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MATHEMATICS LEVEL 2 TEST
Name:
ACT Subject Test Mathematics Level 2
Prepared by
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Ms. Niveen Al-Qsous-0798598447
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MATHEMATICS LEVEL 2 TEST
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Directions:
Read each question carefully. For each multiple-choice question, determine the best
answer to the question from the four answer choices provided.
1. If 𝑥 ≠ ±1, then
(A)
(B)
(C)
(D)
6
𝑥−1
−
7
𝑥+1
=
13−𝑥
𝑥 2 −1
−𝑥−1
𝑥 2 −1
−𝑥−1
𝑥 2 +1
13−𝑥
𝑥 2 +1
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MATHEMATICS LEVEL 2 TEST-Continued
3
2. If 𝑓(𝑥) = 2𝑥 2 − 8𝑥, and 𝑔(𝑥) = √𝑥 2 + 25, then 𝑔(𝑓(−1)) =
(A)
1
(B) 2
(C) 4
(D) 5
|3𝑥 − 2| = −5
3. How many solutions does the given equation above have?
(A) Zero
(B) One
(C) Two
(D) Infinitely many
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MATHEMATICS LEVEL 2 TEST-Continued
B
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𝜃
A
C
4. In the figure above, if sec 𝜃 = 𝑥, then what is the value
of cot 𝜃 ?
(A) √1 − 𝑥 2
(B) √𝑥 2 − 1
(C)
√1−𝑥 2
𝑥
(D)
1
√𝑥 2 −1
4
3
12
8
5. Snacks at a country fair are sold in containers shaped like a cone or a cylinder.
The dimensions, in inches, of each container are shown above. Which statement
about the volumes of the cone and the cylinder is true?
(A)
(B)
(C)
(D)
The volume of the cylinder is about 377 cubic inches greater than the volume of the cone.
The volume of the cylinder is about 377 cubic inches less than the volume of the cone.
The volume of the cylinder is about 25 cubic inches greater than the volume of the cone.
The volume of the cylinder is about 25 cubic inches less than the volume of the cone.
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MATHEMATICS LEVEL 2 TEST-Continued
3𝑚(𝑚 − 1) − (𝑚 + 2)(𝑚 − 2) is equivalent to:
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6.
(A)
(B)
(C)
(D)
2𝑚2 − 3𝑚 − 4
2𝑚2 − 3𝑚 + 4
4𝑚2 − 3𝑚 − 4
4𝑚2 − 3𝑚 + 4
𝜋
7. If 0 < sin−1 𝑥 < , what is the value of sec(sin−1 𝑥)?
2
(A)
(B)
(C)
(D)
𝑥
√𝑥 2 −1
√1 − 𝑥 2
1
√1−𝑥 2
√𝑥 2 −
1
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MATHEMATICS LEVEL 2 TEST-Continued
5
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8. If √𝑥 − 𝑞 = 𝑟, then 𝑥 =
(A)
(B)
(C)
𝑟 5 + 𝑞5
(𝑟 + 𝑞)5
(D)
(𝑟 + 𝑞)5
5
√𝑟 + 𝑞
1
9. Two concentric circles are shown below. The radius
of the larger circle is 10 feet, and the radius of the
smaller circle is 6 feet. One point inside the larger
circle was selected at random. What is the probability
that the selected point would be in the shaded region?
(A)
(B)
(C)
(D)
0.08
0.16
0.36
0.64
10
6
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MATHEMATICS LEVEL 2 TEST-Continued
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𝑦
4
6
.
0
𝑥
𝑂
4
6 10. Which of the following is an equation of the
.
6graph in the figure above?
0
.
𝑦 =𝑥−4
0(A)
(B)
(C)
(D)
𝑦 =4−𝑥
𝑦 =𝑥+4
𝑦 = −𝑥 − 4
11. How many positive 4-digit even integers can
be made by using digits 0, 1, 2, and 3 once?
(A)
(B)
(C)
(D)
6
10
12
24
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MATHEMATICS LEVEL 2 TEST-Continued
2𝑥 2 − 𝑥𝑦 = 1
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𝑦 + 3𝑥 = 1
12. How many solutions are there to the system
of equations above?
