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ACT Math

ACT Math Practice Test: Practice Test 1

Practice Test 1 for ACT Math: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

Given f(x)=x2−2x+1f(x)=x^2-2x+1f(x)=x2−2x+1, what is the value of f(3)f(3)f(3)?

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Question 1

Given f(x)=x2−2x+1f(x)=x^2-2x+1f(x)=x2−2x+1, what is the value of f(3)f(3)f(3)?

  1. 222
  2. 111
  3. 444 (correct answer)
  4. 999

Explanation: To find f(3) where f(x) = x² - 2x + 1, substitute x = 3 into the function. Calculate: f(3) = (3)² - 2(3) + 1 = 9 - 6 + 1 = 4. Notice that f(x) = x² - 2x + 1 = (x - 1)², so f(3) = (3 - 1)² = 4.

Question 2

A temperature conversion is being simplified. If 5x+2=3x+185x + 2 = 3x + 185x+2=3x+18, what is the value of xxx?

  1. 6
  2. 8 (correct answer)
  3. 10
  4. -8

Explanation: This is a linear equation 5x + 2 = 3x + 18 simplifying a temperature conversion, where x is the variable to solve. Subtract 3x from both sides: 2x + 2 = 18. Subtract 2 from both sides: 2x = 16. Divide by 2: x = 8. Therefore, the value of x is 8, matching choice B. Choice C (10) might result from subtracting 2x instead, leading to arithmetic errors.

Question 3

In the diagram, lines mmm and nnn are parallel (⇒). A transversal ttt intersects them. At the intersection with line nnn, the interior angle on the right side of the transversal is labeled 30∘30^\circ30∘. The angle labeled xxx is the alternate interior angle at line mmm.

m  ⇒  ───────────────
        x  \
            \
             \ t
              \
n  ⇒  ───────────────
              30°

If lines mmm and nnn are parallel, what is the measure of angle xxx?

  1. 150∘150^\circ150∘
  2. 30∘30^\circ30∘ (correct answer)
  3. 60∘60^\circ60∘
  4. 90∘90^\circ90∘

Explanation: The angles x and 30° are alternate interior angles formed by parallel lines m and n with transversal t. When two parallel lines are cut by a transversal, alternate interior angles are always equal. Therefore, x = 30°. Choice A (150°) incorrectly treats these as supplementary angles, which would be the relationship for same-side interior angles rather than alternate interior angles.

Question 4

A pizza shop offers 6 toppings. A customer chooses exactly 2 different toppings. Since the order of toppings does not matter, how many different 2-topping pizzas are possible?

  1. 30
  2. 15 (correct answer)
  3. 12
  4. 36

Explanation: Since the order of toppings doesn't matter (pepperoni-mushroom is the same as mushroom-pepperoni), this is a combination problem. We calculate C(6,2) = 6!/(2!×4!) = (6×5)/(2×1) = 30/2 = 15. We choose 2 toppings from 6 available options. Choice A (30) represents the permutation P(6,2) where order would matter.

Question 5

A school principal wants to know what percentage of students at the school eat breakfast. She stands at the cafeteria entrance one morning and asks the first 120 students who walk in whether they ate breakfast. Which characteristic makes this sample biased?

  1. It excludes students who do not enter the cafeteria that morning (correct answer)
  2. It uses a sample size of 120, which is always too small
  3. It asks a yes/no question, which cannot measure breakfast habits
  4. It is conducted in the morning, when students are more honest

Explanation: This sample is biased because it excludes students who do not enter the cafeteria that morning. The principal is only surveying students who come to the cafeteria, which likely overrepresents students who eat breakfast (since they may be coming to get breakfast) and completely misses students who eat at home or skip breakfast entirely. This is a classic example of selection bias where the sampling location systematically excludes certain groups. Choice B incorrectly suggests 120 is always too small - sample size adequacy depends on the population size and desired precision.

Question 6

What is the simplified form of 9b−4(2−b)9b - 4(2 - b)9b−4(2−b)?

  1. 9b−8+4b9b - 8 + 4b9b−8+4b
  2. 13b+813b + 813b+8
  3. 5b−85b - 85b−8
  4. 13b−813b - 813b−8 (correct answer)

Explanation: To simplify this expression, we need to apply the distributive property. First, distribute the -4 to both terms in the parentheses: -4(2 - b) = -8 + 4b. The expression becomes 9b - 8 + 4b. Combining like terms: (9b + 4b) + (-8) = 13b - 8. Choice B incorrectly has +8 instead of -8 as the constant term.

