ACT Math Mastery — 20 Core Exam Questions

📚 Concept Review & Key Formulas

Algebra

Core Algebra & Functions

These topics appear in nearly every ACT Math section. Master these formulas cold.

Quadratic Formula: x = (−b ± √(b²−4ac)) / 2a Discriminant: D = b²−4ac · D > 0 → 2 real roots · D = 0 → 1 repeated root · D < 0 → no real roots Composition: (f∘g)(x) = f(g(x)) Absolute Value: |x − a| < b ↔ −b < x − a < b |x − a| > b ↔ x < a−b or x > a+b Rational Expr: Factor top & bottom, cancel common factors

🧠 Must Memorize
For ax² + bx + c = 0: vertex x = −b/(2a). The sign inside the quadratic formula is ±, never forget both roots.

Quick Example

Solve: x² − 5x + 6 = 0
Factor: (x−2)(x−3) = 0

→ x = 2 or x = 3

Geometry

Geometry & Coordinate Geometry

Circle: Area = πr², Circumference = 2πr Arc Length = (θ/360°)·2πr, Sector Area = (θ/360°)·πr² Similar Triangles: corresponding sides are proportional Distance = √((x₂−x₁)² + (y₂−y₁)²) Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2) Cylinder Volume = πr²h Cone Volume = (1/3)πr²h Parallel lines cut by transversal: alternate interior ∠s equal; co-interior ∠s supplementary

🧠 Trap Alert
Arc length uses the CENTRAL angle in degrees, not radians (unless told otherwise). Cone volume has the 1/3 factor — memorize it!

Trig

Trigonometry

SOH-CAH-TOA: sin θ = Opposite/Hypotenuse cos θ = Adjacent/Hypotenuse tan θ = Opposite/Adjacent Pythagorean Identity: sin²θ + cos²θ = 1 Law of Sines: a/sin A = b/sin B = c/sin C Unit Circle Key Values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3 sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1 sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

🧠 Memory Trick
"All Students Take Calculus" → Quadrants where sin, cos, tan are POSITIVE: All (I), Sin (II), Tan (III), Cos (IV).

Stats & Advanced

Statistics, Logs & Complex Numbers

Mean = Sum of values / Number of values Median = middle value (sort first!) Probability P(A|B) = P(A and B) / P(B) Standard Deviation: measures spread; larger SD = more spread Logarithms: log_b(xy) = log_b(x) + log_b(y) log_b(x/y) = log_b(x) − log_b(y) log_b(xⁿ) = n·log_b(x) log_b(b) = 1, log_b(1) = 0 Complex Numbers: i = √(−1), i² = −1, i³ = −i, i⁴ = 1 (cycle of 4) (a+bi)(c+di) = (ac−bd) + (ad+bc)i

🧠 Powers of i Cycle
i¹=i, i²=−1, i³=−i, i⁴=1, i⁵=i … Divide exponent by 4, use the remainder.

✏️ Exam Questions — Select One Answer Per Question

Algebra Quadratic / Discriminant

MEDIUM

How many real solutions does the equation 3x² − 6x + 3 = 0 have?

✅ Correct!

D = b²−4ac = (−6)²−4(3)(3) = 36−36 = 0
D = 0 → exactly ONE repeated real root: x = 6/(2·3) = 1

❌ Incorrect

D = b²−4ac = 36 − 36 = 0 → D = 0 means exactly one repeated root.
Answer: C — x = 1 (repeated)

Algebra System of Equations

MEDIUM

If 2x + y = 10 and x − y = 2, what is the value of x?

✅ Correct!

Add equations: (2x+y)+(x−y) = 10+2 → 3x = 12 → x = 4
Then y = x−2 = 2. Check: 2(4)+2 = 10 ✓

❌ Incorrect

Add the two equations to eliminate y:
3x = 12 → x = 4. Answer: B

Algebra Absolute Value Inequality

HARD

Which of the following represents the solution to |2x − 4| ≤ 6?

✅ Correct!

|2x−4| ≤ 6 → −6 ≤ 2x−4 ≤ 6
Add 4: −2 ≤ 2x ≤ 10
Divide by 2: −1 ≤ x ≤ 5

❌ Incorrect

For |expr| ≤ k, write −k ≤ expr ≤ k
−6 ≤ 2x−4 ≤ 6 → −1 ≤ x ≤ 5. Answer: D

Algebra Function Composition

HARD

If f(x) = 2x + 1 and g(x) = x² − 3, what is f(g(2))?

