Elizabeth City State University
Elizabeth City, North Carolina27909
2014
STATE REGIONAL MATHEMATICS CONTEST
Level II
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.
Regional State Mathematics Contest
Level II
ECSU 2014
1.
How many real roots does 2( x 2 1) 2 ( x 2 1) 3 0 have?
A. 0
2.
B. 1
Solve for t: 4 2
C. 2
B. 54
If
f ( x)
A. 1/2
6.
7.
C. {t | 4/3 > t > 2}
C. 36
D. 90
E. 45
The volume of a cylinder varies jointly as the height and the square of the base
radius. If a cylinder’s height is doubled and its base radius is halved, then its
volume:
A. is quadrupled
D. remains the same
5.
B. {t | 4/3 < t < 2}
E. {t | t > 2}
The lengths of the three sides of a right triangle are consecutive multiples of
three. What is the area of the triangle?
A. 108
4.
E. 4
3t 5
5.
2
A. {t | t < 4/3 or t > 2}
D. {t | t > 4/3}
3.
D. 3
4
1, find
x
B. -3/7
Solve 27 2 t 1 81t 2 .
A. 3
B. -3
B. is tripled
E. is halved
f 1 (7) , if possible.
C. 4
D. 0
C. -1/2
D. -2
What is the range of the function f(x) =
A. [0, )
B. [0,5]
C. is doubled
C. [0, 23]
E. f
1
does not exist
E. 11/2
8sin 3 x 17 ?
D. [3,5]
E. none of these
8. Suppose a and b are positive integers such that (a 2b)(a b) 10 . What is the
value of 2a b ?
A. 1
B. 2
C. 5
D. 4
E. none of these
9. If log y x log x y 7, then what is the value of (log y x) 2 (log x y ) 2 ?
A. 40
B. 43
C. 45
D. 47
E. 49
10. Let f ( x) be a function such that, for every real number x,
f ( x) 2 f ( x) sin x.
What is the value of f ( ) ?
2
A. 1
B.
1
2
C.
1
2
D. 0
E. none of these
11. Given f ( x) x 2 and g ( x) 3 x , find ( f 1 g 1 )(2).
B. 6
A. 8
12. If 0
4
A. 1
, then
C. 2
D. 6
E. none of these
1 tan 1 sin 2
is equal to
1 tan
cos 2
B. 0
C.
3
2
D.
2
2
E. none of these
13. If sin x cos x 1.2 , then sin 2x
A. .88
B. .88
C. .44
D. .44
E. none of these
14. If f (n 1) f (n) 2n 1 for all positive integers n , and f (1) 1, then f (40) equals
A. 1400
B. 1200
C. 1600
D. 2200
E. none of these
15. The multiplicity of the factor ( x 1) of x5 2 x 4 2 x3 8 x 2 7 x 2 is
A. 1
B. 2
C. 3
D. 4
E. none of these
16. If r and s are the roots of x 2 6 x 2 0, then
A. 6
B. 6
1 1
r s
D. 3
C. 3
E. none of these
17. A drawer contains 64 socks. Each sock is one of 8 colors, and there are 8 socks of
each color. If the socks in the drawer are thoroughly mixed and you randomly
choose two of socks, then what is the probability that these two socks will have the
same color?
A.
1
7
B.
1
8
C.
1
9
D.
7
64
E.
9
64
18. If parallelogram ABCD below, BD = 3 and CD = 5. What is the area of ABCD?
A
B
3
A. 12
B. 15
D
5
D. 18
C. 20
C
E) none of these
19. If the radius of a circle is decreased by 20%, what happens to the area?
A. 10% decrease
D. 40% decrease
B. 20% decrease
E. None of these
C. 36% decrease
4
20. Solve for x in the equation log 4 x 3 3log x (16 x) 7 .
A. 16
B. 27
C. 64
D. 81
E. none of these
21. In the figure below, four circles are drawn with a square whose perimeter is 32.
What is the area of the shaded region?
A. 32-16π
B. 64-16π
C. 64-32π
D. 32π-32
E. None of these
22. What is the sum of the distances AD and BD in the figure shown?
A. 21
B. 23
C 25
D. 27
E. 29
23.What is the area of the shaded portion of this rectangle, given that AD = 6,
CD = 8, AE x
A. 48-3x
B. 48+3x
C. 3x+16
D. 24-3x E. None of these
24. In the figure, the segments of the length a and b lie on the perpendicular to the
diagonals of a square of side 3 . Find a b .
