MATH BOWL 2007
VARSITY 1
SAMPLE
Condense the logarithmic expression and evaluate.
log210 + log28 – log25
18 5 2 8 5 16
Ans.
4
Ans.
24
1.
Simplify as much as possible.
2.
According to a recent poll, 6.6% of respondents had never taken an aspirin. This
means 1 out of _ ? _ had never taken an aspirin.
3.
If the area of
Ans.
ABD equals 27 square units, determine AB .
Ans.
15
AB =9
B
6
A
4.
D
C
A coin jar contains 40 coins, a combination of pennies, nickels and dimes, totaling
$1.42. There are twice as many nickels as dimes. How many pennies are in the jar?
Ans.
5.
22
A string of lights has 20 bulbs, 4 red, 4 blue, 4 green, 4 yellow, and 4 white. In
how many different ways can you arrange the bulbs, so that matching colors are always 5
lights apart?
Ans.
120
6.
The equation of a certain circle is x2 + 6x + y2 – 10y = 2. What is the length of
its diameter?
7.
Ans.
Write a quadratic equation in standard polynomial form for the parabola that has x
intercepts of -3 and 4, and a y intercept of 36.
8.
12
Ans.
y = -3x2 + 3x + 36
The lengths of the sides of this hexagon are consecutive multiples of 4. Its
perimeter is 204 units. If the length of its shortest side equals 4n, then n = ?
Ans.
9.
Find the x intercepts of the graph of the equation f(x) = 3x2 + 4x – 4.
Ans,
10.
6
x= -2, x= 2/3
A =45º, C =75º. Find the length of side AC . Leave your answer in radical
form, rationalizing the denominator if necessary.
B
2cm
A
C
Ans.
6
2x  4
x4
Find the equation for g -1(x).
11.
Let g(x) =
12.
As x increases, which function f grows most rapidly?
A
f(x) = 2
x
B
1
f(x) =  
 4
Ans.
g -1(x) =
4x  4
x2
x
C
f(x) = ex
Ans.
D
f(x) = 3x
B or f(x) =  
 4
1
x
Math Bowl 2007
Varsity 2
SAMPLE
If mC is 150º, express the size of the angle in radian measure, where
5
0 < mC < 2 .
Ans.
6
125  2 10  36
1.
Simplify as much as possible.
Ans.
11
2.
SKIP
3.
If the diagonal of a square is tripled, how many times larger is the area of the new
square, compared to the original?
Ans.
9
4.
The product of the first ___?__ positive integers is 720.
Ans.
6
5.
Simplify as much as possible.
 tan 2  
5
 sec 2  
1  cos 2 
5
6.
Ans.
Find the real-valued solutions to the radical inequality.
2 x  3 1  7
7.
25
Ans.
3 ≤ x < 12 or [3, 12)
If D is the midpoint of BC , find AD . (Drawing not to scale.)
Ans.
B
D *
C
10
A
24
8.
Let an  2n(n  2) . Then a18  ?
Ans.
24
9.
t 
Find the value of t such that cos 1   
 4 3
Ans.
t=2
10.
Solve the equation for x.
Ans.
x=4
log2x + log2(x – 3) = 2
13
11.
ST is tangent to the circle at T. RT is a diameter of the circle. RS =12 and ST = 8.
What is the area of the circle?
Ans.
D or 20 
S
R
A
12.
5
B
T
*
O
8
C
9
D
20 
E
40 
A survey of a group of 50 students showed that 14 were taking Chemistry, 10 were
taking Physics, and 3 were taking both. If a student is chosen at random from the group,
what is the probability that the student will be taking Chemistry or Physics?
Ans.
13.
21
50
40 dogs are entered in a dog show. The entry fee is $50, and the prize is $1000.
Assuming each dog is equally likely to win, what is the expected value for each
contestant?
Ans. -$25.00
Math Bowl 2007
Varsity 3
1.
Solve for x.
log2(log2x)) = 2
Ans.
x = 16
2.
A boat left the harbor on a bearing of 108º. After traveling 9 miles, the boat
changed direction. It then traveled 4 miles on a bearing of 288º. How far is the boat
from the harbor, in miles?
Ans.
5 miles
3.
Carl’s IPOD playlist contains 20 songs, 3 of which are reggae. When he
randomly shuffles the songs, what is the probability that the three reggae songs will
appear first?
4.
Ans.
An equilateral triangle circumscribes a circle of area 12  square centimeters, as
shown. Find the length of a side of the triangle, in centimeters.
5.
1
1140
Ans.
12 cm.
A five foot board is cut into 4 pieces that have a 1:2:3:4 ratio. What is the length,
in feet, of the shortest piece?
Ans.
½ foot
6.
Ans.
5
What is the determinant of the matrix?
 1 1 2 
 0 2 3


 3 4 2
7.
Write the equation(s) of any vertical or horizontal asymptotes of the graph of the
function f (x) 
8.
Let f(x) =
2x 2  x  9
.
3x 2  12
Ans.
x  2 , g(x) = 1/x. State the domain of (f + g) in interval notation.
Ans.
9.
x = 2, x= -2, y = 2/3
[2,0) U(0,)
Find sin(75º). Express your answer as a single fraction.
Ans.
6 2
4
x3
. Find f (4) .
x
or
2 6
4
10.
Let f (x) 
11.
A windmill makes 3 revolutions in 15 seconds. State its angular velocity in
radians per minute.
12.
Let f (x)  3 x  16 , and let g(x)  x 3  16 . Find f (g(2)) .
Ans.
Ans.
20
24  rad/min.
Ans.
14
Ans.

MATH BOWL 2007
VARSITY 4
1.
Find lim (n 2  n) .
n
2.
A certain radioactive isotope has a half-life of 4 days. The amount of the isotope
left after t days is given by the formula A  A0 e
ln 0.5
t
4
How many days until only
1/8 of the original sample remains?
3.
Ans.
How many points with integer coordinates are there in the first quadrant such that
the sum of the coordinates is less than 20?
4.
Ans.
x3
Ans.
1101
Ans.
60 or a2 = 60
The tangent line to the graph of f (x) at the point (4, 3) also passes through the
point (0, 2). Find f (4) .
8.
7
Find the second term of a geometric sequence whose first term is 20, and whose
fourth term is 540.
7.
Ans.
Add the base four numerals, leaving your answer in base four.
312BASE 4 + 123BASE 4
6.
171
If f and g are continuous functions with f (3) = 5, and lim[2 f (x)  g(x)]  3 ,
find g (3).
5.
12 days
Ans.
¼ or 0.25
Find all real solutions to the equation in the interval [0, 2 ) . Express your answer
in radian measure.
sin(2 )  2 cos( )
9.
SKIP
Ans.

 3
,
2 2
3
 3 
n  dx
1  
n 1 
10.
Calculate
Ans.
12
11.
A contestant on a game show chooses 2 of 16 boxes. Three of the boxes contain
money, 5 contain prizes, and 8 are empty. What is the probability that both boxes chosen
are empty?
12.
Ans.
7
30
Find the maximum value of the cost function C = 4x + 5y, subject to the
following constraints:
 x0

 y0
x  y  6

13.
Ans.
30
Find the area of the trapezoid in square centimeters, given that AB  CD
and A  30º  D .
B
3cm
A
C
D
5cm
Ans.
4 3
3