Download ACT PRACTICE MATHEMATICS TEST 60 Minutes – 60 Questions

ACT PRACTICE MATHEMATICS TEST
60 Minutes – 60 Questions
1.
7−3 − 3−7 = ?
4 − −4 → 4 − 4 = 0 → Correct answer is D) 0
The absolute value of a number is always positive.
2.
A consultant charges $45 for each hour she works on a consultation, plus a flat
$30 consulting fee. How many hours of work are included in a $210 bill for a
consultation?
Let x represent the number of hours of work → $45 per hour plus $30 fee must equal $210
45 x + 30 = 210 → 45 x = 180 → x = 4 → Correct answer is G) 4
3.
Vehicle A averages 14 miles per gallon of gasoline, and Vehicle B averages 36
miles per gallon of gasoline. At these rates, how many more gallons of gasoline
does Vehicle A need than Vehicle B to make a 1,008 mile trip?
Gallons for Vehicle A → 1, 008 miles ÷ 14 mpg = 72 gallons
Gallons for Vehicle B → 1, 008 miles ÷ 36 mpg = 28 gallons
Vehicle A ( 72 gallons ) − Vehicle B ( 28 gallons ) = 44 gallons →
Correct answer is C) 44
4.
t 2 − 59t + 54 − 82t 2 + 60t is equivalent to:
t 2 − 59t + 54 − 82t 2 + 60t → 1t 2 − 59t + 54 − 82t 2 + 60t → Add like terms →
1t 2 − 82t 2 − 59t + 60t + 54 = −81t 2 + 1t + 54 → −81t 2 + t + 54 →
Correct answer is J) − 81t 2 + t + 54
When you add like terms, you add the coefficients (the numbers in front of the
variables) but keep the variable part the same.
5.
The figure below is composed of square BCDE and equilateral triangle Δ ABE.
The length of CD is 6 inches. What is the perimeter of ABCDE, in inches?
A
If BCDE is a square, then ED, BC , and EB are all 6 inches.
If + ABE is an equilateral triangle, then EB, EA, and BA are all 6 inches.
Perimeter of ABCDE = AB + BC + CD + DE + EA =
6 + 6 + 6 + 6 + 6 = 30 inches → Correct answer is C) 30
E
B
D
6 inches C
6.
The expression ( 4z + 3 )( z − 2 ) is equivalent to:
Use the FOIL method ( First, Outer, Inner, Last ) to multiply binomials →
( 4 z + 3)( z − 2 ) → 4 z ⋅ z + 4 z ⋅ ( −2 ) + 3 ⋅ z + 3 ⋅ ( −2 ) → 4 z 2 − 8 z + 3z − 6 →
Add like terms → 4 z 2 − 5 z − 6 → Correct answer is J) 4 z 2 − 5 z − 6
7.
If 40% of a given number is 8, then what is 15% of the given number?
Convert 40% to a decimal → 40% = 0.4 → "of" in problem means "multiply" →
40% of a number is 8 → 0.4 ∗ x = 8 → x = 20
What is 15% of this number? → Convert 15% to a decimal → 15% = 0.15 →
15% of 20 → 0.15 ∗ 20 = 3.0 → Correct answer is C) 3.0
is
%
8 40
=
→ =
→
of 100
x 100
Cross multiply → 40 x = 800 → x = 20 = given number →
x
15
=
→ Cross multiply → 100 x = 300 → x = 3 → Correct answer is C) 3.0
20 100
ALTERNATIVE WAY: Use proportion →
8.
The 6 consecutive integers below add up to 447. What is the value of x?
x − 2, x − 1, x, x + 1, x + 2, x + 3
Add these six terms together and set equal to 447 →
( x − 2 ) + ( x − 1) + x + ( x + 1) + ( x + 2 ) + ( x + 3) = 447 → Add like terms →
6 x + 3 = 447 → 6 x = 444 → x = 74 → Correct answer is H) 74
9.
In the standard (x, y) coordinate plane, point M with coordinates (5, 4) is the
midpoint of AB, and B has coordinates (7, 3). What are the coordinates of A?
⎛x +x y +y ⎞
⎛ x + 7 y1 + 3 ⎞
Midpoint = ⎜ 1 2 , 1 2 ⎟ → ( 5 , 4) = ⎜ 1
,
⎟→
2 ⎠
2 ⎠
⎝ 2
⎝ 2
x +7
y +3
→ 10 = x1 + 7 → 3 = x1 , 4 = 1
→ 8 = y1 + 3 → 5 = y1
5= 1
2
2
Correct answer is D) ( 3, 5)
ALTERNATIVE EXPLANATION: Midpoint is halfway between A and B →
7+?