(A)
(B)
(C)
(D)
There are exactly 4 solutions.
There are exactly 2 solutions.
There are exactly 1 solution.
There is no solution.
13. What is the sum of the roots of the equation
𝑥 2 − 2𝑥 − 10 = 2𝑥 − 2 ?
(A)
(B)
(C)
(D)
−4
−2
2
4
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MATHEMATICS LEVEL 2 TEST-Continued
𝑦
𝑥
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6.0
6.
0
14. If the graph shown above represents the graph of 𝑦 = 𝑓(𝑥), then 𝑓(𝑥) =
(A) 𝑥 2 + 5
(B) 𝑥 2 − 5
(C) 5 − 𝑥 2
(D) −5 − 𝑥 2
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MATHEMATICS LEVEL 2 TEST-Continued
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1−𝑛
15. ∑∞
=
𝑛=2 3
(A)
(B)
(C)
(D)
1
9
1
3
1
2
The sum diverges
16. If log 𝑏 5 = 𝑥 and log 𝑏 √3 = 𝑦, then, in terms
of 𝑥 and 𝑦, log 𝑏 45 =
(A)
(B)
(C)
(D)
𝑥 + 4𝑦
𝑥 + 2𝑦
𝑥+𝑦
2𝑥 + 2𝑦
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MATHEMATICS LEVEL 2 TEST-Continued
𝑦
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𝑦
𝑦 = ℎ(𝑥)
𝑦 = 𝑔(𝑥)
16 1
.
0 6.0
𝑥
16
. 1
0 6.0
6
6 6
6
6
6
6
.
.
.
.
.
.
.
0
0
17. The figures above
show the graphs of functions 0𝑔
0
0
0
0
and ℎ. What6 is the domain of 𝑓(𝑥) = 𝑔(ℎ(𝑥))?
6
.
.
0
0
(A) [−1, 4]
(B)
(C)
(D)
𝑥
6
.
0
[−2, 4]
[−2, 1]
[−3, 4]
18. For a class test, the mean score was 71, the median score
was 77, and the standard deviation of the scores was 8.
The teacher decided to add 5 points to each score due to
a grading error. Which of the following statements must
be false for the new scores?
(A) The new mean score is 76.
(B) The new median score is 82.
(C) The new standard deviation of the scores is 13.
(D) The range remains the same.
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MATHEMATICS LEVEL 2 TEST-Continued
19. Which of the following polar equations represents the graph above?
(A) 𝑟(𝜃) = 2 sin 𝜃 cos 𝜃
(B) 𝑟(𝜃) = 2 sin 𝜃
(C) 𝑟(𝜃) = cos 2𝜃
(D) 𝑟(𝜃) = 1 − 2 sin 𝜃
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MATHEMATICS LEVEL 2 TEST-Continued
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𝑦
6
.
0
𝑥
6
20. The graph of 𝑦 = 𝑓(𝑥) is shown. in the figure
above. Which of the following is0 the graph
of 𝑦 = −𝑓(−𝑥)?
(A)
(B)
𝑦
6
.
0
(C)
𝑦
6
.
0
𝑥
6
.
0
𝑦
6
.
0
(D)
𝑥
6
.
0
𝑦
6
.
0
𝑥
6
.
0
𝑥
6
.
0
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MATHEMATICS LEVEL 2 TEST-Continued
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21. Aswarm of bees is spreading from an original region
of 10 square miles. The region the swarm occupies
triples in area each year. At this rate, how long will
it take for the swarm to cover an area of 1,000,000
square miles?