Question 7

When dividing by a complex number, you often use its conjugate. What is the complex conjugate of −6+13i-6+13i−6+13i?

  1. 6−13i6-13i6−13i
  2. −6−13i-6-13i−6−13i (correct answer)
  3. −6+13i-6+13i−6+13i
  4. 6+13i6+13i6+13i

Explanation: The complex conjugate of a complex number a+bia+bia+bi is a−bia-bia−bi, where we change the sign of the imaginary part only. For −6+13i-6+13i−6+13i, the conjugate is −6−13i-6-13i−6−13i. The real part stays the same at −6-6−6, while the imaginary part changes from +13i+13i+13i to −13i-13i−13i.

Question 8

A regular polygon has an exterior angle of 45∘45^\circ45∘ at each vertex. How many sides does the polygon have?

  1. 6
  2. 7
  3. 8 (correct answer)
  4. 9

Explanation: The question asks for the number of sides in a regular polygon with each exterior angle measuring 45°. The sum of exterior angles for any polygon is 360°, and for a regular polygon, each exterior angle is 360/n °. Set 360/n = 45, then n = 360/45 = 8. Interior and exterior angles sum to 180° at each vertex, but here we use the exterior formula directly. Choice A of 6 might confuse with 60° exterior for a hexagon.

Question 9

What is the slope of the line through points (1, 2) and (4, 8)?

  1. 3
  2. 2 (correct answer)
  3. 1
  4. 4

Explanation: We need to find the slope of the line through points (1, 2) and (4, 8). Using the slope formula m = (y₂ - y₁)/(x₂ - x₁), we substitute: m = (8 - 2)/(4 - 1) = 6/3 = 2. The slope is 2, which matches choice B. This means the line rises 2 units for every 1 unit moved to the right. Choice A would result from reversing the numerator and denominator in the slope calculation.

Question 10

A cube has a surface area of 54 square units. What is the length of each side of the cube?

  1. 12 units
  2. 6 units
  3. 3 units (correct answer)
  4. 9 units

Explanation: We need to find the side length of a cube with surface area 54 square units. The surface area formula for a cube is SA = 6s² where s is the side length. Setting up the equation: 6s² = 54, so s² = 9, therefore s = 3 units. Each face of the cube has area s² = 9 square units.

Question 11

Solve for xxx: 8x−6=2(x+3)8x - 6 = 2(x + 3)8x−6=2(x+3)

  1. 4
  2. 3
  3. 1
  4. 2 (correct answer)

Explanation: This is a linear equation with parentheses on one side. To solve 8x−6=2(x+3)8x - 6 = 2(x + 3)8x−6=2(x+3), first distribute the right side: 8x−6=2x+68x - 6 = 2x + 68x−6=2x+6. Subtract 2x from both sides: 6x−6=66x - 6 = 66x−6=6. Add 6 to both sides: 6x=126x = 126x=12. Divide by 6: x=2x = 2x=2. The solution is x=2x = 2x=2.

Question 12

What is tan⁡(0∘)\tan(0^\circ)tan(0∘)?

  1. √3
  2. 1
  3. 0 (correct answer)
  4. √3/2

Explanation: tan(0°) is a standard unit circle value. Using SOH-CAH-TOA, tangent represents the opposite side over the adjacent side. For the angle 0°, tan(0°) = 0. Choice B (1) is actually tan(45°), not tan(0°).

Question 13

What is the range of the data set [11, 14, 18, 21, 25]?

  1. 14 (correct answer)
  2. 11
  3. 25
  4. 18

Explanation: The range is calculated as the difference between the maximum and minimum values. First, identify the maximum value (25) and minimum value (11). Range = maximum - minimum = 25 - 11 = 14. The range indicates the total span of values from the smallest to the largest data point.

Question 14

In the diagram, lines mmm and nnn are parallel and transversal ttt intersects them. Angle AAA is an interior angle below line mmm and to the right of ttt, and angle BBB is an interior angle above line nnn and to the left of ttt.

Which describes the relationship between angles AAA and BBB?

