Download Comphrensive 2007 - Wake Technical Community College

2013 State Math Contest
Wake Technical Community College
Comprehensive Test
1. Tasmania is the 26th largest island in the world. It is shaped like an equilateral triangle. If the length
of the side is approximately 240 miles, which of the following is a reasonable estimate of the size of
the island?
a. 21,600 mi2
b. 23,000 mi2
c. 28,800 mi2
d. 22,000 mi2
e. 25,000 mi2
2. The average of three numbers is 42. The smallest two numbers differ by 1, and the average of the
largest two numbers is 47. What is the difference of the largest and the smallest of the numbers?
a. 29
b. 32
c. 31
d. 30
e. 33
3. A regular octagon is inscribed in a circle of diameter 5 inches. What is the area of the octagon?
b. 50 in2
a. 25 2 in2
4. What is the sum of the solutions of 8x
a. 1
c. 50 2 in2
2
3 x 10
b. −1
 4x
2
x
c. 11
d.
25 2 2
in
2
e. 25 in2
?
d. −11
e. −2.5
5. How many diagonals of a regular decagon do not lie on lines of symmetry?
a. 18
6. Evaluate
a. i
b. 30
c. 25
d. 20
e. 24
c. −1
d. 2
e. i
 5
sin 
 6
  3   5 
 cos 
 tan 

  4   4 .
 2   7 
cos 
 sin

 3   4 
b. 1
1
2013 State Math Contest
Wake Technical Community College
Comprehensive Test
1
7. Determine the sum of the solutions of the equation
1
1
a. 0
b.
4
3
c. 1
 2.
1
1
x 1
d. 
5
3
e. 
4
3

1
 5
8. Evaluate sin arctan   arccsc    .
3
 3

a.
13
5 10
b. 
1
10
c.
9
5 10
d.
5  3 10
5 10
e.
9 10
13
9. If log 2.83  0.4518 , then compute log 283  log 0.000283 .
a. 2.4518
b. 5.9036
10. Determine the solution set of ln
c. −4

d. 6
e. −2
1  sin x    ln 1  sin x    lncos  x  on the interval
0 x  .
 
a.  , 
2 
 
b. 0, 
 2
    
c. 0,    , 
 2 2 
d. 0, 
e.  0, 
11. How many times do the graphs of y  3sin x  and y  3sin 3x  intersect for 0  x  640.5 ?
a. 1920
b. 1921
c. 1922
2
d. 1923
e. 1924
2013 State Math Contest
Wake Technical Community College
Comprehensive Test
12. The polynomial F  x   x 4  x 3  10x 2  5x  25 can be factored as two polynomials G  x  and H  x  .
The coefficients of G  x  and H  x  are integers and their degrees are greater than zero. Compute
G 4  H 4 .
a. 21
b. 26
c. 28
d. 36
e. 18
13. Square ABCD with vertices (0, 0), (2, 0), (0, 2), and (2, 2) is rotated 30° counterclockwise. Determine
the area common to the two squares.
a.
6 3
5
b.
3 3
2
c.
8 3
5
d.
4 3
3
e.
5 3
3
14. A number is chosen at random from the set of all 5-digit numbers containing the digits 1,2,3,4,5
exactly once. Compute the probability that no two adjacent digits are consecutive integers.
a.
1
10
b.
7
60
c.
2
15
d.
3
20
e.
1
6
15. In triangle ABC, let mA  2mB , AC  2 , and BC  3 . Which of the following is the mB ?
3
a. cos 1  
4
4
b. cos 1  
3
16. What is the sum of the solutions of
a. 6
b.
2
3
3
c. sin1  
2
x
2
 3x  4  
c.
5
3
x
2
d. cos 1  
3
2
 5x  4   x  1 ?
d. 
3
4
e. sin1  
3
10
3
e. 5
2013 State Math Contest
Wake Technical Community College
Comprehensive Test
17. What is the behavior of the graph of y 
x2  4
as x approaches 2 from the left (values slightly
x2  4 x  4
less than 2)?
a. y approaches 0 b. y approaches 1 c. y approaches −1 d. y approaches  e. y approaches 
18. Let f  x   sin x  , g  x  
a.
1
2
x 2

, h  x   x  , and k  x   cos  x  . Evaluate
2
6
b. −1
3
2
c.
19. Which of the following is equivalent to
csc  x 
 csc  x   1
2
d. 
3
2
d. csc  x   csc  x   cot  x  
e. 
1
2
?
b. sec  x   sec  x   tan x  
a. csc  x 
     
k  h  g 1  f      .