3+ ?
= 5 → ? = 3,
= 4 → ? = 5 → Correct answer is D) ( 3, 5)
2
2
10.
Rectangle ABCD has vertices A (4,5), B (0, 2), and C (6, –6). These vertices are
graphed below in the standard (x, y) coordinate plane. What are the coordinates
of vertex D?
There are several ways to do this mathematically,
but they all take a lot of time. Here’s the quick and
dirty way. From point B to point A, you go 4 units
to the right and 3 units up. If you start at point C
and go 4 units to the right and 3 units up, you end
up at the point (10, –3).
6
A
4
B
2
5
10
-2
The correct answer is F) (10, –3).
-4
-6
C
-8
11.
Daisun owns 2 sportswear stores (X and Y). She stocks 3 brands of T-shirts (A,
B, and C) in each store. The matrices below show the numbers of each type of Tshirt in each store and the cost for each type of T-shirt. The value of Daisun’s Tshirt inventory is computed using the costs listed. What is the total value of the
T-shirt inventory for Daisun’s 2 stores?
A
B
C
X ⎡ 100 200 150 ⎤
Y ⎢⎣ 120 50 100 ⎥⎦
Cost
A ⎡ $5 ⎤
B ⎢⎢ $10 ⎥⎥
C ⎢⎣ $15 ⎥⎦
This is intended to be a matrix multiplication problem, but you don’t need to
understand matrix multiplication to get the correct answer.
In store X, 100 A shirts * $5/shirt + 200 B shirts * $10/shirt + 150 B shirts * $15/shirt
= $4750.
In store Y, 120 A shirts * $5/shirt + 50 B shirts * $10/shirt + 100 B shirts * $15/shirt
= $2600.
Total inventory = $4750 + $2600 = $7350
Correct answer is E) $7,350
12.
Given the triangle shown below with exterior angles that measure x°, y°, and z°
as shown, what is the sum of x, y and z?
First, find the third angle of the triangle. The three
angles of a triangle add up to 180°. The third angle
must equal 180° – 57° – 72° = 51°.
y°
72°
The x° and the 57° are supplementary, so
x° = 180° – 57° = 123°.
x° 57°
The y° and the 72° are supplementary, so
y° = 180° – 72° = 108°.
The z° and the 51° are supplementary, so
z° = 180° – 51° = 129°.
x + y + z = 123° + 108° + 129° = 360° → Correct answer is J) 360°
51°
z°
ALTERNATE EXPLANATION: The sum of the exterior angles of a convex polygon
is always 360°, regardless of the number of sides.
Use the following information to answer questions #13 – #15.
A poll of 200 registered voters was taken before the election of mayor of
Springdale. All 200 voters indicated which 1 of the 4 candidates they would vote
for. The results of the poll are given in the table below.
Candidate
Blackcloud
Lue
13)
Number of
voters
50
80
Candidate
Gomez
Whitney
Number of
voters
40
30
What percent of the voters polled chose Whitney in the poll?
30 people voted for Whitney out of a total of 200 votes cast. 30 ÷ 200 = 0.15.
Convert 0.15 to a percent by moving the decimal point two places to the right.
0.15 = 15% → Correct answer is A) 15%
is
x
%
30
=
→
=
→
of 100
200 100
Cross multiply → 200 x = 3000 → x = 15 → Correct answer is A) 15%
ALTERNATIVE WAY: Use proportion →
14)
If the poll is indicative of how the 10,000 registered voters of Springdale will
actually vote in the election, which of the following is the best estimate of the
number of votes Lue will receive in the election?
80
x
=
→ 200 x = 800000
200 10000
Divide both sides by 200 → x = 4000 → Correct answer is H) 4,000
Set up a proportion and cross-multiply:
15)
If the information in the table were converted into a circle graph (pie chart),
then the central angle of the sector for Gomez would measure how many
degrees?
There are 360° in a circle. Set up a proportion and cross-multiply:
40
x
=
→ 200 x = 14400 → Divide both sides by 200 → x = 72
200 360
Correct answer is B) 72°
16.
In square ABCE shown below, D is the midpoint of CE .
Which of the following is the ratio of the area of Δ ADE to the area of Δ ADB?
D
Since no side lengths are given, I made up a side length
for the square ABCE of 4 cm.
