by 30 percent, by approximating what percentage should the
height be increased to triple the volume of the can?
B) 54.8
C) 50.0
D) 44.2
E) None of these
A) 77.5
21. An arc of length has length 2π in. and radius 8 in. What is the central angle the arc makes?
A) 45 
B) 90 
C) 180
D) 360
E) None of these
22. Find the amplitude, period, phase shift of y  3 sin(2 x   / 2) .
A) Amplitude =  3 , period =  , phase shift =  / 4
B) Amplitude = 3 , period = 2 , phase shift =  / 4
C) Amplitude = 3 , period =  , phase shift =  / 2
D) Amplitude = 3 , period =  , phase shift =   / 4
E) None of these
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Comprehensive
NC State Math Contest
23. sin 4  is
A) 3 / 8  (1 / 2) cos 2  (1 / 2) cos 4
C) 3 / 8  (1 / 2) cos 2  (1 / 6) cos 4
E) None of these
tan   cot 
is
tan   cot 
A) 2 sin 2   1
ECSU
March 31, 2011
B) 3 / 8  (1 / 2) cos 2  (1 / 4) cos 4
D) 3 / 8  (1 / 2) cos 2  (1 / 8) cos 4
24.
1  cos 
is
sin 
1  sin 
A)
cos 
B) 2 cos 2   1
C) 2 sec 2   1
D) 2 csc 2   1
E) None of these
1  sin 
sin 
E) None of these
25.
B)
sin 
1  cos 
C)
1  cos 
sin 
D)
sin 
1  cos 

is
1  cos 
sin 
A) 2 sin 
B) 2 cos 
C) 2 sec 
D) 2 csc 
E) None of these
1
1

is
1  cos  1  cos 
A) 2  2 sin 2 
B) 2  2 cos 2 
C) 2  2 tan 2 
D) 2  2 cot 2 
E) None of these

 5 
28. The value of cos tan 1    is:
 12  

A) 5/13
B) 3/12
C) 5/12
D) 12/13
E) None of these
26.
27.
C
29. In a triangle ABC, cos  is
2
s( s  a)
s ( s  b)
B)
A)
bc
ca
C)
s ( s  c)
ab
D)
s( s  a )
ab
E) None of these
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Comprehensive
NC State Math Contest
ab
is
c
 A B  A B
sin
 sin

 2   2 
A)
C
C
sin cos
2
2
 A B  A B
cos
 sin

 2   2 
C)
C
C
sin cos
2
2
E) None of these
ECSU
March 31, 2011
30. In a triangle ABC,
 A B  A B
sin
 cos

 2   2 
B)
C
C
sin cos
2
2
 A B  A B
cos
 cos

 2   2 
D)
C
C
sin cos
2
2
31. Two similar cones are cut out of a cylinder. The cylinder has radius r and height h. How much volume is
left of the cylinder?
A)
1 2
r h
3
B)
2 2
r h
3
C)
3 2
r h
4
D)
1 2
r h
2
E) None of these
32. Given the square ABCD with M the midpoint of DC. The ratio of the area of the triangle MEC to that of the
quadrilateral AEMD is:
A) 1 : 3
B) 1 : 4
C) 1 : 5
D) 1 : 6
E) None of these
33. What is the y-intercept of the line that is parallel to y  2 x , and which bisects the area of a rectangle with
corners at ( 0, 0), ( 2, 0 ), ( 2, 6 ), and ( 0, 6 ) ?
A) 1
B) 1.5
C)  1
D)  2
E) None of these
6
Comprehensive
NC State Math Contest
ECSU
March 31, 2011
34. If log a x  4 , then log1 / a ( x / a)  ?
A)  4
C)  3
B) 1
D) 1 / 4
E) None of these
35. The floor of a square hall is tiled with unbroken square tiles. Along the two diagonals there are 111 tiles
altogether. How many tiles are there in the hall?
A) 3025
B) 3136
C) 3249
D) 4225
E) None of these
36. A palindromic number is a natural number that is the same when written forwards or backwards. For
instance, 323 is palindromic number. How many three-digit palindromic numbers are there?
A) 100
B) 81
C) 90
D) 99
E) None of these
37. For all real numbers x, except x  0 and x  1, the function f is defined by
 x  1
f

 x 1 x
2
Suppose 0     / 2 . What is f cos  ?
A) tan  
2
B) sin  

2

C) cos 
2
D) cot  
38. How many subsets X exist such that {1,2,3}  X  {1,2,3,4,5,6,7} ?
A) 16
B) 112
C) 120
D) 128
2
E) csc  
2
E) None of these
39. If the number 848 848 is written out as an ordinary base 10 numeral, then what is the digit in the ones place?
A) 2
B) 4
C) 6
D) 8
E) None of these
40. Find the sum of all the three-digit palindromic numbers.
A) 49500
B) 50400
C) 45900
D) 49950
E) None of these
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Comprehensive
Answers
Question
Answer
1
D
2
B
3
D
4
A
5
B
6
B
7
A
8
C
9
C
10
B
11
C
12
B
13
D
14
B
15
B
16
B
17
E
18
B
19
C
20
A
21
A
22
D
23
D
24
A
25
B
26
D
27
D
28
D
29
C
30
B
31
B
32
C
33
A
34
C
35
B
36
C
37
A
38
A
39
C
40
A
NC State Math Contest
ECSU
March 31, 2011
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