Algebra I Contest 2005
1. How many zeros does y = -a(x - h)2 – k have if a, h, and k are positive constants?
a. 3
d. 0
b. 2
e. cannot be determined
c. 1
2. Suppose you are given a line with slope 3 that goes through point (4, 17). A second
line goes through points (1, -5), (2, 1), and (3, 7). At what point do these two lines
intersect?
a. (5.3, 21)
d. (-2, -1)
−17 −11
16
(
,
)
( ,21)
8
8
3
b.
e.
c. (3, 14)
3. A man left an estate of $64,000 to three children. The eldest child received three
times as much as the youngest. The middle child received $14,000 more than the
younger. How much money did the middle child receive?
a. $10000
d. $24000
b. $15600
e. $30000
c. $19600
4. Suppose you have a sequence of numbers where the first term is 3 and the sequence
increases by 8 for each consecutive term. What is the 33rd term in the sequence?
a. 251
d. 275
b. 259
e. 283
c. 267
5. Which of the following statements best describe the relationship between the two
lines 2y + 18 = x and 2y + 4x = -18.
I.
They are parallel.
II.
They intersect at (0, -9)
III.
They are perpendicular.
a. I only
b. II only
c. III only
d. Both I and II
e. Both II and III
6. Two executives in cities 400 miles apart drive to a business meeting at a location on
the line between their cities. They meet after 4 hours. If one car travels 20 miles per
hour faster than the other, find the speed of the slower car.
a. 40 mph
b. 50 mph
c. 60 mph
d. 70 mph
e. 80 mph
7. The functions y = 0.1(x3 +13x2 – 33x – 45) and y = x + 3 share how many solutions in
common?
a. 3
b. 2
c. 1
d. 0
e. none of the above
8. Two cars start at the same point and travel eight miles in opposite directions. Then
they both make right turns (90 degrees) and drive 6 miles and stop. How far apart are
the cars when they stop?
a. 16 miles
d. 20 miles
b. 32 miles
e. 0 miles
c. 28 miles
9.
10.
11.
Simplify the following expression:
35(2b + 1)9 2
(
)
7(2b + 1) −1
a. 5(2b+1)8
b. 5(2b+1)16
c. 25(2b+1)16
d. 5(2b+1)20
e. 25(2b+1)20
The cost of producing q widgets is given by the formula
C = C0 +mq
where C0 and m are positive constants. If the quantity of widgets produced is
doubled, then the cost of production
a. more than doubles
d. stays the same
b. exactly doubles
e. cannot be determined
c. increases but less than doubles
The amount of garbage produced in the United States from 1960 to 1990 was
reported to be:
years
millions of
tons of
garbage in
US
1960
1965
1970
1975
`
1980
1985
1990
90
105
120
130
150
165
180
If G represents the number of tons of garbage in the US (in millions) and t
represents the number of years since 1960, which of the following functions best
fits the trend of the data between 1960 and 1990?
a. G = -3t2 + 90
b. G = 3t + 90
c. G = 5t – 90
d. G = 5t2 – 90
e. G = -3t + 90
12.The crime rate for a given city (number of crimes/number of people in the city) last
year was 6%. The number of crimes in a city, C, tends to vary directly with the size
of the city’s population, P. Which of the following formulas best describes the
number of crimes in the city as a function of its population?
a. P = 6C
d. C = 0.06P
b. P = 0.06/C
e. C = 0.06/P
c. C = 6P
13. A developer has planned out the subdivision he is building on rectangular grid
paper. He has one road running along the path y = 3x + 2. He would like a second
road to run parallel to the first road through the point (1,7) on his grid. Which of the
following best describes the equation of the path of the second road?
a. y = -3x - 4
d. y = 3x + 4
b. y = -3x + 10
e. y = 3x - 20
c. y = (-1/3)x - 21
14. Matrix K shows the weights of four men and four women at the beginning of a diet
designed to produce weight loss. Matrix M shows the weights after the diet.
160 158 172 193 Men


