Additional Practice Test 2: ACT Math Study Guide

This form deliberately disrupts the ACT's usual overall increase in difficulty so recovery can be practiced when an early item feels expensive. It remains 45 questions in 50 minutes and keeps broad content coverage, but it should be treated as a stress drill instead of a simulation of official difficulty ordering. Geometry, trigonometry, and visual interpretation are intentionally more visible.

Time: 50 minutes. Questions: 45. Calculator permitted throughout.

Work in order unless a question becomes locally expensive. Choose the best answer and move on when the route is not developing. Do not leave any question unanswered. A useful pacing checkpoint is roughly 15 questions complete by 14–16 minutes and roughly 30 complete by 31–34 minutes, preserving 16–19 minutes for the final 15. This form intentionally places a few expensive questions earlier than expected; a hard early item should be skipped before it creates a panic-speed cascade.

▸ Questions

Question 1

A package weighs 2.4 kilograms. How many grams does it weigh?

A. 24
B. 240
C. 2,400
D. 24,000

Question 2

If 5x+2=32, what is x?

F. 5
G. 7
H. 8
J. 6

Question 3

A triangle has angles 48 degrees and 67 degrees. What is the third angle?

A. 55
B. 75
C. 85
D. 65

Question 4

For f(x)=x²+2x, what is f(−3)?

F. 9
G. 15
H. 3
J. −3

Question 5

The table shows the number of books read by 8 students: 0,1,1,2,3,3,3,7. What is the median?

A. 1.5
B. 2
C. 2.5
D. 3

Question 6

A square has side length 7. A second square is created by doubling every side length. How many times as large is the area of the second square?

F. 4
G. 2
H. 7
J. 14

Question 7

Which expression is equivalent to √72?

A. 6√2
B. 3√8
C. 8√3
D. 12√2

Question 8

A line passes through (2,5) and (8,17). What is its slope?

F. 1/2
G. 2
H. 3
J. 6

Question 9

The graph shown is a dilation of triangle ABC about the origin with scale factor 2. If A=(2,1), what is A'?

$Figure from the book$

A. (4, 2)
B. (1, 0.5)
C. (2, 2)
D. (4, 1)

Question 10

A bank account starts with $500 and grows by 4% each year. Which expression gives the balance after t years?

F. 500+0.04t
G. 500(0.04)^t
H. 500(1.04)^t
J. 500(1.4)^t

Question 11

A spinner has 8 equal sectors numbered 1 through 8. What is the probability of landing on a prime number?

$Figure from the book$

A. 1/4
B. 3/8
C. 5/8
D. 1/2

Question 12

If a+bi = 7−3i, what is a+b?

F. 7
G. 4
H. 10
J. −10

Question 13

The diagram shows two parallel lines cut by a transversal. One exterior angle is 118 degrees. What is the acute angle formed at the other intersection?

$Figure from the book$

A. 52
B. 72
C. 118
D. 62

Question 14

A theater sold adult tickets for $12 and student tickets for $8. A total of 150 tickets produced $1,560 in revenue. How many adult tickets were sold?

F. 90
G. 60
H. 75
J. 120

Question 15

A box plot is shown. Which statement must be true?

$Figure from the book$

A. Exactly 25% of data lie below the minimum.
B. About 50% of data lie between Q1 and Q3.
C. The mean equals the median.
D. There are no outliers.

Question 16

The function p(x)=|x−4|+2 has minimum value:

F. 2
G. 0
H. 4
J. 6

Question 17

A circle has center O. Chord AB subtends a central angle of 140 degrees. An inscribed angle ACB intercepts the same arc AB. What is angle ACB?

$Figure from the book$

A. 35
B. 140
C. 280
D. 70

Question 18

A delivery company charges a fixed fee plus a constant amount per mile. A 6-mile delivery costs $17 and a 14-mile delivery costs $29. What is the fixed fee?

F. $6
G. $10
H. $12
J. $8

Question 19

Which statement is equivalent to log₃(81)=4?

A. 3+4=81
B. 4³=81
C. 3⁴=81
D. 81³=4

Question 20

A cone has radius 6 and height 9. What is its volume?

$Figure from the book$

F. 54π
G. 108π
H. 162π
J. 324π

Question 21

A gym membership is discounted 20%, then 8% sales tax is applied to the discounted price. If the original price is $250, what is the final price?

A. $200
B. $208
C. $216
D. $230

Question 22

The quadratic x²+kx+16=0 has exactly one real solution. What are the possible values of k?

F. k=8 or k=−8
G. k=4 only
H. k=−4 only
J. k=8 only

Question 23

The graph of y=f(x) is translated 3 units right and 5 units down. Which function represents the translated graph?

A. f(x+3)+5
B. f(x−3)−5
C. f(x+3)−5
D. f(x−3)+5

Question 24

Two chords intersect inside a circle. One chord is split into segments of lengths 4 and 9. The other is split into 6 and x. What is x?