E
4 cm
DE = 1/2 ∗ 4 cm = 2 cm
Area of + ADE = 1/2 ∗ 2 cm ∗ 4 cm = 4 cm 2
Area of + ADB = 1/2 ∗ 4 cm ∗ 4 cm = 8 cm
Ratio of + ADE to + ADB =
17.
2
C
4 cm
A 4 cm B
4 cm 2 1
= → Correct answer is G) 1:2
8 cm 2 2
Which of the following is the slope of a line parallel to the line y =
in the standard (x, y) coordinate plane?
Parallel lines have the same slope.
Slope-intercept form: y = mx + b, where m = slope and b = y-intercept
2
2
2
y = x − 4 → m = , b = −4 → slope = , y -intercept = −4
3
3
3
2
Correct answer is E)
3
2
x−4
3
18.
Janelle cut a board 30 feet long into 2 pieces. The ratio of the lengths of the 2
pieces is 2 : 3. What is the length, to the nearest foot, of the shorter piece?
If the pieces are in the ratio of 2 : 3, call the
shorter piece 2x and the longer piece 3x.
3x
2x
30 feet
2 x + 3x = 30 → 5 x = 30 → x = 6 → Shorter piece = 2 x = 2 ∗ 6 = 12 feet
Correct answer is H) 12
19.
What is the smallest integer greater than 58 ?
Use your calculator to evaluate 58 → 58 ≈ 7.61 → First integer > 7.61 is 8
Correct answer is C) 8
20.
Sergio plans to paint the 4 walls of his room with 1 coat of paint. The walls are
rectangular, and, according to his measurements, each wall is 10 feet by 15 feet.
He will not need to paint the single 3-foot-by-5-foot rectangular window in his
room and the 3½-foot-by-7-foot rectangular door. Sergio knows that each gallon
of paint covers between 300 and 350 square feet. If only 1-gallon cans of paint
are available, which of the following is the minimum number of cans of paint
Sergio needs to buy to paint his walls?
Area of each rectangular wall = 10 ft ∗15 ft = 150 ft 2 → 4 walls = 4 ∗150 ft 2 = 600 ft 2
Area of window = 3 ft ∗ 5 ft = 15 ft 2 , Area of door = 3.5 ft ∗ 7 ft = 24.5 ft 2
Total area to be painted = 600 ft 2 − 15 ft 2 − 24.5 ft 2 = 560.5 ft 2
1 can of paint covers 300 ft2 - 350 ft2, 2 cans of paint covers 600 ft2 - 700 ft2
1 can is not enough to cover 560.5 ft2, but 2 cans are enough
Correct answer is G) 2
21.
What values of x are solutions for x 2 + 2 x = 8?
This is a quadratic equation, so it can be solved by factoring or the Quadratic
Formula. In either case, the equation must be set equal to 0 first.
x 2 + 2 x = 8 → x 2 + 2 x − 8 = 0 → ( x + 4 )( x − 2 ) = 0 → x + 4 = 0 or x − 2 = 0 →
x = −4 or x = 2 → Correct answer is A) − 4 and 2
22.
For all a > 1, the expression
3a 4
equals:
3a 6
3a 4
3∗ a ∗ a ∗ a ∗ a
1
1
1
=
=
= 2 → Correct answer is K) 2
6
3a
3∗ a ∗ a ∗ a ∗ a ∗ a ∗ a a ∗ a a
a
ALTERNATIVE WAY: Reduce 3 in top and bottom, reduce powers by subtracting
3a 4
1
1
exponents → 6 = a 4−6 = a −2 = 2 → Correct answer is K) 2
3a
a
a
23.
24.
If point M has a nonzero x-coordinate and a nonzero y-coordinate and the
coordinates have opposite signs, then point M must be located in which of the 4
quadrants below?
Ordered pairs with opposite signs
will appear in Quadrant II and
Quadrant IV.
Correct answer is E) II or IV only Quadrant II
x negative, y positive
Quadrant I
x positive, y positive
Quadrant III
x negative, y negative
Quadrant IV
x positive, y negative
The fixed costs of manufacturing basketballs in a factory are $1,400.00 per day.
The variable costs are $5.25 per basketball. Which of the following expressions
can be used to model the cost of manufacturing b basketballs in 1 day?
$1,400.00 is spent every day, regardless of the number of basketballs made.
The cost of manufacturing 1 basketball is $5.25, so the cost of manufacturing b
basketballs is $5.25 * b, or $5.25b.