✅ Correct!

Step 1: g(2) = 2²−3 = 4−3 = 1
Step 2: f(g(2)) = f(1) = 2(1)+1 = 3

❌ Incorrect

Always evaluate INSIDE first: g(2)=1, then f(1)=3. Answer: C

Algebra Rational Expression

HARD

Simplify: (x² − 9) / (x² − x − 6), where x ≠ 3 and x ≠ −2

✅ Correct!

Numerator: x²−9 = (x−3)(x+3)
Denominator: x²−x−6 = (x−3)(x+2)
Cancel (x−3): result = (x+3)/(x+2)

❌ Incorrect

Factor both: (x−3)(x+3) / [(x−3)(x+2)] = (x+3)/(x+2). Answer: B

Algebra Exponential Equation

HARD

If 4^x = 32, what is the value of x?

✅ Correct!

4 = 2², so 4^x = 2^(2x). Also 32 = 2^5.
2^(2x) = 2^5 → 2x = 5 → x = 5/2

❌ Incorrect

Convert to same base: 4^x = 2^(2x) = 32 = 2^5 → 2x=5 → x=5/2. Answer: D

Geometry Circle — Arc & Sector

HARD

A circle has radius 9. A sector has a central angle of 80°. What is the area of the sector? (Use π ≈ 3.14)

✅ Correct!

Sector Area = (θ/360°)·πr²
= (80/360)·π·81
= (2/9)·3.14·81
= (2/9)·254.34 ≈ 56.52

❌ Incorrect

Formula: (80/360)·π·9² = (2/9)·81π ≈ 56.52. Answer: C

Geometry Similar Triangles

MEDIUM

Two similar triangles have corresponding sides in the ratio 3:5. If the shorter triangle has a side of 12, what is the corresponding side of the larger triangle?

✅ Correct!

Ratio 3:5, shorter side = 12
3/5 = 12/x → x = 12·5/3 = 60/3 = 20

❌ Incorrect

Set up proportion: 3/5 = 12/x → x = 20. Answer: B

Geometry 3D Volume — Cylinder + Cone

VERY HARD

A solid consists of a cylinder topped with a cone. Both have radius 3. The cylinder has height 8 and the cone has height 4. What is the total volume? (Leave answer in terms of π)

✅ Correct!

Cylinder: πr²h = π·9·8 = 72π
Cone: (1/3)πr²h = (1/3)·π·9·4 = 12π
Total = 72π + 12π = 84π

❌ Incorrect

Cylinder πr²h = 72π. Cone (1/3)πr²h = 12π. Total = 84π. Answer: A

Geometry Coordinate — Distance

MEDIUM

What is the distance between the points (1, 2) and (7, 10)?

✅ Correct!

d = √((7−1)²+(10−2)²) = √(36+64) = √100 = 10

❌ Incorrect

√(6²+8²) = √(36+64) = √100 = 10. Answer: C

Geometry Parallel Lines & Angles

MEDIUM

Two parallel lines are cut by a transversal. One of the co-interior (same-side interior) angles measures 65°. What is the measure of the other co-interior angle?

✅ Correct!

Co-interior (consecutive interior) angles are SUPPLEMENTARY:
65° + x = 180° → x = 115°

❌ Incorrect

Co-interior angles sum to 180°: 180−65 = 115°. Answer: B

Trigonometry SOH-CAH-TOA / Elevation

HARD

A surveyor stands 50 meters from the base of a building. The angle of elevation to the top of the building is 60°. How tall is the building? (Use tan 60° = √3)

✅ Correct!

tan(angle) = Opposite/Adjacent
tan 60° = height/50
height = 50·tan 60° = 50√3 meters

❌ Incorrect

tan 60° = h/50 → h = 50·√3. Answer: D

Trigonometry Law of Sines

VERY HARD

In triangle ABC, angle A = 30°, angle B = 45°, and side a (opposite angle A) = 10. Using the Law of Sines, find side b (opposite angle B). (Use sin 30° = 0.5, sin 45° = √2/2 ≈ 0.707)

✅ Correct!

