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GRISSOM MATH TOURNAMENT
APRIL 17, 2010
COMPREHENSIVE TEST
1. Solve for x: 8(2x – 5) + 11 = 2(7x + 4) – 13.
A. -14
2. Evaluate:
B. 1
C. 12
D. 28
E. 30
4x 2  8x  12
.
lim
x 3  27
x 3
A. 0
B.
1
9
C.
4
27
D.
16
27
E. 1
3. Find the 10th term in the arithmatic series 2010, 1095, 1080, …
A. 1860
B. 1875
C. 1885
D. 1890
E. 1895
4. How many distinct positive solutions does the equation x 4  6x 3  11x 2  60x  100  0 have?
A. 0
B. 1
C. 2
D. 3
E. 4
D. 18
E. 20
5. Evaluate: 2 sin 30   4 tan 45   6 cos 60   8 sin 90  .
A. 12
B. 14
C. 16
6. Find the maximum possible area of a rectangle with perimeter equal 14 and integral side lengths.
A. 6
B. 10
C. 11
D. 12
E. 14
1
3


7. Evaluate: log 3 7 2 log 27 9 log 7 3 .
A.
4
9
B.
9
14
C.
49
3
8. Find the product of the coefficients of polynomial q(x) if q( x ) 
A. -900
B. -600
C. -210
9. For what values of x is x 3  x 2  2x ?
A.  1,1  2, 
B.  1,0  2, 
D.  ,1  0,2 E.  ,1  0,2
C. 0,1  2, 
D. 9
E. 27
2x 4  11x 3  16x 2  28x  49
.
x7
D. 210
E. 600
10. Solve the following for x: 4 2 x 1  129  4 x  32  0.
A.
1
4
B. -1
5
2
C. -1,
D. -1, 5
E. 16
11. If the sum of the squares of two positive integers is 205, and if the product of the two numbers is 78, find
the absolute value of the difference between the numbers.
A. 5
B. 6
C. 7
12. Given that sin A = 3/5 and cos B = -12/13, where
A.
B.

2
D. 8
A
E. 9
3
and 0  B   , evaluate cos (A + B).
2
C.
D.
E.
13. Which of the following are odd functions? f ( x )  x x  3 , g(x)  3x 3  5x  7 , and
h ( x )  x | 4x | .
A. f only
B. f and g
C. f and h
D. g and h
E. h only
14. The coordinates of point A are (-4, 8), and the coordinates of point B are (5, -4). Write an equation of the
line perpendicular to line segment AB and passing through point C on AB such that AC is twice CB.
A. 3x – y = 6
B. 3x – 4y = 6
C. 4x + 3y = 6
D. 4x – 3y = 6
E. 2x – y = 3
 3 
x   1 have on the interval x  0,6  ?
 2 
15. How many x-intercepts does the graph of y  8 sin 
A. 4
B. 6
C. 9
D. 12
E. 18
D. 15
E. not possible
16. If 3x + y – z = 7 and x + 2y + 3z = 4, determine x + y + z.
A. -1
B. 3
C. 5
17. Lainie, Molly, and Shannon are playing a game involving two fair six-sided dice. If Lainie rolls a sum total
of four or lower, she will win. If the sum is nine or higher, Molly will win. Otherwise Shannon will win.
What is the probability Shannon wins?
A.
2
9
B.
1
3
C.
4
9
D.
5
9
E.
2
3
18. Travis drops a baseball 63 feet from the top of Gravelpit Point. The baseball rebounds 1/3 of its
original height each time it bounces. What is the total distance in feet the baseball travels before
coming to a stop?
A. 42
B. 84
C. 126
D. 160
E. 252
19. Find the volume of a tetrahedron with vertices at (2, 5, 8), (3, 8, -16), (6, 4, 8), and (4, 3, 8).
A. 40
B. 60
C. 80
D. 120
E. 240
20. Square ABCD has side length 6 cm. The four sides are trisected by points M, N, P, Q, R, S, T, and U.
Point M lies on AB closer to point A, point P lies on BC closer to B, point R lies on CD closer to C, and
point T lies on DA closer to D. Square MPRT is drawn. If the four sides of MPRT are trisected, another
square formed, and this process is continued infinitely, find the sum of the areas of the squares.
A.
B.
 2n
10
21. Evaluate:
2
 3n  5
C.
D.
E.
C. 775
D. 885
E. 930

n 1
A. 500
B. 545
22. The Scout, Soldier, Pyro, Demoman, Heavy, Engineer, Sniper, Spy, and Medic are all seated at a circular
table. The Medic must sit between the Heavy and Soldier and the Pyro cannot sit next to the Spy. How
many distinct arrangments can be made? (Arrangements made by rotating the table are not distinct.)
A. 960
B. 1200
C. 1440
D. 1920
E. 8640
23. Find the length of the line segment joining the points of intersection of the graphs with polar equations
r = 2 cos θ and r = 2 sin θ.
A.
2
2
B. 1
C.
2
D. 2
E. 2 2
24. What is the sum of the solutions on the interval –π < x 2π to the equation:
 4 cos 2 4x   4 sin( 4x)  4  0 ?
A.
9
4
B.
9
2
C.
7
2
D.

8
E.
3
4
25. A game is played in which two dice are rolled and the sum of the values is divided by 4. The player
receives $5 if the remainder is 3 and pays $2 if the remainder is 0, 1, or 2. Find the expected value (in
dollars) of the game.
A. 
1
4
B. 
1
8
C.
5
18
D.
13
18
E.
17
6
TB1: If A is the sum of the roots of x2 – 5x – 11 = 0, B is the product of the roots of 2x2 + 5x + 1 = 0, and C is
the number of distinct arrangements of the letters in the word GRISSOM, find BC/A.
TB2: How many prime numbers between 500 and 700 end in 3?
TB3: What is the largest number less than 2010 that has a remainder of 2 when divided by 5 and a remainder
of 5 when divided by 7?