ACT Mathematics · Advanced Practice

20 Hard Questions

All Major Topics · Official Difficulty · Detailed Solutions
Pre-Algebra Algebra I & II Geometry Trigonometry Statistics Functions
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Each question reflects real ACT difficulty. Work through concepts first, then tackle the problems. Select an answer to receive immediate feedback.

01
Algebra & Functions
Questions 1–5

Key Concepts to Memorize

Quadratic Formula: x = (−b ± √(b²−4ac)) / 2a
Vertex of parabola: (−b/2a, f(−b/2a))
Discriminant: b²−4ac > 0 → 2 real roots; = 0 → 1; < 0 → no real roots
Absolute value: |x − a| = b → x = a+b or x = a−b
Function composition: (f∘g)(x) = f(g(x))
★ MEMORIZE Completing the square: x² + bx = (x + b/2)² − (b/2)²
Worked Example
If f(x) = 2x² − 3x + 1 and g(x) = x + 2, find f(g(1)).
g(1) = 3; f(3) = 2(9) − 9 + 1 = 10 ✓
02
Geometry & Coordinate Geometry
Questions 6–10

Key Concepts to Memorize

Circle: (x−h)² + (y−k)² = r² → center (h,k), radius r
Distance: d = √((x₂−x₁)² + (y₂−y₁)²)
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Similar triangles: corresponding sides proportional
Arc length = (θ/360°) × 2πr; Sector area = (θ/360°) × πr²
★ MEMORIZE 30-60-90 sides: 1 : √3 : 2  |  45-45-90 sides: 1 : 1 : √2
Worked Example
A circle has equation x² + y² − 6x + 4y = 12. Find the center and radius.
(x−3)² + (y+2)² = 25 → center (3,−2), radius 5 ✓
03
Trigonometry
Questions 11–14

Key Concepts to Memorize

sin²θ + cos²θ = 1 (Pythagorean identity)
tan θ = sin θ / cos θ
sin(A+B) = sinA cosB + cosA sinB
cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
Law of Sines: a/sinA = b/sinB = c/sinC
Law of Cosines: c² = a² + b² − 2ab·cosC
★ MEMORIZE SOH-CAH-TOA  |  ASTC (All Students Take Calculus)
Worked Example
If sinθ = 3/5 and θ is in Quadrant II, find cosθ.
cos²θ = 1 − 9/25 = 16/25; Q II → cosθ = −4/5 ✓
04
Statistics & Probability
Questions 15–17

Key Concepts to Memorize

Mean = Σx / n
Median = middle value (odd n) or avg of two middle (even n)
P(A and B) = P(A) × P(B) [independent events]
P(A or B) = P(A) + P(B) − P(A and B)
Combinations: C(n,r) = n! / (r!(n−r)!)
Permutations: P(n,r) = n! / (n−r)!
★ MEMORIZE Adding a constant k to all data: mean shifts by k, std dev unchanged
Worked Example
A bag has 4 red, 3 blue balls. P(two red in a row without replacement)?
4/7 × 3/6 = 12/42 = 2/7 ✓
05
Advanced Algebra & Number Theory
Questions 18–20

Key Concepts to Memorize

Exponential: aˣ · aʸ = aˣ⁺ʸ; (aˣ)ʸ = aˣʸ; a⁰ = 1
Logarithms: log_a(xy) = log_a(x) + log_a(y)
log_a(xⁿ) = n·log_a(x)
log_a(x) = log(x)/log(a) [change of base]
Arithmetic sequence: aₙ = a₁ + (n−1)d
Geometric sequence: aₙ = a₁ · rⁿ⁻¹
Sum of arithmetic series: S = n/2 · (a₁ + aₙ)
★ MEMORIZE Remainder theorem: f(a) = remainder when f(x) ÷ (x−a)
Worked Example
Solve: log₂(x+3) + log₂(x−1) = 5
log₂((x+3)(x−1)) = 5 → (x+3)(x−1) = 32 → x² + 2x − 35 = 0 → x = 5 ✓ (x = −7 rejected)
⬇ PRACTICE QUESTIONS ⬇
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