(A) 19.93 years
(B) 16.61 years
(C) 12.58 years
(D) 10.48 years
22. If 𝑓(𝑥) = 𝑥 6 + 2𝑥 3 + 1, then 𝑓(𝑥) can be factored as:
(A)
(B)
(C)
(𝑥 − 1)2 (𝑥 2 + 𝑥 + 1)2
(𝑥 + 1)2 (𝑥 2 − 𝑥 + 1)2
(𝑥 3 + 1)(𝑥 3 − 1)
(D)
(𝑥 2 + √2𝑥 + 1)
3
3
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MATHEMATICS LEVEL 2 TEST-Continued
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23. What is the remainder when 𝑥 3 − 19𝑥 2 + 11𝑥 + 107 is divided
by (𝑥 + 2) ?
(A)
(B)
(C)
(D)
0
1
2
3
24. A function 𝑓 is defined by the table below. For which
of the following values of 𝑥 does 𝑓(𝑓(𝑥)) = 6 ?
𝑥
𝑓(𝑥)
(A)
(B)
(C)
(D)
1
5
2
3
3
6
4
2
5
1
6
0
0
2
3
5
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7
5
MATHEMATICS LEVEL 2 TEST-Continued
25. How many lines can be drawn in the 𝑥𝑦-plane that
𝑥2
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do not intersect the graph of 𝑦 =
(A)
(B)
(C)
(D)
𝑥 2 −4
?
None
Two
Three
Infinitely many
26. If the graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑓 −1 (𝑥) are
identical, then each of these graphs must be
symmetric about the
(A)
(B)
(C)
(D)
𝑦-axis
𝑥-axis
Line 𝑦 = −𝑥
Line 𝑦 = 𝑥
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MATHEMATICS LEVEL 2 TEST-Continued
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27. The probability of getting 80% or more of the
questions correct on a 10-question true/false
exam merely by guessing is
(A)
(B)
(C)
(D)
5
32
1
16
7
128
7
32
3
28. If 𝑧 = [ √4 (cos
𝜋
𝜋
3
+ 𝑖 sin )] = 𝑎 + 𝑖𝑏,
4
4
then, in terms of 𝑎 and 𝑏, 𝑧 =
(A)
(B)
(C)
(D)
−2√2 + 𝑖2√2
2√2 − 𝑖2√2
2√2 + 𝑖2√2
−2√2 − 𝑖2√2
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MATHEMATICS LEVEL 2 TEST-Continued
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29. If 𝑎𝑏 ≠ 0, then 𝑥 = 𝑎 sec 𝜃 and 𝑦 = 𝑏 tan 𝜃 represent the
parametric equations of a(n):
(A)
(B)
(C)
(D)
Ellipse
Hyperbola
Parabola
Circle
30. What are the domain and range of the function 𝑓(𝑥) =
−1
𝜋𝑥
−2?
(A)
Domain: 𝑥 ≠ 0 , Range: 𝑓(𝑥) ≠ −2
(B)
Domain: 𝑥 ≠ 𝜋 , Range: 𝑓(𝑥) = 2
(C)
Domain: 𝑥 ≠ 0 , Range: 𝑓(𝑥) is all real numbers
(D)
Domain: 𝑥 is all real numbers, Range: 𝑓(𝑥) is all real numbers
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MATHEMATICS LEVEL 2 TEST-Continued
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𝐵
6
.
0
√42
3
6.
0
𝜃
𝐴
6.
60
6.0
4
6.0
.
31. In Δ𝐴𝐵𝐶 shows above, sin 𝜃 =
0
(A)
(B)
(C)
(D)
32.
6
.
0
−1
−0.706
1
0.706
sin 𝑥 csc 𝑥
sec 𝑥 cos 𝑥
(A)
(B)
(C)
(D)
𝐶
=
cos 2 𝑥 − sin2 𝑥
sin2 𝑥 − cos 2 𝑥
sin2 𝑥 + cos 2 𝑥
tan2 𝑥
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MATHEMATICS LEVEL 2 TEST-Continued
33. The graph of 𝑦 = 𝑓(𝑥) has four 𝑥-intercepts and one 𝑦-intercept.
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Which of the following functions could be 𝑓(𝑥)? (𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓,
and 𝑔 are real numbers).