  1. Corresponding angles
  2. Alternate interior angles (correct answer)
  3. Vertical angles
  4. Complementary angles

Explanation: Angles A and B are alternate interior angles formed by the transversal crossing the parallel lines. Alternate interior angles are congruent when the lines are parallel, meaning they have equal measures. This relationship arises from the properties of parallel lines and transversals. The description places them on alternate sides of the transversal within the interior region. Choice A might be selected by mistaking their positions for corresponding angles.

Question 15

Which polynomial is equivalent to 3(x2−2x+1)3(x^2 - 2x + 1)3(x2−2x+1)?

  1. 3x2+6x+33x^2 + 6x + 33x2+6x+3
  2. 3x2−2x+13x^2 - 2x + 13x2−2x+1
  3. 3x2−6x+33x^2 - 6x + 33x2−6x+3 (correct answer)
  4. 3x2−2x+33x^2 - 2x + 33x2−2x+3

Explanation: We need to expand 3(x² - 2x + 1) using the distributive property. Multiplying each term inside the parentheses by 3: 3(x² - 2x + 1) = 3x² - 6x + 3. This matches choice A exactly.

Question 16

In △ABC\triangle ABC△ABC and △DEF\triangle DEF△DEF, ∠A≅∠D\angle A \cong \angle D∠A≅∠D and ∠B≅∠E\angle B \cong \angle E∠B≅∠E. Also, AB=10AB=10AB=10 and DE=15DE=15DE=15. What is the scale factor from △ABC\triangle ABC△ABC to △DEF\triangle DEF△DEF?

  1. 23\frac{2}{3}32​
  2. 32\frac{3}{2}23​ (correct answer)
  3. 53\frac{5}{3}35​
  4. 35\frac{3}{5}53​

Explanation: The triangles are similar by AA similarity criterion since two pairs of corresponding angles are congruent. When triangles are similar, all corresponding sides are proportional with the same scale factor. The scale factor from triangle ABC to triangle DEF equals DE/AB = 15/10 = 3/2. This means each side of triangle DEF is 3/2 times the corresponding side of triangle ABC.

Question 17

Which vector represents the displacement from point A(2,−1)A(2,-1)A(2,−1) to point B(−3,5)B(-3,5)B(−3,5)?

  1. ⟨−5,6⟩\langle -5,6\rangle⟨−5,6⟩ (correct answer)
  2. ⟨5,−6⟩\langle 5,-6\rangle⟨5,−6⟩
  3. ⟨−1,4⟩\langle -1,4\rangle⟨−1,4⟩
  4. ⟨1,−4⟩\langle 1,-4\rangle⟨1,−4⟩

Explanation: We need the displacement vector from A(2, -1) to B(-3, 5). The displacement formula is: vector AB equals ⟨x₂ minus x₁, y₂ minus y₁⟩. Calculating: vector AB equals ⟨-3 minus 2, 5 minus (-1)⟩ equals ⟨-5, 6⟩. The vector points from the starting point A to the ending point B.

Question 18

What is the probability of drawing a spade or a club from a standard deck of cards?

  1. 13/52
  2. 1/4
  3. 1/2 (correct answer)
  4. 1/13

Explanation: A standard deck has 52 cards with 13 spades and 13 clubs, giving us 26 favorable outcomes total. Since drawing a spade and drawing a club are mutually exclusive events, P(spade or club) = P(spade) + P(club) = 13/52 + 13/52 = 26/52 = 1/2. Choice B (1/4) represents the probability of drawing cards from just one suit.

Question 19

A scatterplot shows points representing (x, y) pairs for four measurements. The plotted points are (1,4)(1, 4)(1,4), (2,6)(2, 6)(2,6), (3,5)(3, 5)(3,5), and (4,7)(4, 7)(4,7).

Which point on the graph represents the condition x=3x=3x=3?

  1. (1,4)(1,4)(1,4)
  2. (2,6)(2,6)(2,6)
  3. (3,5)(3,5)(3,5) (correct answer)
  4. (4,7)(4,7)(4,7)

Explanation: The question asks which point represents the condition x=3x=3x=3. Looking at the four plotted points (1,4)(1,4)(1,4), (2,6)(2,6)(2,6), (3,5)(3,5)(3,5), and (4,7)(4,7)(4,7), the point where x=3x=3x=3 is (3,5)(3,5)(3,5).

