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If t = –2, then t3 – 2t2 + 5t – 3 = ?
1. –29
2. –27
3. –13
4. –9
5. 7
Answer
Plug in for t:
(–2)3 – 2(–2)2 + 5(–2) – 3
–8 – 8 – 10 – 3
–29
What is the length, in feet, of the
circumference of a circle whose
diameter is 12 feet?
1.
2.
3.
4.
5.
6π
12π
6π2
36π
36π2
Answer
Using the formula circumference = π ×
diameter, you get that the circumference =
π × 12 = 12
If 3(x + 2) + 3 = 6x, what is the
value of x?
1. -3
2. -5/3
3. 1/3
4. 5/3
5. 3
Answer
Distribute and then solve:
3x + 6 + 3 = 6x
9 = 3x
x=3
If a drawer contains 7 navy socks, 4 white
socks, and 9 black socks, what is the
probability that the first sock randomly drawn
out of the drawer will not be white?
1. 1/5
2. 1/4
3. 7/20
4. 4/5
5. 16
Answer
There is a total of 20 socks, and 16 socks
are navy or black, and, therefore, not
white. The probability that the first sock
drawn will not be white is 16 out of 20;
16/20 reduces to 4/5 .
Which of the following shows the
prime factorization for 720?
1. 24 × 32 × 5
2. 2 × 3 × 5
3. 24 × 3 × 15
4. 24 × 45
5. 23 × 90
Answer
The answer must be expressed as the
product of only prime numbers. (This
makes choices #3, #4, and #5 wrong.) 720
= 2 × 360 = 2 × 2 × 180 = 2 × 2 × 2 × 90 =
2 × 2 × 2 × 2 × 45 = 2 × 2 × 2 × 2 × 3 × 3 ×
5 = 24 × 32 × 5, thus choice A is correct.
Notice that choice B represents a value of
30.
The number of gallons of paint needed to cover the
exterior of a house with one coat of paint is estimated by
the formula [10n(l + w) – 9n] ÷ 350, where n is the number
of stories, l is the length of the house in feet, and w is the
width of the house in feet. Approximately how many gallon
cans of paint should somebody buy in order to paint one
coat on the exterior of a 30 × 50-foot 2-story house?
1. 3
2. 4
3. 5
4. 6
5. 7
Answer
Plug in the variables: We know the number of
stories, n, will be 2. We choose the larger
number, 50, to represent the length, l. The width,
or w, is 50.
[10n(l + w) – 9n] ÷ 350 becomes:
[10(2)(30 + 50) – 9(2)] ÷ 350
[20(80) – 18] ÷ 350
(1600 – 18) ÷ 350
1582 ÷ 350
4.52
Since the paint must be purchased in gallon
cans, 5 cans are necessary.
Which coordinate pair is a solution
to the inequality 12 – 3y > 6x + 3?
1. (1,1)
2. (2,1)
3. (1,2)
4. (-1,-2)
5. (2,-1)
Answer
Plugging in (1, 1) gives 9 > 9, which is not a true
statement.
Plugging in (2, 1) gives 9 > 15, which is not a
true statement.
Plugging in (1, 2) gives 6 > 9, which is not a true
statement.
Plugging in (–1, – 2) gives 18 > –3, which IS a
true statement.
Plugging in (2, –1) gives 15 > 15, which is not a
true statement.
Therefore (–1, –2) is the only correct solution.
Which linear equation has a slope
of – 1/2 and a y-intercept of 6?
1. x = 6y – 1/2
2. y = 6x – 1/2
3. x = – 1/2 y + 6
4. y = – 1/2 x + 6
5. None of these
Answer
Slope-intercept form for a linear equation
is y = mx + b, where m is the slope and b
is the y-intercept.
The equation y = – 1/2 x + 6 has a slope of
–1/2 and a y-intercept of 6.
|–5 · 2| – |(–3)2| = ?
1. -19
2. -1
3. 1
4. 4
5. 19
Answer
Simplify:
|–10| - |9|
Absolute values result in a positive
result:
10 – 9
1
The price of a $220 saw was
reduced to $154. What was the
percent discount?
1. 30%
2. 43%
3. 66%
4. 67%
5. 70%
1
2
3
4
5
Answer
The percent discount is the difference in
the prices divided by the original price:
($220 – $154) ÷ $220
$66 ÷ $220
0.3
Multiply by 100 to get a percent:
30%
For all x, (3x – 4)^2 = ?