Law of Sines: a/sin A = b/sin B
10/sin30° = b/sin45°
10/0.5 = b/(√2/2)
20 = b/(√2/2)
b = 20·(√2/2) = 10√2

❌ Incorrect

a/sinA = b/sinB: 10/0.5 = b/(√2/2) → b = 20·(√2/2) = 10√2. Answer: A

Trigonometry Unit Circle — Exact Values

HARD

What is the exact value of sin(225°)?

✅ Correct!

225° = 180° + 45° → Quadrant III (sin is NEGATIVE)
Reference angle = 45°, sin 45° = √2/2
sin 225° = −√2/2

❌ Incorrect

225° is in Q3 (both sin & cos negative). Reference = 45°. sin 225° = −√2/2. Answer: C

Trigonometry Pythagorean Identity

HARD

If sin θ = 3/5 and θ is in Quadrant II, what is cos θ?

✅ Correct!

sin²θ + cos²θ = 1
(3/5)² + cos²θ = 1
9/25 + cos²θ = 1
cos²θ = 16/25
cos θ = ±4/5
Q II: cos is NEGATIVE → cos θ = −4/5

❌ Incorrect

cos²θ = 1−9/25 = 16/25 → cosθ = ±4/5. In QII cos is negative → −4/5. Answer: E

Statistics Mean, Median, Mode

MEDIUM

The data set is: {4, 7, 7, 9, 13, 15, 15, 15, 20}. Which of the following is true?

✅ Correct!

Sorted: {4,7,7,9,13,15,15,15,20} (9 values)
Mean = (4+7+7+9+13+15+15+15+20)/9 = 105/9 ≈ 11.67
Median = 5th value = 13
Mode = 15 (appears 3 times)
Mean(≈11.67) < Median(13) ✓ — E is TRUE

❌ Incorrect

Sorted: {4,7,7,9,13,15,15,15,20}
Mean = 105/9 ≈ 11.67, Median = 13, Mode = 15
Mean(11.67) < Median(13) → Answer: E

Statistics Conditional Probability

VERY HARD

A bag has 4 red and 6 blue marbles. Two marbles are drawn WITHOUT replacement. What is the probability that the second marble is red, given the first marble drawn was red?

✅ Correct!

After 1 red drawn: 3 red remain out of 9 total
P(2nd red | 1st red) = 3/9 = 1/3
Note: C (3/9) and D (1/3) are equivalent. C = 3/9 is the unreduced form shown.

❌ Incorrect

After removing 1 red: 3 red left, 9 total. P = 3/9. Answer: C

Statistics Standard Deviation Concept

HARD

Two classes take the same exam. Class A has scores: {70, 72, 74, 76, 78}. Class B has scores: {50, 60, 74, 88, 98}. Both have the same mean of 74. Which statement is true?

✅ Correct!

Class A deviations from mean 74: {−4,−2,0,+2,+4} → small spread
Class B deviations from mean 74: {−24,−14,0,+14,+24} → large spread
Larger deviations = larger SD → Class B has higher SD

❌ Incorrect

Class B scores are far more spread from the mean (74). SD measures spread → Class B's SD is larger. Answer: D

Advanced Logarithm Properties

VERY HARD

Which expression is equivalent to log₂(8x³)?

✅ Correct!

log₂(8x³) = log₂(8) + log₂(x³) [product rule]
= log₂(2³) + 3·log₂(x) [power rule]
= 3 + 3log₂(x)

❌ Incorrect

log(ab)=log a+log b; log(xⁿ)=n·log x; log₂8=3.
Result: 3 + 3log₂x. Answer: A

Advanced Complex Numbers

VERY HARD

What is the simplified form of (3 + 2i)(1 − i)?

✅ Correct!

(3+2i)(1−i) = 3·1 + 3·(−i) + 2i·1 + 2i·(−i)
= 3 − 3i + 2i − 2i²
= 3 − i − 2(−1) [since i²=−1]
= 3 − i + 2
= 5 − i

❌ Incorrect

FOIL: 3−3i+2i−2i² = 3−i−2(−1) = 3−i+2 = 5−i. Answer: B

0/20

Total Score

—

Correct

Wrong

Accuracy

Answer Key

Full Solutions & Explanations

Study these explanations to master every concept

Question 01 — Discriminant

✅ Answer: C — Exactly 1 real solution (repeated)

For 3x²−6x+3=0: D = b²−4ac = (−6)²−4(3)(3) = 36−36 = 0. When D = 0, the quadratic has exactly one repeated real root. Solving: x = 6/(2·3) = 1.