The total manufacturing cost per day is fixed cost ($1,400.00) + variable cost
($5.25b) → Correct answer is K) $1,400.00 + $5.25b
25.
In the figure below, where Δ ABC ~ Δ KLM, lengths are given in centimeters.
What is the perimeter, in centimeters, of Δ ABC? NOTE: The symbol ~ means
“is similar to.”
L
12.5
The ratio of the perimeters of the two triangles is the
same as the ratio of any two corresponding sides.
B
A
The perimeter of Δ KLM = 12.5 + 7.5 + 15 = 35.
3
AC
Perimeter of + ABC
3
p
=
→
=
→
C
7.5 35
KM Perimeter of + KLM
Cross multiply → 7.5 p = 105 → p = 14 → Correct answer is B) 14
26.
If
3 7
a 7
=
K
7.5
15
M
3 7
is true, then a = ?
7
3 7 3 7
=
→ If the numerators are equal, then the denominators must be equal →
7
a 7
a 7
7
7
7∗ 7
7 7
=
→a=
→a=
→a=
→a= 7
7
7
7
7
7∗ 7
Correct answer is G) 7
a 7 =7→
ALTERNATIVE WAY: Cross multiply → a 7 ∗ 3 7 = 3 7 ∗ 7 → 3a ∗ 7 = 21 7 →
21a = 21 7 → a = 7 → Correct answer is G) 7
27.
A hot-air balloon 70 meters above the ground is falling at a constant rate of 6
meters per second while another hot-air balloon 10 meters above the ground is
rising at a constant rate of 15 meters per second. To the nearest tenth of a
second, after how many seconds will the 2 balloons be the same height above the
ground?
Distance between upper balloon and lower
balloon = 70 m – 10 m = 60 m
Distance formula → Distance = Rate ∗ Time
Rate of upper balloon = 6 m/sec
Distance covered by upper balloon →
D = r ∗ t → D = 6 ∗ t → D = 6t
Rate of lower balloon = 15 m/sec
Distance covered by lower balloon →
D = r ∗ t → D = 15 ∗ t → D = 15t
Dupper + Dlower = 60 → 6t + 15t = 60 → 21t = 60 →
t = 2.86 seconds → Correct answer is C) 2.9
60 m
70 m
10 m
28.
A hiking group will go from a certain town to a certain village by van on 1 of 4
roads, from the village to a waterfall by riding bicycles on 1 of 2 bicycle paths,
and then from the waterfall to their campsite by hiking on 1 of 6 trails. How
many routes are possible for the hiking group to go from the town to the village
to the waterfall to their campsite?
Town-village roads (4) * village-waterfall paths (2) * waterfall-campsite trails (6) =
4 ∗ 2 ∗ 6 = 48 possible routes → Correct answer is J) 48
29.
Cube A has an edge length of 2 inches. Cube B has an edge length double that of
Cube A. What is the volume, in cubic inches, of Cube B?
If cube B has a edge length double that of Cube A, then Cube B has an edge length
of 4 in. In a cube, the length, the width, and the height are all the same.
V = Length ∗ Width ∗ Height → V = 4 in ∗ 4 in ∗ 4 in = 64 in 3 →
Correct answer is E) 64
30.
A formula used to compute the current value of a savings account is
A = P(1 +r)n, where A is the current value; P is the amount deposited; r is the
rate of interest for 1 compounding period, expressed as a decimal; and n is the
number of compounding periods. Which of the following is closest to the value of
a savings account after 5 years if $10,000 is deposited at 4% annual interest
compounded yearly?
P = 10000, r = 0.04 ( 4% as a decimal ) , n = 5
A = 10000 ∗ (1 + 0.04 ) → A = 10000 ∗ (1.04 ) → A = 10000 ∗1.2166529024 →
A = $12,166.53 → Correct answer is G) $12,167
5
31.
5
A right circular cylinder is shown in the figure below, with dimensions given in
centimeters. What is the total surface area of this cylinder, in square
centimeters? NOTE: The total surface area of a cylinder is given by 2πr2 + 2πrh
where r is the radius and h is the height.
Diameter of cylinder = 20 → Radius of cylinder = 10
Surface Area = 2π r 2 + 2π rh →
A = 2 ∗ π ∗102 + 2 ∗ π ∗10 ∗ 20 → A = 2 ∗ π ∗100 + 400π →
A = 200π + 400π → A = 600π →
Correct answer is D) 600π
20
20
32.
Given f(x) = 4x + 1 and g(x) = x2 – 2, which of the following is an expression for
f[g(x)]?