A.
3 2
2
B. 3
C
2 3
2
D.
2 3
3
E. none of these.
25. A square and an equilateral triangle have equal parameters. The area of the
triangle is 9 3 square inches. What is the length of the diagonal of the
square?
5 2
9 2
A.
B. 4 2 C. 9 2 D.
E. None of these
2
2
26. An 8''×12'' paper is folded so that the upper right corner touches the middle of
the opposite (left ) side. Find the lendth of the fold.
A 4 15 ''
B. 10''
C. 10
5
''
12
D. 5 5 ''
E. none of these
27. Find the area of the quadrilateral ABCD with the lengths as shown in the
figure.
A. 54 sq units
B.130 sq units
C. 114 sq units D. 168 sq units E. None of these
28. The smallest angle in the right triangle with sides 8, 15, and 17 is
approximately:
A.
27.8 o
B. 82.4 o
C. 7.6 o
D. 51. 5 o
E. 58.1 o
29. A 16'' diameter pizza is cut into 12 equal slices. What is the distance around
one slice?
A.
16
3
4
5
3
B. 16
C. 16
4
3
D. 16
E. none of these
30. Modify the right triangle with sides 11, 60 and 61 by doubling the length of its
shortest side, but keeping its perimeter and larger angle unchanged. What is the
length of the new hypotenuse?
A. 52
4
5
B. 55
C. 57
1
5
D. 2885
E. none of these
31. Given the following two circles, find the equation of the line connecting their
centers.
Circle I: x 2 4 x y 2 6 y 12 0 Circle II: x 2 8 x y 2 0
A. 4 x 6 y 12 B. 3x 2 y 12 C. x 2 y 4
D. 3x 2 y 12 E. none of these
32. An arbelos as shown in the figure, in the region formed by three mutually
tangent circles whose centers are collinear. If the radii of two smaller circles
a and b , h is the high of the larger circle at the tangent point of other two
circles. What is the area of the arbelos ?
A.
h 2
4
B.
h a 2 b 2
2
C.
h(a b)
2
D. 2ab E. none of these
33. Equilateral triangle ABC is inscribed in a circle with center O, as shown. If the
radius of the circle is 2, what is the area of triangle ABC?
A. 3
B. 2 3
C. 3 2
D. 3 3
E. None of these.
34. A spherical orange with radius 2 units is sliced into eight by three
perpendicular planes through its center so that peel of each piece look like
the figure. Determine the distance around the edge of the peel in the picture
(on the right).
A. 6 units
B.
4
units
3
C. 3 units
D. 6 units
E. none of these
35. In the figure below, If AE 1 , what is the sum of the of triangles
ABC and CDE if AECD is a parallelogram?
A.
1
3
B.
1
2
C. 1
D. 2
E. none of these
36. Two circles are drawn internally tangent to one another, as shown. The
segments given are parts of the extended diameters of the smaller circle. If
x 12 and y 5 what is the length of the larger circle’s radius.
A. 18
1
2
B. 12
1
2
C. 4 15
D. 17
E. 21
1
4
37. If f ( x) is a function that satisfies f (2 x 1) 2 f ( x) 1 for all x, and if
f (0) 2 , then f (3)
A. 5
B. 9
C. 11
D. 13
E. 1
38. Three mutually tangent unite circles are circumscribed by a fourth circle
as shown in the figure. Find the area of shaded region of larger circle.
A. (36 3 62)
24 3
D.
3
4 3 2
B.
3
2 2
2
C.
3
E. none of these
39. Suppose the roots of the quadratic equation x 2 ax b 0 are sin15 and
cos15 . What is the value of a 4 b 2 ?
35
A. 1
B. 1
C.
D. 1 2
E. none of these
16
40. A circle has the same center as an ellipse and passes through the foci F1 and F2 ,
as shown in the picture below. The two curves intersect at four points. Let P be any
one of the points of intersection. If the major axis of the ellipse is 15 and the area of
the triangle PF1 F2 is 26, the distance between the foci is
A. 10
B. 13
C. 11
D. 12
E) cannot be determined
0