Question 20

A line represents a constant-rate change in elevation. The line passes through points (−2,3)(-2,3)(−2,3) and (4,0)(4,0)(4,0). What is the slope of the line through points (−2,3)(-2,3)(−2,3) and (4,0)(4,0)(4,0)?

  1. 12\dfrac{1}{2}21​
  2. −12-\dfrac{1}{2}−21​ (correct answer)
  3. −2-2−2
  4. 222

Explanation: We need to find the slope of the line through points (-2,3) and (4,0). Using the slope formula m = (y₂-y₁)/(x₂-x₁), we calculate m = (0-3)/(4-(-2)) = -3/6 = -1/2. The slope represents the rate of elevation change, which is -1/2 units per unit of horizontal distance. Choice A shows the positive version of the correct answer, while choices C and D show integer values that would result from calculation errors.

Question 21

Triangles KLM and NOP are similar with a scale factor of 4:34:34:3. If KL=12KL = 12KL=12, what is NONONO?

  1. 6
  2. 16
  3. 8
  4. 9 (correct answer)

Explanation: Triangles KLM and NOP are similar with a scale factor of 4:3 from KLM to NOP. This means sides of triangle NOP are 3/4 the length of corresponding sides in triangle KLM. The sides KL and NO correspond to each other, with KL = 12. Setting up the proportion: NO/KL = 3/4, we get NO/12 = 3/4. Cross-multiplying: 4 × NO = 3 × 12, so 4 × NO = 36, giving NO = 9.

Question 22

Two triangles have side lengths AB=6AB=6AB=6, BC=8BC=8BC=8, AC=10AC=10AC=10 and DE=9DE=9DE=9, EF=12EF=12EF=12, DF=15DF=15DF=15. Which statement is true about their relationship?

  1. They are congruent by SSS.
  2. They are similar by SSS, with scale factor 32\frac{3}{2}23​ from △ABC\triangle ABC△ABC to △DEF\triangle DEF△DEF. (correct answer)
  3. They are similar by SAS, with scale factor 23\frac{2}{3}32​ from △ABC\triangle ABC△ABC to △DEF\triangle DEF△DEF.
  4. They are not similar because only two sides are proportional.

Explanation: The triangles are similar by SSS similarity criterion because all three pairs of corresponding sides are proportional. Checking the ratios: DE/AB = 9/6 = 3/2, EF/BC = 12/8 = 3/2, and DF/AC = 15/10 = 3/2. Since all three ratios equal 3/2, the triangles are similar with scale factor 3/2 from triangle ABC to triangle DEF. This means each side of triangle DEF is 3/2 times the corresponding side of triangle ABC.

Question 23

What is f(0)f(0)f(0) for the piecewise function f(x)={2x−5if x<−1x2+1if x≥−1f(x) = \begin{cases} 2x - 5 & \text{if } x < -1 \\ x^2 + 1 & \text{if } x \geq -1 \end{cases}f(x)={2x−5x2+1​if x<−1if x≥−1​?

  1. 1 (correct answer)
  2. 2
  3. 3
  4. 0

Explanation: For x = 0, we check the intervals: Is 0 < -1? No. Is 0 ≥ -1? Yes. Since x = 0 satisfies the second condition, we use f(x) = x² + 1. Substituting x = 0: f(0) = (0)² + 1 = 0 + 1 = 1. Choice B would result from using the first piece incorrectly.

Question 24

The function B=800−5nB = 800 - 5nB=800−5n models the number of books in a library, where BBB is the number of books and nnn is the number of days books are borrowed. How many books are left after 50 days?

  1. 750
  2. 550 (correct answer)
  3. 1000
  4. 5500

Explanation: This library model B=800−5nB = 800 - 5nB=800−5n shows books remaining B after n days of borrowing. The y-intercept 800 is the initial number of books, and slope -5 means 5 books are borrowed (removed) per day. After 50 days: B=800−5(50)=800−250=550B = 800 - 5(50) = 800 - 250 = 550B=800−5(50)=800−250=550 books remain. Choice A adds instead of subtracts.

Question 25

A circle has area 16π16\pi16π. What is the radius of the circle?

  1. 888
  2. 444 (correct answer)
  3. 222
  4. 161616

Explanation: We need to find the radius when the area is 16π. Using A = πr², we have 16π = πr², so r² = 16, giving r = 4. Choice A incorrectly doubles the radius, while choice C gives half the radius, and choice D squares the radius.