To evaluate f ⎡⎣ g ( x ) ⎤⎦ , replace every instance of x in f ( x ) with algebraic expression
from g ( x ) . → f ⎡⎣ g ( x ) ⎤⎦ = f ( x 2 − 2 ) = 4 ( x 2 − 2 ) + 1 = 4 x 2 − 8 + 1 = 4 x 2 − 7 →
Correct answer is H) 4 x 2 − 7
33.
The table below shows the total number of goals scored in each of 43 soccer
matches in a regional tournament.
Total number of
goals in a match
0
1
2
3
4
5
6
7
Number of
matches
with this total
4
10
5
9
7
5
1
2
Total goals scored in 4 matches : 4 ∗ 0 = 0
Total goals scored in 10 matches : 10 ∗1 = 10
Total goals scored in 5 matches : 5 ∗ 2 = 10
Total goals scored in 9 matches : 9 ∗ 3 = 27
Total goals scored in 7 matches : 7 ∗ 4 = 28
Total goals scored in 5 matches : 5 ∗ 5 = 25
Total goals scored in 1 matches : 1 ∗ 6 = 6
Total goals scored in 2 matches : 2 ∗ 7 = 14
Total goals scored in 43 matches : 0 + 10 + 10 + 27 + 28 + 25 + 6 + 14 = 120 goals
Average number of goals: 120 ÷ 43 = 2.791 goals/game → Correct answer is B) 2.8
34.
Lines a, b, c, and d are shown below and a || b. Which of the following is the set of
all angles that must be supplementary to ∠ x?
∠ x and ∠ 1 are supplementary, because they
form a linear pair
∠ x and ∠ 2 are supplementary, because they
form a linear pair
∠ 2 ≅ ∠ 10, because they are corresponding ∠s
formed by parallel lines
∠ 1 ≅ ∠ 9, because they are corresponding ∠s
formed by parallel lines
Correct answer is H) {1, 2, 9, 10}
35.
( 3x )
3
(3x )
3 3
36.
3
a
b
x 1
2 3
8 9
10 11
12 13
14 15
4 5
6 7
c
d
is equivalent to:
= 33 ∗ ( x 3 ) = 27 x 9 → ( x a ) = x a∗b → Correct answer is E) 27 x 9
3
b
Which of the following is equivalent to the inequality 4x – 8 > 8x + 16?
4 x − 8 > 8 x + 16 → −4 x − 8 > 16 → −4 x > 24 → x < −6 → Correct answer is F) x < −6
When you divide both sides of an inequality by a negative number, the direction of
the inequality arrow is reversed.
37.
As shown in the standard (x, y) coordinate plane below, P (6, 6) lies on the circle
with center (2, 3) and radius 5 coordinate units. What are the coordinates of the
image of P after the circle is rotated 90° clockwise about the center of the circle?
Slope of a line =
rise
run
8
P (6, 6)
6
3 ( rise ) 3
Slope of line joining center and P =
=
4 ( run ) 4
Point P is rotated 90° clockwise → The line that
joins the center and this new point must be ⊥
to the previous line. Lines that are ⊥ have slopes
that are opposite reciprocals. Opposite reciprocal of
−4 ( rise )
4 ( rise )
3
4
is − , either
or
4
3
3 ( run )
−3 ( run )
4
2
(2, 3)
5
10
-2
−4 ( rise )
from center gives point ( 5, − 1) ← 90° clockwise
3 ( run )
4 ( rise )
from center gives point ( −1, 7 ) ← 90° counter-clockwise
−3 ( run )
Correct answer is C) ( 5, − 1)
38.
For right triangle Δ KLM below, what is sin ∠ M?
Calculate m using Pythagorean Theorem → 102 + m 2 = 122 →
K
100 + m = 144 → m = 44 → m = 44 → m = 44
2
sine =
2
2
12 cm
opposite leg
44
→ sin ∠ M =
→
hypotenuse
12
Correct answer is K)
44
12
m
M
10 cm
L
39.
In the figure below, B lies on AC , BD bisects ∠ ABE , and BE bisects ∠ CBD.
What is the measure of ∠ DBE ?
D
If BD bisects ∠ ABE , then ∠ ABD ≅ ∠ DBE.
E
If BE bisects ∠ CBD, then ∠ DBE ≅ ∠ EBC.
If ∠ ABD ≅ ∠ DBE and ∠ DBE ≅ ∠ EBC ,
then ∠ ABD ≅ ∠ EBC.