1. 6x – 8
2. 9x2 + 16
3. 9x2 - 16
4. 9x2 – 12x + 16
5. 9x2 – 24x + 16
1
2
3
4
5
Answer
(3x – 4)2 can be rewritten as
(3x – 4)(3x – 4). Multiply using the
distributive property or "FOIL" (first terms,
outside terms, inside terms, last terms):
9x2 – 12x – 12x + 16
9x2 – 24x + 16
In a seminar, the ratio of men to
women is 5:9. If there are 56 people
in the seminar, how many are
women?
1. 4
2. 7
3. 20
4. 26
5. 36
1
2
3
4
5
Answer
A ratio of 5 parts men to 9 parts women,
means there are 14 parts in all. Thus, the
number of women in the seminar are
9/14 of the total number of people in the
seminar. Therefore, the number of women
in the seminar is 9/14 of 56, or:
9/14 × 56 = 36.
(2·3√ 9)3 = ?
1. 18
2. 24
3. 54
4. 72
5. 216
1
2
3
4
5
Answer
(2·3√ 9)3 is equivalent to 23 · (3√ 9)3.
Because a cube root and a cube are
inverses, (3√ 9)3 simplifies to 9.
This results in 23 · 9 = 8 · 9 = 72.
If V = x^3 + y[r(3s + 2) + 4r] and x =
2, y = 3, r = 5, and s = 4, what is
the value of V?
1. 269
2. 272
3. 275
4. 278
5. 281
1
2
3
4
5
Answer
V= x^3 + y[r(3s + 2) + 4r]
= (2)^3 + 3[5(3 × 4 + 2) + 4 × 5]
= 8 + 3[5(14) + 20)]
= 8 + 3[70 + 20]
= 8 + 3[90] = 8 + 270
= 278
Remember to follow the order of operations:
Parentheses, exponents, multiplication/division,
addition/subtraction—otherwise known as
PEMDAS.
What is the slope of the line
determined by the equation
15y – 10x = –9?
1. 3/2
2. 2/3
3. 3/5
4. -2/3
5. -3/2
1
2
3
4
5
Answer
We need to make the equation look like y = mx +
b, and our answer will be m, the slope. Solve the
equation for y to write the equation in slopeintercept form:
15y = 10x – 9
y = ( 10/15)x – 9/15
y = 2/3 x + 3/5
The coefficient of the x term, 2/3 , is the slope of
the line.
If –x > y and |x| > y, which values for x and y
make both statements true?
I. (0, 0)
II. (–2, 3)
III. (–5, 4)
IV. (–2, –3)
V. (5, –5)
1.
2.
3.
4.
5.
1
2
All except II.
All of them
III. And IV.
IV. Only
I., II., and IV.
3
4
5
Answer
Plug in each of the possible answers to
see which make true statements:
I. –0 > 0 is false.
II. 2 > 3 is false.
III. 5 > 4 is true and |–5| > 4 is also true.
IV. 2 > –3 is true and |2| > –3 is also true.
V. –5 > –5 is false.
After a computer is discounted by 20%,
its price is $960. What was the original
price of the computer before the
discount?
1. $768
2. $1066
3. $1120
4. $1200
5. $1220
1
2
3
4
5
Answer
After the computer is discounted by 20%,
its price is 80% of its original price, p.
p × 0.80 = $960
p = $960 ÷ 0.80
p = $1200
Therefore, its original price was $1200.
In the standard (x, y) coordinate plane,
point P is located at (16, 12) and point Q
is located at (11, 7). What is the distance
between points P and Q?
1. √ 5
2. 5
3. 5 √ 2
4. 10
5. 50
1
2
3
4
5
Answer
Using the formula,Distance
Distance =
=
If the circumference of a circle
measures 10 feet, what is its area
in square feet?
5
10
15
25
30
1.
2.
3.
4.
5.
1
2
3
4
5
Answer
Using the formula, Circumference = ×
diameter, you get 10 = × diameter, or
the diameter of the circle is 10 feet. Thus,
its radius is 5 feet, and
its Area = ×
Π
radius2 = × 52 = 25 square feet.
What is the x-intercept of the line
given by the equation
5x + 3y = 12?
1. 12
2. 4
3. 12/5
4. -3/5
5. -5/3
1
2
3
4
5
Answer
The x-intercept occurs when y = 0, so plug
in 0 for y and solve for x:
5x + 3(0) = 12
5x = 12
x = 12/5
For what value of b in the equation
(2z + b)(z – 3) = 0 will a solution for
z be –4?
1. -8
2. 1
3. 3
4. 8
5. 32/3
1
2
3
4
5
Answer
To solve this quadratic equation for z, both
factors would be set equal to 0:
(2z + b) = 0 and (z – 3) = 0
We will focus on the first factor that
contains b.