132
143
119
157

 Women
K=
154 148 163 178  Men


132
154
112
136

 Women
M=
Which matrix expression below would result in a matrix that gives the weight losses
of all eight people on the diet?
a. (-1)(K+M)
b. 8M-8K
c. K-8M
d. (-1)K+ (-1)M
e. (-1)M+K
15. Mary’s algebra teacher gave her the following instructions:
Take any number, multiply it by 3, add 49, and divide the result by 7.
Subtract 7 from the quotient, divide the new result by 3. Tell me your
answer and I will tell you what your original number was.
Mary chose an algebraic expression instead of a number at the beginning
and yet Mary’s teacher was still able to guess her original expression.
If Mary’s answer at the end of the process was 2a + b, what was her original
algebraic expression?
14a + 7b
3
2a + b
7
d.
b. 14a + 7b – 98/3
2a + b
3
e.
a.
c. 14a + 7b
Algebra I Contest 2005
16. Marc goes to the store with exactly $1.00 in change. He has at least one of each coin
less than a half-dollar coin, but he does not have a half-dollar coin. What is the least
number of coins he could have?
a. 4
d. 10
b. 5
e. 11
c. 7
17. Suppose we are given that f(x) is an increasing linear function. We will make a new
linear function g(x) = 57*f(x). Which of the following values associated with f(x)
will always remains the same for the new function g(x)?
a. The slope
d. Both the slope and the yb. The y-intercept
intercept
c. The x-intercept
e. None of the above
18. The viscosity of motor oil (in lbs•sec/in2) is a measure of its effectiveness as a
lubricant in the engine of a car. The relationship between a certain motor oil’s
viscosity, v, and its temperature T (in °F), was found to be
v(T) = 75.6 – 0.2937T
Which of the following interpretations best describes the vertical intercept (or yintercept) of this equation in terms of the motor oil’s viscosity and temperature?
a.
b.
c.
d.
e.
The viscosity decreases 0.2937 lbs•sec/in2 per 1°F increase in temperature
The viscosity increases 75.6 lbs•sec/in2 per 1°F increase in temperature
The viscosity was 75.6 lbs•sec/in2 at 1°F.
The viscosity was 75.6 lbs•sec/in2 at 0°F.
The viscosity was 0 lbs•sec/in2 at 75.6°F.
19. Given the function S(x) = 74,741(1.17)x, where S represents the average annual major
league baseball player’s salary x years since 1976. According to this model, in what
year was the average annual major league baseball player’s salary double the average
salary in 1976?
a. 1978
d. 1994
b. 1980
e. 2004
c. 1984
20. A rod of length l was cut into two pieces of equal length. Then one of the pieces was
halved again (and so on). Find the length of the smallest piece, s, after n cuts in terms
of l and n.
a.
b.
c.
d.
e.
s = l/2 + n
s = n/2 + l
s = l + (1/2)n
s = l (1/2)n
s = n (1/2)l
21. Factor completely:
x 3 − 2x 2 + 16(2 − x)
a. (2 − x)(x + 4)(x − 4)
b. (x − 2)(x + 4)(x − 4)
2
c. (x − 2)(x + 4)
d.
e.
x 2 (x − 2) − 16(2 − x)
(x + 4)2 (2 − x)
22. Choose the phrase that best fills in the blank. “Linear functions are ____________.”
a. not polynomials
b. only degree 0 polynomials
c. only degree 1 polynomials
d. degree 0 or degree 1 polynomials
e. degree 0, degree 1, or degree 2 polynomials
23. Tommy goes to the Game Store to buy some used video games. The store has a sale
on used games of $15 each (limit three games) and 15% off the regular price for any
additional used games purchased over the first three games. The used games are
regularly $25 each. Tommy purchases 5 games. He has $100. How much change (if
any) will he receive from the clerk after he purchases his 5 games?
a. $47.50
d. $7.00
b. $29.50
e. He gets no change.
c. $12.50
24. Which of the following points is not a solution to the following system of
inequalities:
3x + 2y ≤ 6
2x – 5y ≥ 10
a. (2, -7)
d. (0, 3)
b. (0, -2)
e. (3, -2)
c. (-2, -3)
25. A fence around a pool has length equal to 7 m more than its width. The area enclosed
by the fence is 44 m2. What is the length of the shortest side?
a. 4 m
b. 7 m
c. 7.5 m
d. 11 m
e. 15.5 m
26. Guess my number: I am a number between 1000 and 3000. I am a perfect cube.
The sum of my digits is a prime number that is two more than another prime number.
What number am I?
a. 1156
d. 2197
b. 1331
e. 2744
c. 1728
Algebra I Contest 2005
27. Evaluate the following expression at. M = 2
M 2 (4 − 2M) − (4M − M 2 )(2M)
M4
a. 0
b. 1
c. -1
d.
e.
-2
2
0 0
−18 −33




0 0
3
12 


28. If the solution to the operation 3A – 2B is
where B =
, find the
matrix A that makes this relationship true.
12 22 


−2 −8
a. 
−36 −66


6
24


18 33 
d.


−3 −12
b. 
−12 −22


2
8


36 66 
e.


−6 −24
c. 
29. Toothpicks of the same size are used to make hexagonal trains.
Find a rule that represents the relationship between the number of hexagons, h, and
the number of toothpicks, t, used in the train.
a. t = 6h + 1
d. t = h + 6
b. t = 5h + 6
e. t = 6h + 6
c. t = 5h + 1
30. A snowboard distributor misread an order for snowboards and made way too many.
The season is now over and they are overstocked. They want to reduce their stock by
decreasing the original price of $350 by 15% each week. How long does it take for
the price of one of these snowboards to drop below $50 (assuming the supply of
snowboards last that long)?
a. 12 weeks
d. 21 weeks
b. 13 weeks
e. none of the above
c. 20 weeks