∠ ABD + ∠ DBE + ∠ EBC = 180°
A
B
If these three angles add up to 180° and are all
congruent, then each angle equals 180° ÷ 3 = 60° → Correct answer is B) 60°
40.
C
If there are 8 x 1012 hydrogen molecules in a volume of 4 x 104 cubic centimeters,
what is the average number of hydrogen molecules per cubic centimeter?
Molecules 8 × 1012
=
= 2 ×108 ( 8 ÷ 4 = 2, 1012 ÷ 104 = 1012− 4 = 108 )
4
Volume
4 ×10
Correct answer is H) 2 × 108
41.
In the figure below, a radar screen shows 2 ships. Ship A is located at a distance
of 20 nautical miles and bearing 170°, and ship B is located at a distance of 30
nautical miles and bearing 300°. Which of the following is an expression for the
straight-line distance, in nautical miles, between the 2 ships? NOTE: For Δ ABC
with side of length a opposite ∠ A, side of length b opposite ∠ B, and side length
c opposite ∠ C, the law of cosines states that c2 = a2+b2 – 2ab * cos ∠ C.
Side a is opposite ∠ A → a = 30
Side b is opposite ∠ B → b = 20
∠ C = 300° − 170° = 130°
c 2 = a 2 + b 2 − 2 ∗ a ∗ b ∗ cos C →
c 2 = 30 2 + 20 2 − 2 ∗ 30 ∗ 20 ∗ cos130° →
N
B
c 2 = 302 + 202 − 2 ∗ 30 ∗ 20 ∗ cos130° →
30
c = 302 + 202 − 2 ∗ 30 ∗ 20 ∗ cos130° →
Correct answer is B) 20 + 30 − 2 ( 20 )( 30 ) cos130°
2
2
C
c
130°
20
A
170°
300°
42.
What rational number is halfway between
1
1
and ?
5
3
1 3
1 5
3
5
4
4
=
→ Correct answer is J)
and = → Halfway between
and
is
5 15
3 15
15
15
15
15
43.
In isosceles trapezoid ABCD, AB is parallel to DC , ∠ BDC measures 25°,
and ∠ BCA measures 35°. What is the measure of ∠ DBC ?
If ∠ BDC = 25°, then ∠ ACD = 25°
∠ DXC = 180° − 25° − 25° = 130°
∠ BXC = 180° − 130° = 50°
∠ XBC = 180° − 50° − 35° = 95°
∠ XBC and ∠ DBC are the same angle
Correct answer is B) 95°
A
B
X
130°
25°
D
50°
95°
35°
25°
C
ALTERNATIVE WAY: If ∠ BDC = 25°, then ∠ ACD = 25°
∠ BCD = 25° + 35° = 60°
∠ DBC = 180° − 25° − 60° = 95°
44.
In the figure below, the area of the larger square is 50 square centimeters and
the area of the smaller square is 18 square centimeters. What is x, in
centimeters?
Area of square = side2
Area of large square = 50 → side 2 = 50 → side 2 = 50 →
side of large square = 50 = 25 ∗ 2 = 5 2
Area of small square = 18 → side 2 = 18 → side2 = 18 →
side of small square = 18 = 9 ∗ 2 = 3 2
x = side of large square − side of small square = 5 2 − 3 2 = 2 2
Correct answer is G) 2 2
45.
Which of the following is a rational number?
A rational number is any number that can be expressed as a fraction with an integer in
both the numerator and denominator. The majority of square roots are irrational,
which eliminates 2, π , and 7.
5
5
5
=
=
→ Again, most square roots are irrational.
25
5
25
64
64 8
=
= → This number is rational. Correct answer is E)
49
49 7
46.
64
49
If a < b, then |a – b| is equivalent to which of the following?
Just use trial & error with a set of numbers: a = 3, b = 4 → a < b
Does a − b = a + b ? → 3 − 4 = −1 = 1, 3 + 4 = 7 → NO
Does a − b = − ( a + b ) ? → 3 − 4 = −1 = 1, − ( 3 + 4 ) = −7 → NO
Does a − b = a − b ? → 3 − 4 = −1 = 1,
3 − 4 = −1 = i → NO
Does a − b = a − b ? → 3 − 4 = −1 = 1, 3 − 4 = −1 → NO
Does a − b = − ( a − b ) ? → 3 − 4 = −1 = 1, − ( 3 − 4 ) = − ( −1) = 1 → YES
Correct answer is K) − ( a − b )
47.