Question 02 — System of Equations

✅ Answer: B — x = 4

Add the equations: 3x = 12 → x = 4. Substitute back: y = x−2 = 2. Verify: 2(4)+2 = 10 ✓.

Question 03 — Absolute Value

✅ Answer: D — −1 ≤ x ≤ 5

|2x−4| ≤ 6 → −6 ≤ 2x−4 ≤ 6 → add 4: −2 ≤ 2x ≤ 10 → divide by 2: −1 ≤ x ≤ 5.

Question 04 — Function Composition

✅ Answer: C — 3

Always evaluate inner function first: g(2) = 4−3 = 1, then f(1) = 2(1)+1 = 3.

Question 05 — Rational Expressions

✅ Answer: B — (x+3)/(x+2)

Factor: (x−3)(x+3) / (x−3)(x+2). Cancel common factor (x−3): result = (x+3)/(x+2).

Question 06 — Exponential Equations

✅ Answer: D — x = 5/2

Convert to the same base 2: 4^x = (2²)^x = 2^(2x) and 32 = 2^5. So 2x = 5 → x = 5/2.

Question 07 — Circle: Sector Area

✅ Answer: C — 56.52

Formula: (θ/360)·πr². = (80/360)·π·81 = (2/9)·81π ≈ (2/9)·254.34 ≈ 56.52.

Question 08 — Similar Triangles

✅ Answer: B — 20

Ratio 3:5 → proportion 3/5 = 12/x → cross multiply: 3x = 60 → x = 20.

Question 09 — 3D Volume

✅ Answer: A — 84π

Cylinder: πr²h = π(9)(8) = 72π. Cone: (1/3)πr²h = (1/3)π(9)(4) = 12π. Total = 84π.

Question 10 — Distance Formula

✅ Answer: C — 10

d = √((7−1)²+(10−2)²) = √(36+64) = √100 = 10. (This is a 6-8-10 right triangle.)

Question 11 — Parallel Lines

✅ Answer: B — 115°

Co-interior (same-side interior) angles formed by parallel lines and a transversal are supplementary: they add to 180°. 180−65 = 115°.

Question 12 — Angle of Elevation

✅ Answer: D — 50√3 m

tan(angle) = opposite/adjacent → tan 60° = h/50 → h = 50·√3.

Question 13 — Law of Sines

✅ Answer: A — 10√2

a/sinA = b/sinB → 10/0.5 = b/(√2/2) → 20 = b/(√2/2) → b = 20·(√2/2) = 10√2.

Question 14 — Unit Circle

✅ Answer: C — −√2/2

225° = 180°+45° → Quadrant III. In Q3, sine is negative. Reference angle = 45°, sin45° = √2/2 → sin225° = −√2/2.

Question 15 — Pythagorean Identity

✅ Answer: E — −4/5

cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25 → cosθ = ±4/5. In Quadrant II, cosine is negative → cosθ = −4/5.

Question 16 — Statistics

✅ Answer: E — Mean < Median

Data: {4,7,7,9,13,15,15,15,20}. Mean = 105/9 ≈ 11.67. Median = 5th value = 13. Mode = 15. Since 11.67 < 13, Mean < Median is TRUE.

Question 17 — Conditional Probability

✅ Answer: C — 3/9

After drawing 1 red marble: 3 red remain out of 9 total. P = 3/9 = 1/3. Without replacement changes the sample space.

Question 18 — Standard Deviation

✅ Answer: D — Class B has higher SD

Class A deviations from 74: ±4, ±2, 0 (small). Class B deviations: ±24, ±14, 0 (large). Larger spread = larger standard deviation. Class B's scores are much more dispersed.

Question 19 — Logarithms

✅ Answer: A — 3 + 3log₂(x)

log₂(8x³) = log₂(8) + log₂(x³) = 3 + 3log₂(x). (Since log₂(8) = log₂(2³) = 3, and log₂(x³) = 3log₂(x) by the power rule.)

Question 20 — Complex Numbers

✅ Answer: B — 5 − i

FOIL: (3+2i)(1−i) = 3−3i+2i−2i² = 3−i−2(−1) = 3−i+2 = 5−i. Key: i² = −1.