$Figure from the book$

F. 3
G. 8
H. 6
J. 12

Question 25

A quantity is inversely proportional to x. If y=18 when x=4, what is y when x=12?

A. 6
B. 3
C. 18
D. 54

Question 26

Simplify (x⁴y⁻²)/(x⁻¹y³).

F. x⁵/y⁵
G. x³/y
H. x⁵y
J. 1/(x⁵y⁵)

Question 27

A scatterplot shows a strong negative linear association between x and y. Which correlation coefficient is most plausible?

$Figure from the book$

A. −0.92
B. −0.15
C. 0.11
D. 0.89

Question 28

A quadratic function has vertex (2,−5) and passes through (3,−2). Which function could represent it?

F. y=−3(x−2)²−5
G. y=3(x−2)²−5
H. y=3(x+2)²−5
J. y=(x−2)²−5

Question 29

In right triangle ABC, angle C=90 degrees, AC=9, and BC=40. What is sin(A)?

$Figure from the book$

A. 9/40
B. 40/41
C. 9/41
D. 41/40

Question 30

A school has 480 students. If 35% take a language course and 25% of those students take French, how many students take French?

F. 42
G. 60
H. 120
J. 168

Question 31

Solve √(x+2)=x.

A. x=−1 only
B. x=2 only
C. x=2 or x=−1
D. no real solution

Question 32

An arithmetic series has first term 5, common difference 3, and 12 terms. What is its sum?

F. 198
G. 246
H. 258
J. 276

Question 33

A tangent and a secant are drawn from the same exterior point of a circle. The tangent length is 12. The external part of the secant is 8. What is the whole secant length?

$Figure from the book$

A. 18
B. 12
C. 16
D. 24

Question 34

A data set is approximately normal with mean 100 and standard deviation 12. About what percent of values lie between 88 and 112?

F. 34%
G. 95%
H. 68%
J. 99.7%

Question 35

The matrices A=[[2,1],[0,3]] and B=[[4],[5]]. What is AB?

A. [[8],[15]]
B. [[9],[15]]
C. [[13],[5]]
D. [[13],[15]]

Question 36

A rectangular prism has volume 360 cubic inches. Its length is 12 inches and width is 5 inches. What is its height?

F. 4
G. 5
H. 8
J. 6

Question 37

A line is perpendicular to 3x−2y=8. What is the slope of the perpendicular line?

A. −3/2
B. −2/3
C. 2/3
D. 3/2

Question 38

A function satisfies f(f(x))=x for every x in its domain. Which statement must be true?

F. f is constant.
G. f has no zeros.
H. f is quadratic.
J. f is its own inverse.

Question 39

A triangle has sides 7 and 10 with included angle 60 degrees. What is the third side?

A. √39
B. √109
C. √79
D. 17

Question 40

A poll estimates 46% support with margin of error ±3%. Which statement is best supported?

F. A majority definitely supports the proposal.
G. 50% is outside the reported plausible interval.
H. The exact population proportion must be 46%.
J. The margin of error proves the sample had no bias.

Question 41

The polynomial x⁴−5x²+4 has how many real zeros?

A. 0
B. 2
C. 4
D. 3

Question 42

A piecewise function is p(x)=2x+1 for x<3 and p(x)=x²−4 for x≥3. What is p(3)?

F. 7
G. 5
H. 9
J. 13

Question 43

A circle of radius 5 is inscribed in a square. What fraction of the square's area is occupied by the circle?

$Figure from the book$

A. π/5
B. π/4
C. 5π/4
D. 25π/4

Question 44

A machine produces 120 parts in 8 minutes. It then increases its rate by 20%. At the new constant rate, how many parts does it produce in 15 minutes?

F. 225
G. 240
H. 300
J. 270

Question 45

The line y=x−1 intersects the circle x²+y²=13 at two points. What is the sum of the x-coordinates of the intersection points?

A. −1
B. 1
C. −2
D. 2

▸ Answers and Explanations

Show the answers and explanations

1: C. 1 kilogram=1,000 grams, so 2.4(1,000)=2,400 grams.

2: J. 5x=32−2=30, so x=6.

3: D. Triangle angles sum to 180°: 180−48−67=65°.

4: H. f(−3)=(−3)²+2(−3)=9−6=3.

5: C. With 8 values, the median is the average of the 4th and 5th values in order: (2+3)/2=2.5.

6: F. Area scales by the square of the linear factor: 2²=4 times as large.

7: A. 72=36·2, so √72=√36·√2=6√2.

8: G. Slope=(17−5)/(8−2)=12/6=2.

9: A. A dilation about the origin multiplies both coordinates by the scale factor: (2,1)→(2×2, 1×2)=(4,2).

10: H. Repeated percent growth uses the form initial×(1+r)^t: 500(1.04)^t.

11: D. The primes among 1–8 are 2, 3, 5, and 7: 4 of the 8 equally likely outcomes, so P=4/8=1/2.