Tom has taken 5 of the 8 equally weighted tests in his U. S. History class this
semester, and he has an average of exactly 78.0 points. How many points does he
need to earn on the 6th test to bring his average score up to exactly 80.0 points?
Tom has taken 5 tests and scored an average of 78.0 points. Therefore, he has scored
a total of 5 ∗ 78.0 = 390 points. If Tom wants to average 80.0 points after the 6th test,
he must score 6 ∗ 80.0 = 480 points.
480 ( points needed ) − 390 ( points scored ) = 90 ( points on 6th test )
Correct answer is A) 90
48.
In the complex plane, the horizontal axis is called the real axis and the vertical
axis is called the imaginary axis. The complex number a + bi is graphed in the
complex plane is comparable to the point (a, b) graphed in the standard (x, y)
coordinate plane. The modulus of the complex number a + bi is given by
a 2 + b2 . Which of the complex numbers z1 , z2 , z3, , z4 , and z5 below has the
greatest modulus?
imaginary axis
z1
In the standard ( x, y ) coordinate plane,
a 2 + b 2 represents the distance from the origin
to the point ( a, b ) . The modulus essentially
does the same for the complex plane.
Which point looks the farthest from the origin?
z1 looks the farthest away from the origin.
Correct answer is F) z1
49.
z5
z2
real axis
O
z4
z3
In the real numbers, what is the solution of the equation 82x + 1 = 41 – x?
To solve this exponential equation, you must convert both sides of the equation to the
same base.
8 = 23 and 4 = 22 → 82 x +1 = 41− x → ( 23 )
2 x +1
= ( 22 )
1− x
→ 26 x +3 = 22 − 2 x →
When you raise a power to another power, you keep the base the same and you
multiply the exponents.
1
1
6 x + 3 = 2 − 2 x → 8 x + 3 = 2 → 8 x = −1 → x = − → Correct answer is C) −
8
8
50.
⎛1 ⎞
The graph of the trigonometric function y = 2cos ⎜ x ⎟ is shown below.
⎝2 ⎠
The function is:
2
A function is even if f ( x ) = f ( − x ) .
When looking at the graph,
f (1) ≈ 1.1 and f ( −1) also ≈ 1.1.
f (1) = f ( −1) , so the function is even.
-5
Correct answer is F) even ( that is, f ( x ) = f ( − x ) for all x ) .
5
-2
51.
An integer from 100 through 999, inclusive, is to be chosen at random. What is
the probability that the number chosen will have 0 as at least 1 digit?
List all the numbers from 100 to 199 that contain 0 as a digit:
100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 120, 130, 140, 150, 160, 170,
180, 190 = 19 numbers
How many numbers between 200 and 299 contain 0 as a digit? Also 19 numbers.
How many numbers between 300 and 399 contain 0 as a digit? Also 19 numbers.
How many numbers between 400 and 499 contain 0 as a digit? Also 19 numbers.
How many numbers between 500 and 599 contain 0 as a digit? Also 19 numbers.
How many numbers between 600 and 699 contain 0 as a digit? Also 19 numbers.
How many numbers between 700 and 799 contain 0 as a digit? Also 19 numbers.
How many numbers between 800 and 899 contain 0 as a digit? Also 19 numbers.
How many numbers between 900 and 999 contain 0 as a digit? Also 19 numbers.
How many numbers between 100 and 999 contain 0 as a digit? 9 * 19 = 171
How many total numbers exist between 100 and 999? 999 – 100 + 1 = 900
Desired outcomes 171
171
=
→ Correct answer is D)
Probability =
Possible outcomes 900
900
52.
In the figure below, line q in the standard (x, y) coordinate plane has equation
–2x + y = 1 and intersects line r, which is distinct from line q, at a point on the
x-axis. The angles, ∠a and ∠b, formed by these lines and the x-axis are
congruent. What is the slope of line r?
4
rise
run
1
Slope of line q =
= 1 ÷ 0.5 = 2
0.5
1
Slope of line r =
= 1 ÷ ( −0.5) = −2
−0.5
Correct answer is F) − 2
Slope of a line =
r
b
2
-2
q
a
O
53.
In the right triangle below, 0 < b < a. One of the angle measures in the triangle
⎡
⎛a⎞
⎛ a ⎞⎤
is tan−1 ⎜ ⎟ . What is cos ⎢tan−1 ⎜ ⎟ ⎥ ?
