Grissom Mathematics Tournament
Comprehensive Test
March 15, 2003
1. What is the geometric mean of 4! and 3! divided by the arithmetic mean of 4! and 3!?
A.
2
3
B.
3
4
4
5
C.
D. 1
E. 2
D. 213034
E. 231034
2. What is 20036 in base 4?
A. 123034
B. 132034
C.133024
A.
1
2
1
i
2
B.
7
8
2 i
.
3 i
1 :
3. Write in a + bi form where i =
1
i
8
7
8
C.
5
i
8
D.
7
8
5
i
8
E.
7 1
i
10 10
B
4. Find the area of
triangle ABC:
8
30
A
0
C
9
A. 9 3
5. Evaluate:
A. 13.5
B. 18
lim
x
3
C. 18 2
D. 18 3
E. 36
C. 27.5
D. 28.5
E. Undefined
x 3 27
5x
2x 6
B. 18.5
6. What is the 2003rd digit to the right of the decimal point in the decimal equivalent
5
of ?
7
A. 1
B. 2
C. 4
D. 5
E. 8
7. A ball is dropped from a height of 4096 ft. Each time it bounces, it rebounds to 1/16 of its
previous height. Find the distance the ball has traveled in feet if it is caught at the highest
point after the third bounce.
A. 4640
B. 4641
C. 4642
D. 4643
E. 4644
8. If (1,2) is one of the points of intersection of the graphs of ax2 + by2 = 5 and
bx2 + ay2 = 5, then find a + b.
A. 1
B. 2
C. 3
D. 4
E. 5
9. Given parallelogram ABCD, with AB = 10, AD = 12, and BD = 2 13 , find AC.
B. 2 13
A. 10
C. 12 3
4 7
,
13 13
10. Find the sum of x on the interval
A. 0
B.
11. Evaluate:
2
A.
b
a
B.
xy
y2
for which 2 cos2 x 1
D.
3
E.
2
sin x .
2
3
1
2
12. If x 2
E. cannot be determined
1
2
A. 1
C.
6
D. 2 109
1
2 ...
5
4
C.
a and x 3
B.
b2
1
a2
y3
3
2
b then xy
C.
b2
3a 2
b
3
D.
7
4
E. 2
D.
b2 a 3
3a 2
E.
?
a2
2b 3
3a
13. Eva writes the letters of the word “Alabama” on cards, one letter per card. She puts all the
cards in a hat. What is the probability that she draws three straight cards with a’s without
replacement if a is not distinguished from A?
A.
1
12
B.
1
7
4
35
D.
C. 1
D.
C.
4
7
E.
24
35
E.
1
9
14. If F(a ) 1 log3 a 2 , find F 1 ( 1) .
A. 3
B.
3
1
3
15. What is the area enclosed by the graph of the conic with the equation
25x 2 36 y 2 100 x 216 y 476 0 ?
A. 0
B. 10
C. 15
D. 30
E. 45
16. Find the sum of the solutions on the interval (0,2 ] to the equation:
2 sin 2 x 2 cos2 x
0
A. 2
B. 3
C. 4
2
3
17. Simplify: tan Arc tan
A. - 3
x 8
A. –1.25
x
D. 5
E. 6
C. 0
D. 1
E.
C. –1.75
D. 2
E.
C. 2,007,006
D. 4,010,006
E. 4,012,009
Arc tan 5 .
B. - 1
18. Evaluate:
sin 3 x sin x
.
cos x
3
2
.
12x 35
2
B. -1.5
2003
x3
x 0
2003
19. Simplify completely:
.
x
x 0
A. 2003
B. 2,005,003
20. Simplify: i 2
A.
1024i
729
2
B.
10
1
2 3
512
243
1
i
2
12
C.
, where i=
1.
1024
729
D.
512i
243
E.
1024
729
21. Which of the following is an equation of the plane containing the point (20, 0, 3) that is
parallel to the plane containing the points (2, 1, 1), (4, 4, -1), and (2, -1, 2)?
A. x 2y 4z 32
D. x 4y 2z 14
B. 2x y 4z 28
E. x y 2z 26
C. 4x 2y 1z
77
22. If log2 10
A,
ln10
ln 7
B , and 11c
10 , which of the following expressions is equivalent
to log154 ?
A. A B C
23. Find xy if
A. –4
2
B. ABC
x 2y
1
and 0.2
8
B. –1
C.
1
ABC
x y
D.
A B C
ABC
E.
AB AC BC
ABC
125 .
C. 0
D. 1
E. 4
24. How many ordered sextuplets (a, b, c, d, e, f) exist such that a, c, and e are contained in the
set of solutions to the inequality 2 < |x| < 9; b, d, and f are positive integers; and
48 a b c d e f ?
A. 24
B. 36
C. 48
D. 72
E. 144
25. If A is the region enclosed by the graph of the parametric equations: x =
5
sec
and
5
where 0< < 2 and B is the area of the region described by |y| < [x] where [x] is
csc
the greatest integer function, then which of the following expressions is equivalent to the
area of A B?
y=
A. 25
B. 25 /4
C. 25 /4 – 4
D. 25 sin
1
3
5
TB1. What is the coefficient of the x 2 y 3 z 2 term in the expansion of ( x
E. 25 sin
1
3
-12
5
y 2z ) 7 ?
TB2. Let d be the distance between the points with polar coordinates 17,
2
9
and 8,
8
.
9
Find the value of d2.
1 1
1 1
TB3. In the first hundred terms of the harmonic series 1, , ,...., ,
, how many of the
2 3
99 100
terms have a repeating decimal when written in decimal form?