(A)
𝑥 3 + 𝑎𝑥 + 𝑐
(B)
𝑥 5 + 𝑎𝑥 4 + 𝑏𝑥 3 + 𝑐𝑥 2 + 𝑑𝑥 + 𝑒
(C)
𝑥 6 + 𝑎𝑥 5 + 𝑏𝑥 4 + 𝑐𝑥 3 + 𝑑𝑥 2 + 𝑒𝑥 + 𝑓
(D)
𝑥 7 + 𝑎𝑥 6 + 𝑏𝑥 5 + 𝑐𝑥 4 + 𝑑𝑥 3 + 𝑒𝑥 2 + 𝑓𝑥 + 𝑔
2
𝑥
−2
6.
0
6.
34. The graph shown above has equation 𝑦 = 2 sin 𝑥. If the graph is
0
𝜋
shifted units to the right, which of the following could be the
3
equation of the shifted graph?
𝜋
(A) 2 sin (𝑥 + )
3
𝜋
(B) 2 sin (𝑥 − )
3
(C) 2 sin 𝑥 +
(D) 2 sin 𝑥 −
𝜋
3
𝜋
3
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MATHEMATICS LEVEL 2 TEST-Continued
2𝑥 − 3𝑦 = 4
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3𝑦 = 2𝑥 − 8
35. If the system of equations above is transferred to the matrix equation
𝐴𝑋 = 𝐵, where 𝐴 is the coefficient matrix, 𝑋 is the variable matrix,
and 𝐵 is the constant matrix, what is the determinant of matrix 𝐴?
(A)
(B)
(C)
(D)
0
1
3
6
36. If the frequency of a periodic function is defined
as the reciprocal of the period, what is the
𝜋
frequency of 𝑓(𝑥) = −5 cos 4 (3𝑥 + )?
3
(A)
(B)
(C)
(D)
3
𝜋
3
2𝜋
6
𝜋
10
𝜋
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MATHEMATICS LEVEL 2 TEST-Continued
37. In the 𝑥𝑦-plane, if 0 ≤ 𝑥 ≤ 2𝜋, the graph of 𝑦 = sin 𝑥 and 𝑦 = −cos 𝑥
M
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will intersect in quadrant(s):
(A)
I only
(B)
I and II
(C)
II and IV
(D)
I and III
38. A hyperbola has foci with coordinates (4, 2) and (−6, 2). The graph
of the hyperbola passing through point (3,
6+4√7
).
3
Which of the
following is the standard equation of the hyperbola?
(A)
(B)
(C)
(D)
(𝑦−2)2
16
(𝑦−2)2
9
(𝑥+1)2
16
(𝑥+1)2
9
−
−
−
−
(𝑥+1)2
9
(𝑥+1)2
16
(𝑦−2)2
9
(𝑦−2)2
16
=1
=1
=1
=1
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MATHEMATICS LEVEL 2 TEST-Continued
39. Which of the following is an equation of a circle that
M
s.
M Ni
r. ve P
Ba en r
ha A epa
a l-Q re
To s d
ba ou by
il s (
(0 07
79 9
09 85
36 98
76 44
6) 7)
has a center of (1,0) and a diameter of 6?
(A)
(𝑥 − 1)2 + 𝑦 2 = 9
(B)
(𝑥 − 1)2 + 𝑦 2 = 36
(C)
𝑥 2 +(𝑦 − 1)2 = 9
(D)
𝑥 2 +(𝑦 − 1)2 = 36
6
40. Which of the following is equivalent to √𝑥 + 𝑦 ?
1
1
(A)
𝑥6 + 𝑦6
(B)
(𝑥 + 𝑦)6
(C)
𝑥6 + 𝑦6
(D)
(𝑥 + 𝑦)6
1
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MATHEMATICS LEVEL 2 TEST-Continued
41. In the standard 𝑥𝑦-plane, line 𝑙 passes through
M
s.
M Ni
r. ve P
Ba en r
ha A epa
a l-Q re
To s d
ba ou by
il s (
(0 07
79 9
09 85
36 98
76 44
6) 7)
points (0, −1) and (2, 1). Line 𝑚 passes through
points (𝑥, 4) and (0, 5). What is the value of 𝑥 if
⃡
𝑙⊥𝑚
⃡ ?
