GACS Dwight Love Mathematics Tournament
March 17, 2007
Comprehensive Written Test
1. Find the sum of all the 5th roots of -32.
a) -32
b) 1
2. Solve over R:
a)
c) 0
d) -2
e) NOTA
x2 −1
≤0.
x3
( ∞, −1] U ( 0,1]
b) [ −1,1]
c) [1, ∞ ) U ( −∞, −1]
d)
( −∞, −1] U [0, ∞ )
e) NOTA
3. Find the sum of all of the real solutions of the following equation:
(4
a) ½
x
) (
4
− 16 + 16 x − 4
b) 7/2
) = (4
4
x
+ 16 x − 20
)
c) 3
4
d) 5/2
e) NOTA
4. Find the sum of all values of m such that no solutions exist for the following system of
equations.
x − 2 y + 5 z = −363
2 x + 8 y + mz = 3636
− mx + 4 y + 2 z = −36363
a) -14
b) -20
c) -18
e) NOTA
3log 27 a + log 243 b5 = 6
5. Find b.
a) 729
d) -32
log 27 b3 + log 9 a 7 = 9
b) 729 3
c)
2187
2
d) 243 3
e) NOTA
6. Find the distance between the points A and B with polar coordinates A(7,13°) and B(11,47°).
a)
170 + 3
b)
93
c)
170 − 77 3
d)
247
e) NOTA
GACS Dwight Love Mathematics Tournament
March 17, 2007
Comprehensive Written Test
7. Simplify.
2 +1
a)
cos 47° + cos 2°
sin 47° − sin 2°
b) 2
c)
2 −1
d)
2+ 2
2
e) NOTA
8
4
8. Find the coefficient of the term a 6b6 c 2 in the expansion of  a + ( b + c )  .


a) 784
b) 224
c) 1568
d) 3136
 sin 25° cos 25° 
9. Find the value of the determinant of 
 .
 − cos 25° sin 25° 
2
a) 2
b)
c) 1
d) -1
2
e) NOTA
9
10. Find the conjugate of the multiplicative inverse of
a) 2
b)
2
2
c) 1 − i
e) NOTA
1 1
+ i.
2 2
d) 1 + i
E) NOTA
11. Quadrilateral ABCD lies in a plane with vertices A(0,0), B(52,0), C(15,12) and D(0,12).
The quadrilateral is revolved about the x-axis. Find the volume of the solid generated.
a) 2313π
b) 3036π
c) 5712π
d) 7479π
e) NOTA
12. The sum of the first dozen terms of an arithmetic sequence is 120. The sum of the second
dozen terms is 408. Find the sum of the third dozen terms.
a) 936
b) 528
c) 1224
d) 696
e) NOTA
( −1) ( n ) + ...
1 2
3
4
13. Find the sum. − +
−
+ ... +
5 25 125 625
5n
n +1
a)
1
8
b)
5
36
c)
5
16
d) 1
e) NOTA
GACS Dwight Love Mathematics Tournament
March 17, 2007
Comprehensive Written Test
14. Which of the following best describes the graph of r 2 =
a) hyperbola
b) ellipse
10
?
3cos 2θ − 4sin 2θ
c) parabola
d) circle
e) NOTA
15. The lengths of the sides of a triangle are 3, 5, and 7. If a circle is circumscribed about this
triangle, find the area of the largest triangle that can be inscribed in this circle.
a)
25 3
4
b)
49 3
4
c)
35 2
2
d)
15 7
2
e) NOTA
16. A group of 200 high school seniors are taking various AP courses. 110 are taking AP
Physics, 120 are taking AP English, 130 are taking AP Calculus, 80 are studying AP Calculus
and AP Physics, 60 are studying AP English and AP Calculus, and 50 are studying AP English
and AP Physics. If everyone is taking at least one class, how many are taking all three?
a) 10
b) 20
c) 30
d) 50
e) NOTA
2
2
2
2
1  1   2   3 
 3n  
17. Evaluate: lim   +   +   + ... +   
x →∞ n
 n  
 n   n   n 
a) ∞
b) 0
c) 9
d)
1
3
e) NOTA
18. The street map below shows the only routes in a rectangular coordinate system. Using this
map, how many different shortest paths, along streets only, are there from A to B?
B
A
a) 927
b) 100
c) 250
d) 297
e) NOTA
GACS Dwight Love Mathematics Tournament
March 17, 2007
Comprehensive Written Test
19. Grace rolls a fair 6-sided die and a fair 8-sided die and adds the two numbers she gets. What
are the odds against her rolling a 9?
a) 1:8
b) 8:1
c) 1:7
d) 7:1
e) NOTA
f ( x) = x 2 − 3969
Set A = domain of f
Set B = all integers divisible by 2, 3 or 5
20.
How many natural numbers are in ( A U B ) ?
C
a) 17
b) 16
c) 34
d) 32
e) NOTA
21. In a certain triangle, if the measures of two of its angles are θ and (θ + 90° ) and the sides
opposite these angles are x and y respectively, then tan θ =?
a)
x+ y
y
b)
x
y
c)
x+ y
x
d)
1− x
y
e) NOTA
( )
22. F ( x ) = ln x 2 , x > 0; G ( x ) = e 2 x , x ≥ 0. Find the domain of G o F .
a) x > 0
b) x > 0
23. Simplify
3+ 2 2 + 3− 2 2.
a) 2
b) 6
c) x ≥ 1
c) 2 2
d) x ≥ 1
e) NOTA
d) 3
e) NOTA
24. Find a vector perpendicular to the plane that passes through the points P(1,4,6), Q(-2,5,-1)
and R(1,-1,1).
a)
−40,15, −15
b)
−30, −15, −15
c)
−40, −15,15
d)
−30,15,15
e) NOTA
25. Eliminate the parameter t and write as a single fraction.
x=
3t
,
1+ t3
y=
3t 2
1+ t3
a) x3 − y − 3 x = 0 b) x 3 + y 3 − 3 xy = 0 c) x + y − 3 = 0 d) x + y 2 − xy = 0 e) NOTA
GACS Dwight Love Mathematics Tournament
March 17, 2007
Comprehensive Written Test
Tiebreakers
Name:__________________School:________________________ ID Number:________ TO
BE COUNTED, THIS PAGE MUST BE DETACHED FROM YOUR TEST AND CLIPPED
TO YOUR ANSWER SHEET. YOUR PROCTOR HAS PAPER CLIPS.
T1. Let [x] denote the greatest integer less than or equal to the real number x, and let
{ 3}2 − 2{ 2}2
{x} = x – [x] be called the fractional part of x. For z =
, find z.
{ 3} − 2{ 2}
T1. ___________________
T2. Evaluate:
∞
1
∑ n(n + 1)
n =3
T2.____________________
T3. The surface area and the volume of a sphere are both 3 digit integers times π. If r is the
radius of the sphere, how many integral values can be found for r?
T3._____________________
GACS Dwight Love Mathematics Tournament
March 17, 2007
Comprehensive Written Test
Answer Key
1. C
2. A
3. B
4. C
5. D
6. B
7. A
8. A
9. C
10.D
11. E, 3936π
12. D
13. B
14. A
15. B
16. C
17. C
18. E, 262
19. D
20. A
21. B
22. D
23. C
24. C
25. B
TB1.-2
1
TB2.
3
TB3. 2