⎝b⎠
⎝ b ⎠⎦
⎣
opposite leg
adjacent leg
tan =
& cos =
adjacent leg
hypotenuse
Which acute angle in the triangle has a tangent ratio
a
a
of → tan ( ∠ A) =
b
b
⎡
b
⎛ a ⎞⎤
→
cos ⎢ tan −1 ⎜ ⎟ ⎥ = cos [ ∠ A] =
2
⎝ b ⎠⎦
⎣
a + b2
b
Correct answer is D)
a 2 + b2
C
a
B
b
a 2 + b2
A
Use the following information to answer questions #54 – #56.
The radio signal from the transmitter site of
radio station WGGW can be received only with
a radius of 52 miles in all directions from the
transmitter site. A map of the region of coverage
of the radio signal is shown to the right in the
standard (x, y) coordinate plane, with the transmitter
site at the origin and 1 coordinate unit representing
1 mile.
54.
y
52
x
O
Which of the following is closest to the area, in square miles, of the region
covered by the radio signal?
Area of a circle = π ∗ Radius 2 → A = π ∗ 52 2 → A ≈ 8494.9 mile 2
Correct answer is J) 8,500
55.
52
Which of the following is an equation of the circle shown on the map?
The equation of a circle with its center at the origin is x 2 + y 2 = Radius 2 →
The radius is 52 miles → Correct answer is E) x 2 + y 2 = 522
56.
The transmitter site of radio station WGGW and the transmitter of another radio
station, WGWB, are on the same highway 100 miles apart. The radio signal from
the transmitter site of WGWB can be received only within a radius of 60 miles in
all directions from the WGWB transmitter site. For how many miles along the
highway can the radio signals of both stations be received? NOTE: Assume the
highway is straight.
Distance from WGGW to point D =
100 − 60 = 40 m
Distance from WGGB to point C =
100 − 52 = 48 m
The road from D to C represents when
both signals can be heard
Distance from D to C =
100 − 40 − 48 = 12 m →
Correct answer is G) 12
57.
60 m
52 m
WGGW
40 m
D
C
48 m
WGWB
The graphs of the equations y = x – 1 and y = (x – 1)4 are shown in the standard
(x, y) coordinate plane. What real values of x, if any, satisfy the inequality
(x – 1)4 < x – 1?
The graph of y = x − 1 is drawn in purple.
2
The graph of y = ( x − 1) is drawn in blue.
Where on the graph is the blue portion below,
or less than, the purple portion? This happens
between 1 and 2. Correct answer is E) 1 < x < 2
4
-2
58.
For every positive 2-digit number, x, with tens digit t and units digit u, let y be
the 2-digit number formed by reversing the digits of x. Which of the following
expressions is equivalent to x – y?
Just use trial & error with a set of numbers: x = 72, y = 27, t = 7, u = 2
x − y = 72 − 27 = 45
Does x − y = 9 ( t − u ) → x − y = 45, 9 ( t − u ) = 9 ( 7 − 2 ) = 9 ( 5 ) = 45 → YES
Does x − y = 9 ( u − t ) → x − y = 45, 9 ( t − u ) = 9 ( 2 − 7 ) = 9 ( −5) = −45 → NO
Does x − y = 9t − u → x − y = 45, 9t − u = 9 ∗ 7 − 2 = 63 − 2 = 61 → NO
Does x − y = 9u − t → x − y = 45, 9u − t = 9 ∗ 2 − 7 = 18 − y = 11 → NO
Does x − y = 0 → x − y = 45 → NO
Correct answer is F) 9 ( t − u )
59.
In the figure below, the vertices of Δ ABC has (x, y) coordinates (4, 5), (5, 3), and
(1, 3) respectively. What is the area of Δ ABC?
1
∗ base ∗ height →
2
Base of triangle = 4 units, Height of triangle = 2 units
1
A = ∗ 4 ∗ 2 = 4 → Correct answer is A) 4
2
Area of a triangle =
60.
6
4
2
A (4, 5)
2
C (1, 3)
B (5, 3)
5
10
The sum of an infinite geometric series with first term a and common ratio
a
r < 1 is given by
. The sum of a given infinite geometric series is 200, and
1− r
the common ratio is 0.15. What is the second term of this series?
a
a
a
→ 200 =
→ 200 =
→ 170 = a
1− r
1 − 0.15
0.85
In a geometric series, each successive term is produced by multiplying the previous
term by the common ratio. First term is 170, so second term = 170 * 0.15 = 25.5.
Correct answer is F) 25.5
Sum =