12: G. a=7 and b=−3, so a+b=7+(−3)=4.

13: D. The acute angle supplementary to the given 118° exterior angle is 180−118=62°. Parallel-line angle relationships carry that acute measure to the other intersection.

14: F. Let a=adult tickets and s=student tickets. a+s=150 and 12a+8s=1,560. Substituting s=150−a: 12a+8(150−a)=1,560, so 4a=360, giving a=90.

15: B. The box from Q1 to Q3 always contains the middle 50% of the data, by definition of the quartiles. The other three statements are not guaranteed by a box plot alone.

16: F. Absolute value is minimized when its argument is 0: at x=4, |x−4|=0, so the minimum is p(4)=0+2=2.

17: D. An inscribed angle is half the central angle intercepting the same arc: 140÷2=70°.

18: J. Slope=(29−17)/(14−6)=12/8=1.5 dollars per mile. The fixed fee b satisfies 17=1.5(6)+b=9+b, so b=8.

19: C. A logarithm states the exponent needed to produce the argument: log₃(81)=4 means 3⁴=81.

20: G. V=(1/3)πr²h=(1/3)π(6)²(9)=(1/3)π(36)(9)=108π.

21: C. After the discount: 250(0.80)=200. After the tax: 200(1.08)=216.

22: F. Exactly one real solution means the discriminant is 0: k²−64=0, so k²=64, giving k=±8.

23: B. Translating right 3 replaces x with x−3; translating down 5 subtracts 5 from the whole function: f(x−3)−5.

24: H. By the intersecting chords relationship, the products of the segments are equal: 4(9)=6x, so 36=6x, giving x=6.

25: A. Inverse proportion means xy=k is constant: k=18(4)=72. When x=12, y=72/12=6.

26: F. Subtract exponents for the same base: x^(4−(−1))=x⁵ in the numerator, and y^(−2−3)=y⁻⁵=1/y⁵.

27: A. A strong negative linear association corresponds to a correlation coefficient close to −1, making −0.92 the most plausible choice.

28: G. Vertex form is y=a(x−2)²−5. Substituting the point (3,−2): −2=a(1)²−5, so a=3.

29: B. The hypotenuse is AB=√(9²+40²)=√1,681=41. Relative to angle A, the opposite side is BC=40, so sin A=40/41.

30: F. 480(0.35)=168 students take a language course. Of those, 168(0.25)=42 take French.

31: B. Squaring both sides: x+2=x², so x²−x−2=0, giving (x−2)(x+1)=0 and candidates x=2 or x=−1. Checking the original equation: √4=2 works, while √1=1 does not equal −1. Only x=2 is valid.

32: H. a₁₂=5+11(3)=5+33=38. Sum=12(5+38)/2=6(43)=258.

33: A. By the power-of-a-point relationship, tangent²=(external part)(whole length): 12²=8L, so 144=8L, giving L=18.

34: H. 88 and 112 are exactly one standard deviation below and above the mean of 100. By the empirical rule, about 68% of values fall within 1 standard deviation.

35: D. Multiply row by column: first row gives 2(4)+1(5)=13; second row gives 0(4)+3(5)=15. So AB=[[13],[15]].

36: J. V=lwh, so 360=12(5)h=60h, giving h=6.

37: B. Rewriting 3x−2y=8 in slope-intercept form: −2y=−3x+8, so y=(3/2)x−4, with slope 3/2. A perpendicular line has slope equal to the negative reciprocal: −2/3.

38: J. If applying f twice always returns the original input x, then f undoes itself, meaning f is its own inverse.

39: C. By the Law of Cosines: c²=7²+10²−2(7)(10)cos(60°)=49+100−70=79, so c=√79.

40: G. The plausible interval is 46%±3%, or 43% to 49%. Since 50% falls outside that interval, it is not consistent with the reported margin of error. The interval does not prove the exact population value or rule out bias by itself.

41: C. Let u=x²: u²−5u+4=(u−1)(u−4)=0, so u=1 or u=4, giving x²=1 or x²=4, and x=±1 or x=±2: four real zeros.

42: G. At x=3, the condition x≥3 applies, so use the second piece: p(3)=3²−4=9−4=5.

43: B. The square's side equals the circle's diameter: 2(5)=10. Square area=100; circle area=π(5)²=25π. The fraction is 25π/100=π/4.

44: J. The original rate is 120/8=15 parts per minute. The new rate is 15(1.20)=18 parts per minute. In 15 minutes: 18(15)=270 parts.

45: B. Substituting y=x−1 into the circle's equation: x²+(x−1)²=13, giving 2x²−2x−12=0, or x²−x−6=0. By Vieta's formulas, the sum of the roots is −(coefficient of x)=1, which is faster than solving for both roots individually.

Additional Practice Test 2

▸ Questions

▸ Answers and Explanations

TIMED MINI-SECTION