2z + b = 0
Substitute z = –4 into the equation:
–8 + b = 0
Solve for b:
b=8
If the edge of a cube is doubled,
the volume of the cube will be
increased by what factor?
1. 2
2. 4
3. 6
4. 8
5. 16
1
2
3
4
5
Answer
The volume of cube is given by the
formula Volume = edge^3. Therefore, if the
edge is doubled, you get Volume = (2 ×
edge)^3 = 2^3 × edge^3 = 8 × edge^3.
Thus, the volume is increased by a factor
of 8.
A train travels at a constant rate of
30 miles per hour. How long will it
take the train to travel 100 miles?
1. 3 hours and 15 minutes
2. 3 hours and 20 minutes
3. 3 hours and 30 minutes
4. 3 hours and 33 1/3 minutes
5. 3 hours and 40 minutes
1
2
3
4
5
Answer
Using the formula Distance = Rate × Time,
you get 100 = 30 × Time,
or Time = 100/30 = 3 1/3 hours = 3 hours
and 20 minutes.
If 2(x – 1)^2 + 5 = 13, what are the
two possible values of x?
1. 1 and –3
3 and –1
3. 5 and –3
4. 1 + √ 2 and 1 – √ 2
5. There is no solution.
2.
1
2
3
4
5
Answer
2(x – 1)^2 + 5 =13
2(x – 1)^2=8
(x – 1)^2=4
x – 1= ± √4
x–1=±2
x – 1 = 2 or x – 1 = –2
x = 3 or x = –1
If |2x – 3| >= 20 which shows the
solutions for x?
1. -17/2 = x = 23/2
2. X
-17/2 or x 23/2
3. X=23/2
4. x 17/2
5. There is no solution.
1
2
3
4
5
Answer
The inequality |2x - 3| 20 is the same as
the two inequalities 2x - 3 20 or
2x + 3 20.
Solving the first inequality, you get
2x 23, or x
23/2. Solving the second
inequality, you get 2x 17, or x -17/2 .
These are both solutions.
At a recent sold-out concert of a 300-seat
theater, two types of tickets were sold: regular
tickets for $8 each and discount tickets for $4
each. If $1,600 in tickets were sold, how many
were regular tickets?
1. 50
2. 100
3. 150
4. 200
5. 2,500
1
2
3
4
5
Answer
Let x = the number of regular tickets sold.
Thus, 300 - x = the number of discount
tickets sold. Since $1,600 in tickets were
sold, you get 8x + 4(300 - x) = 1,600.
Solving for x, you get 8x + 1,200 - 4x
1,600, or 4x = 400, and x = 100.
Which of the following is an
equivalent expression for
(5x+4)/x – (x+1)/3x ?
1.
2.
3.
4.
5.
1
4x + 3
(4x+3)/3x
(14x+13)/14x
(14x+11)/3x
(4x+3)/3x^2
2
3
4
5
Answer
multiplying by 1 doesn't change the value
of the expression, we will multiply by 1,
represented as 3/3.
So, multiplying both the numerator and
denominator of the first fraction by 3, you
get
=?
-4
-1
-1/4
1/4
4
1.
2.
3.
4.
5.
1
2
3
4
5
Answer
A number raised to a negative exponent
can be represented by writing 1 over that
number to the positive exponent. This
means we can bring the 4^–2 in the
numerator down to the denominator as
4^2, and we can bring the 2^–3 in the
denominator up to the numerator as 2^3:
Kathy has test scores of 88, 90, 91, and 78.
What must she score on her next test to have
an average of 84 for all five tests?
1. 71
2. 73
3. 75
4. 77
5. 4
1
2
3
4
5
Answer
The sum of the first four tests is 347. In
order to have an average of 84 on all five
tests, the sum of those tests must equal 5
× 84 = 420. Therefore, the fifth test score
must be 420 - 347 = 73.
In the standard (x, y) coordinate plane, a
rectangle has the coordinates (6, 5), (6, –2), (–3,
5), and (–3, –2). What is the length of a diagonal
of the rectangle?
1. 4
2. √130
3. 16
4. 130
5. 256
1
2
3
4
5
Answer
Explanation: Plot the points. Then, find
the distance between two opposite points,
(6, 5) and (–3, –2). (-3,5) (6,5)(3, -2) (6, -2)
Use the Distance formula
If 3y^2 + 24y + 15 = 0, what are the
possible values of y?
