Official ACT Prep · Advanced Level

ACT Math
Practice Test

All Topics · Hard & Very Hard
Full Solutions & Concept Review

20
Questions
35
Minutes
6
Topics

Concepts · Key Formulas · 20 Questions · Full Solutions

Step 1 — Concept Review

Core Concepts & Key Formulas

1. Pre-Algebra & Number Properties
  • LCM(a,b) = |a·b| / GCD(a,b)
  • For consecutive integers n, n+1: their product is always even
  • Remainder theorem: if f(x) is divided by (x−k), remainder = f(k)
  • Prime factorization: every integer > 1 is a unique product of primes
  • Even × Even = Even; Odd × Odd = Odd; Even + Odd = Odd
% change = (New − Old) / Old × 100 distance = rate × time work = rate × time
📝 Quick Example
If 3x ≡ 12 (mod 5), find x mod 5.
3x = 12 → 3x mod 5 = 12 mod 5 = 2 → 3x ≡ 2 (mod 5) → x ≡ 4 (mod 5)
Answer: x ≡ 4 (mod 5)
2. Algebra & Functions
  • Quadratic formula: x = (−b ± √(b²−4ac)) / 2a
  • Discriminant Δ = b²−4ac: Δ>0 two real roots; Δ=0 one root; Δ<0 no real roots
  • Sum of roots = −b/a; Product of roots = c/a (Vieta's formulas)
  • Absolute value: |x−k| < d ⟺ k−d < x < k+d
  • Exponential: a^m · a^n = a^(m+n); (a^m)^n = a^(mn)
(a+b)² = a² + 2ab + b² (a−b)² = a² − 2ab + b² a² − b² = (a+b)(a−b) a³ − b³ = (a−b)(a²+ab+b²)
📝 Quick Example
If f(x) = 2x² − 3x + 1, find f(f(1)).
f(1) = 2(1) − 3(1) + 1 = 0 → f(0) = 2(0) − 0 + 1 = 1
Answer: f(f(1)) = 1
3. Coordinate Geometry
  • Distance: d = √[(x₂−x₁)² + (y₂−y₁)²]
  • Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
  • Slope: m = (y₂−y₁)/(x₂−x₁); perpendicular slope = −1/m
  • Circle: (x−h)²+(y−k)²= r² → center (h,k), radius r
  • Parabola vertex form: y = a(x−h)²+k → vertex (h,k)
Slope-intercept: y = mx + b Point-slope: y − y₁ = m(x − x₁) Standard: Ax + By = C
📝 Quick Example
Circle x²+y²−4x+6y = 3. Find center and radius.
Complete square: (x−2)²+(y+3)² = 16 → center (2,−3), r = 4
Answer: center (2,−3), r = 4
4. Plane Geometry & Solid Geometry
  • Triangle: Area = ½bh; sum of angles = 180°
  • Similar triangles: corresponding sides proportional; angles equal
  • 30-60-90: sides 1 : √3 : 2 (short leg : long leg : hypotenuse)
  • 45-45-90: sides 1 : 1 : √2
  • Circle: C = 2πr; A = πr²; arc length = rθ (θ in radians)
  • Cylinder: V = πr²h; Sphere: V = (4/3)πr³
  • Inscribed angle = ½ central angle subtending same arc
Hero's formula: A = √[s(s-a)(s-b)(s-c)], s=(a+b+c)/2 Sector area = (θ/2π)πr² = ½r²θ
5. Trigonometry
  • SOH-CAH-TOA: sin=opp/hyp; cos=adj/hyp; tan=opp/adj
  • sin²θ + cos²θ = 1; tan²θ + 1 = sec²θ
  • sin(A+B) = sinA·cosB + cosA·sinB
  • cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ
  • Law of Sines: a/sinA = b/sinB = c/sinC
  • Law of Cosines: c² = a²+b²−2ab·cosC
  • Reference angles: sin(π−θ)=sinθ; sin(π+θ)=−sinθ
Unit circle key: sin30°=½; sin45°=√2/2; sin60°=√3/2 Amplitude A, Period T=2π/B for y=A·sin(Bx+C)
6. Statistics & Probability
  • Mean = (sum of values) / (number of values)
  • Median = middle value when sorted
  • Standard deviation measures spread from mean
  • P(A or B) = P(A)+P(B)−P(A and B)
  • P(A and B) = P(A)×P(B) if independent
  • Combinations: C(n,r) = n! / [r!(n−r)!]
  • Permutations: P(n,r) = n! / (n−r)!
Expected value E(X) = Σ[x · P(x)] Weighted mean = Σ(value × weight) / Σweight
📝 Quick Example
In a group of 6 people, how many ways can a president and VP be chosen?
P(6,2) = 6!/(6−2)! = 6×5 = 30
Answer: 30 ways
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