(A)
−1
(B)
0
(C)
1
(D)
2
42. Which of the following functions has a graph
that is symmetrical about the origin?
(A)
𝑓(𝑥) = 𝑥 2 + 1
(B)
𝑓(𝑥) = cos 𝑥
(C)
𝑓(𝑥) =
(D)
𝑓(𝑥) = 𝑥 3 + 𝑥 − 1
1
𝑥3
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MATHEMATICS LEVEL 2 TEST-Continued
𝑏⃑
M
s.
M Ni
r. ve P
Ba en r
ha A epa
a l-Q re
To s d
ba ou by
il s (
(0 07
79 9
09 85
36 98
76 44
6) 7)
𝑎⃑
6.0
6.0
The Horizontal
43. In the figure above, vector 𝑎⃑ has magnitude 10 and
6.0
makes an angle of 30° with the horizontal. Vector
𝑏⃑ has magnitude 8 and makes an angle of 120° with
the horizontal. What is the angle that the vector
𝑐⃑ = 𝑏⃑ − 𝑎⃑ makes with the horizontal?
(A)
(B)
(C)
(D)
68.66°
171.34°
188.66°
−68.66°
44. The inner (dot) product of two vectors 𝐴⃑ and 𝐵⃑ is given as
𝐴⃑. 𝐵⃑ = −|𝐴⃑||𝐵⃑|. What is the angle between the two vectors?
(A)
(B)
(C)
(D)
0°
45°
90°
180°
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MATHEMATICS LEVEL 2 TEST-Continued
1
2 sin 𝜃
is equivalent to
M
s.
M Ni
r. ve P
Ba en r
ha A epa
a l-Q re
To s d
ba ou by
il s (
(0 07
79 9
09 85
36 98
76 44
6) 7)
45. The polar equation 𝑟 = −
which of the following equations in the 𝑥𝑦plane?
1
(A)
𝑦=−
(B)
𝑥=−
(C)
𝑥 2 + (𝑦 + 1)2 = 1
(D)
(𝑥 + 1)2 + 𝑦 2 = 1
2
1
2
46. The quadratic equation 𝑦 = −(𝑥 + 2)2 + 4
has a maximum value at 𝑥 =
(A)
−4
(B)
−2
(C)
2
(D)
4
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MATHEMATICS LEVEL 2 TEST-Continued
M
s.
M Ni
r. ve P
Ba en r
ha A epa
a l-Q re
To s d
ba ou by
il s (
(0 07
79 9
09 85
36 98
76 44
6) 7)
47. Which of the following graphs does NOT show 𝑦 as a function of 𝑥?
(A)
(B)
𝑦
6
.
0
(C)
𝑦
6
.
0
𝑥
6
.
0
𝑦
6
.
0
(D)
𝑥
6
.
0
𝑦
6
.
0
𝑥
6
.
0
𝑥
6
.
0
1 , 𝑥<0
48. A function 𝑓(𝑥) is defined by 𝑓(𝑥) = {2 − 𝑥 , 0 ≤ 𝑥 ≤ 1.
0 , 𝑥>1
What is the value of 𝑓(𝑓(3))?
(A)
(B)
(C)
(D)
0
1
2
3
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MATHEMATICS LEVEL 2 TEST-Continued
𝑥 2 +𝑥−6
𝑥+3
, what number does 𝑓(𝑥) approach as
M
s.
M Ni
r. ve P
Ba en r
ha A epa
a l-Q re
To s d
ba ou by
il s (
(0 07
79 9
09 85
36 98
76 44
6) 7)
49. If 𝑓(𝑥) =
𝑥 approaches −3?
(A) −5
(B) −3
(C) 0
(D) 𝑓(𝑥) increases without bound
50. For which of the following values of 𝑥 in the interval:
[0, 2𝜋] is 𝑓(𝑥) =
5
2 cos 𝑥+3
undefined?
(A) 0.84
(B) 2.30
(C) 3
(D) There is no such value of 𝑥
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