    6 
e.
c. sec  x 
cos  x   sin x   1
sin x 
20. Two cars drive around a racetrack, each at its own constant speed. When they go in opposite
directions, they pass each other every 30 seconds. When they travel in the same direction, they
meet every two and a half minutes. If the distance around the track is a mile, what is the rate of the
slower car in miles per hour?
a. 45 mph
b. 50 mph
c. 48 mph
4
d. 46 mph
e. 52 mph
2013 State Math Contest
Wake Technical Community College
Comprehensive Test
21. A Ferris Wheel completes a revolution in 80 seconds and has a diameter of 200 feet. At its lowest
point, the wheel is two feet above the ground. If you get into a seat at the bottom of the ride, and
then the ride begins immediately with no stops, how far from the ground are you after 25 seconds?
a. 102  50 2  2 ft
b. 102  50 2  2 ft
d. 102  50 1  2 ft
c. 102  50 1  2 ft
e. 102  50 3  2 ft
22. Team A and B play a series of games. The first to win two games wins the series and Team A has a
70% chance of winning any game. What is the probability that Team A wins the series?
a. 0.784
b. 0.700
c. 0.657
d. 0.616
e. 0.637
23. The library in Gotham City has between 1000 and 2000 books. Twenty-five percent of the books are
1
1
fiction,
of the books are biographies, and
of the books are atlases. How many books are
13
17
either biographies or atlases? (These are three different types of books, i.e. no intersection)
a. 221
b. 235
c. 442
d. 270
e. 240
24. In right triangle ABC, AC is the hypotenuse and D is the midpoint of AC . If BD  6n  2 and
DC  4n  8 , what is the length of AC ?
a. Cannot be determined
b. 56
c. 5
d.
104
5
e. 14
25. Three young men raced on bikes from Sydney to Wollongong. Peter finished the race before the
rider of the red bike. The rider of the blue bike took 6.5 hours to finish. The total time required for all
three bikers was 21 hours. The rider on the black bike finished 30 minutes before Max. If the third
rider Liam didn’t ride the black bike and wasn’t first, how long did it take Liam to finish the race?
a. 7.5 hours
b. 7 hours
c. 8.5 hours
5
d. 8 hours
e. 6.5 hours
2013 State Math Contest
Wake Technical Community College
Comprehensive Test
SHORT ANSWER
Place the answer in the appropriate space.
66. In Yahtzee, a player rolls 5 six-sided dice up to three times. After the first roll the player can choose
to roll fewer than 5 of the dice. On the first roll, a player rolls three fours and picks up the other two
dice to reroll them. The player is trying for “Yahtzee” which is when all five dice show the same
number. What is the probability that the player rolls a “Yahtzee” in the next two rolls?
67. Let an  be a sequence of positive integers such that an  an1  an2 for n  3 . If a8  82 and
a11  348 , determine a9 .
68. A rectangular solid has edges that add up to 76 inches, the total surface area is 206 square inches,
and the volume is 165 cubic inches. What is the length of the smallest edge?
69. Mile High HS has four floors. Jeff plans to burn 3024 calories by taking the stairs to and from all his
classes. His 1st, 3rd, 5th, and 7th period classes are on the fourth floor; his 2nd, 4th, and 6th period
classes are on the first floor. Jeff takes 18 seconds to walk from floor to floor. Assuming he burns 15
calories per minute of stair walking, how many days will he need to burn 3024 calories?
70. Triangles ACD and BCD are inscribed in a semicircle with diameter CD . Let AD  14 , BD  40 ,
CD  50 , and AB  25 . Calculate the area of the union of the interiors of triangles ACD and BCD.
6
2013 State Math Contest
Wake Technical Community College
Comprehensive Test
Answer Key
1. e
2. a
3. d
4. d
5. b
6. a
7. e
8. b
9. d
10. b
11. c
12. e
13. d
14. b
15. a
16. c
17. e
18. d
19. c
20. c
21. a
22. a
23. e
24. b
25. c
121
1296
67. 133
68. 3
69. 28
70. 673.5
66.
7