1. -8±2√11
2. -4±2√11
3. -4 ±√130
4. ±6√11
5. ± √11
1
2
3
4
5
Answer
First, reduce the equation by dividing all
terms by 3 to give y^2 + 8y + 5 = 0. Using
the quadratic formula.
where a and b are the coefficients of y^2
and y, respectively, and c is the constant.
What is the slope of a line
perpendicular to the line
8x + 4y =12?
-2
-1/2
1/2
2
12
1.
2.
3.
4.
5.
1
2
3
4
5
Answer
Solving for y to write 8x + 4y =12 in slopeintercept form, you get:
4y = -8x + 12
y = -2x + 3
Therefore, slope of this line is the
coefficient of the x term, or -2. The slope of
a line perpendicular to this line is the
negative reciprocal of -2, or 1/2 .
What is the x-intercept of the circle
(x – 3)^2 + (y + 4)^2 = 16?
1. (3,0)
2. (4,0)
3. (0,-3)
4. (0,-4)
5. The circle has no x-intercept
1
2
3
4
5
Answer
The general equation for a circle is (x a)^2 + (y - b)^2 = r^2, where (a, b) is the
center of the circle, and r is the radius of
the circle. Therefore, in the circle (x – 3)^2
+ (y + 4)^2 = 16, the center of the circle is
the point (3, -4), and the radius is 4. Thus,
the since the center is 4 units below the xaxis, and the radius is 4, the circle will
touch the x-axis 4 units directly above the
center, at (3, 0).
If 4^(2x + 1) = 1/64 , x = ?
1. -2
2. -3/2
3. 0
4. 3/2
5. 2
1
2
3
4
5
Answer
Substitute 4^–3 for 1/64 :
4(2x + 1) = 4^–3
Set the exponents equal to each other and
solve for x:
2x + 1 = –3
2x= –4
x= –2
What is the area, in square inches,
of an isosceles trapezoid with side
lengths 5, 5, 10, and 18 inches?
1. 21
2. 38
3. 42
4. 56
5. 70
1
2
3
4
5
Answer
Because the trapezoid is isosceles, it can be
broken into a rectangle and two right triangles,
with the two side lengths of 5 applying to the
legs of the trapezoid. Because the difference in
the bases is 8, half that will be the length of the
leg of each of the triangles. Therefore, the right
triangles will have a hypotenuse of 5 and a leg
of 4. The Pythagorean Theorem shows that the
length of the second leg is 3. The area of the two
right triangles is equivalent to a 3 by 4 rectangle,
and the other rectangle that makes up the
trapezoid is 3 by 10. This gives areas of 12 and
30, which adds up to 42.
The points (3, –5), (1, –4), and
(–5, k) lie on the same line. What is
the value of k?
1. -4
2. -3
3. -2
4. -1
5. 0
1
2
3
4
5
Answer
Because all the points are on the same line, the slope between any two
points will be the same. Find the slope between the first two points:
m = (–5 –(–4)) / (3 – 1)
m = - 1/2
Set up an equation using the slope and a second pair of points and solve for
k:
–1 / 2 = (– 4 – k) / (1 – (–5))
–1 / 2 = (– 4 – k) / 6
–6 = 2(– 4 – k)
–6 = –8 – 2k
2 = –2k
k = –1
If the height of an equilateral
triangle is 9, what is the perimeter of
the triangle?
1.
18 √3
(81 x √2 )/4
3. 27
4. 27 x √2
2.
5.
1
54 √3
2
3
4
5
Answer
The height of an equilateral triangle creates two
30-60-90 triangles, with the height being the
long leg of the special triangle. Using the special
qualities of a 30-60-90 triangle,
the short leg is
which is rationalized to
Because the short leg is half the base, double
the short leg to find the base of the triangle,
giving
Multiply that length by 3 to get the perimeter,
For the circle represented by the
equation x^2 + 6x + y^2 - 2y = -1 in the
standard (x, y) coordinate plane, what
are the coordinates of the center?
1. (-3,1)
2. (-3,-1)
3. (3,-1)
4. (3,1)
5. (-3,1)
1
2
3
4
5
Answer
The general equation for a circle is (x - a)^2 + (y
- b)^2 = r^2, where (a, b) is the center of the
circle, and r is the radius of the circle. In order to
write the equation x^2 + 6x y^2 - 2y = -1 in this
form, first add 9 to both sides of the equation to
get x^2 + 6x + 9 y^2 - 2y = -1 + 9. This is the
same as (x + 3)^2 + y^2 -2y = 8. Then add 1 to
both sides of the equation. This gives (x + 3)^2 +
y^2 - 2y + 1 = 8 + 1,
which is the same as (x + 3)^2 + (y - 1)^2 = 9.
Therefore, the coordinates of the center of the
circle are (-3, 1).
A machine can produce N cans per minute.
Another machine can produce 4 times as
many cans per minute. How many minutes
will it take the two machines working together
to produce 1,000 cans?
1. 500N
2. 250/N
3. 250N
4. 200/N
5. 200N
1
2
3
4
5
Answer
Since the first machine can produce N cans per
minute, the other machine can produce 4N cans
per minute. Thus, working together, the two
machines can produce 5N cans per minute. We
set up a proportion:
Cross multiply to get:
Finally, isolate x:
x= 1000/5N
x= 200/N
5Nx = 1000
What is the smallest integer that is
greater than
?
1. 1
2. 2
3. 3
4. 4
5. 5
1
2
3
4
5
Answer
You can multiply to get:
The total is 1.5 + 1 + 0.75 = 3.25. The
smallest integer larger than 3.25 is 4.
If 3x^2 + 7x + 2 = 0, what is one
possible value of x?
1. -2
2. -1
3. 0
4. 1/3
5. 3
1
2
3
4
5
Answer
You can either factor this by trial and error,
or you can actually solve it using the more
general formulas for factoring a quadratic
polynomial into (ax + b)×(cx + d).
Here 3x^2 + 7x + 2 = (3x + 1)×(x + 2)
As this is zero only when one of the
factors is zero, only x = -2 is a possible
solution.
If the number 114 is written as a
product of N numbers, where
each of the N numbers is prime
and where the number 1 is not
included, what is the value of N?
1
2
3
4
5
1.
2.
3.
4.
5.
1
2
3
4
5
Answer
Rewriting 114 as the product of prime
numbers means breaking it into factors
such that the factors themselves cannot
be further subdivided. The factors 2, 3 and
19 combine to give 114, so the value of N
must be three.
Let the function "deuce(x)" mean to add 2
to x. The process can be repeated, so that
"deuce^2(x)" means to "deuce" the result
of "deuce(x)". If deuce^n (2) is greater
than 101, what is the smallest possible
integer value of n?
102
101
100
51
50
1.
2.
3.
4.
5.
1
2
3
4
5
Answer
The wording might be a little tricky, but all
that's being described is a series 2, 4, 6,
8.... If 4 is the result of the first step, then
102 is reached after 50 steps.
0 < n^-1 + 3m < 4
What is one possible pair of values
for (n, m)?
(-1, 0)
(-1, 1)
(1, -1)
(1, 1)
( 1/2, 1)
1.
2.
3.
4.
5.
1
2
3
4
5
Answer
Many of the answers are very close. But
only the pair (-1, 1) works.
(-1)^-1 + 3(1) = -1 + 3 = 2.
Which value is largest?
1. 150% of 32
2. 120% of 40
3. 80% of 60
75% of 64
50% of 98
4.
5.
1
2
3
4
5
Answer
All the other choices were equal to 48. But
50% of 98 is 49, so this is the correct
choice.
Two years ago, a rabbit breeder had 75
rabbits. Since then 500 rabbits have been
born, 220 have died and 185 have been sold.
How many rabbits does the breeder have now
expressed as a percentage of the number she
had two years ago?
21 %
44 %
133 %
200 %
227 %
1.
2.
3.
4.
5.
1
2
3
4
5
Answer
She starts with 75 rabbits. 500 are born,
220 die and 185 are sold. The new
number of rabbits is:
75 + 500 - 220 - 185 = 170 and 170 is
indeed 227 % of 75.
What is the value of
1. 1,000,000
2. 100,000
10,000
1,000
100
3.
4.
5.
1
2
3
4
5
Answer
The number inside the radical is 10^6 =
1,000,000. If you know that 1000 is the
square root of one million you're done. You
might notice that taking the square root of
a number is equivalent to raising it to the
power 1/2.
So (10^6)^1/2 = 10^3 = 1000.
How many positive integers less
than 62 have 8 as a factor?
1. 9
2. 8
3. 7
4. 6
5. 2
1
2
3
4
5
Answer
The first thing to notice is that 8*8 = 64. So
the number has to be less that 8. But 8*7 =
56, which is less than 62. That means that
there are 7 positive integers less than 62
with 8 as a factor
For which of the following is the
sum of the digits in the tenths and
hundredths places the greatest?
1. 510.823
2. 620.439
3. 126.750
982.453
5. 563.929
4.
1
2
3
4
5
Answer
For the number 126.750, the sum of the
digit in the tenths place (7) and the digit in
the hundredths place (5) is 12. This is a
greater sum than for any of the other
numbers shown.