ACT Math Elite Practice | 20 Hard Questions

1

Pre-Algebra & Number Theory

LCM(a,b) = (a × b) / GCD(a,b) Remainder Theorem: a mod m = r Percent change = (new − old)/old × 100%

If n² ends in 6, then n ends in 4 or 6. If n² ends in 5, n ends in 5.

For consecutive integers: sum of n terms starting at a = n(2a + n − 1)/2

Q: What is the LCM of 12 and 18?

GCD(12,18) = 6 → LCM = (12×18)/6 = 36 ✓

2

Algebra & Linear Equations

Slope: m = (y₂−y₁)/(x₂−x₁) Point-slope: y − y₁ = m(x − x₁) Systems: substitution or elimination Absolute value: |x−a| = b → x = a±b

Two lines are parallel if slopes are equal; perpendicular if m₁ × m₂ = −1

Direct variation: y = kx. Inverse variation: y = k/x

Q: Line through (2,3) and (6,11). Find slope.

m = (11−3)/(6−2) = 8/4 = 2 ✓

3

Quadratics & Polynomials

Quadratic formula: x = [−b ± √(b²−4ac)] / 2a Discriminant: b²−4ac → (+) 2 real, (0) 1 real, (−) no real roots Vertex: x = −b/(2a), y = f(−b/2a) Vieta's: x₁+x₂ = −b/a, x₁·x₂ = c/a

If roots are r and s: quadratic is (x−r)(x−s) = x²−(r+s)x+rs

Completing the square: x² + bx = (x + b/2)² − (b/2)²

Q: Sum of roots of 2x²−6x+4=0?

Sum = −(−6)/2 = 3 ✓

4

Functions & Graphs

f(g(x)): apply g first, then f Domain: all valid inputs; Range: all outputs Even: f(−x) = f(x) [symmetric y-axis] Odd: f(−x) = −f(x) [symmetric origin]

Vertical line test: if any vertical line crosses graph twice → NOT a function

For f(x) = √(x−a): domain is x ≥ a. For 1/(x−a): domain is x ≠ a

Q: If f(x) = 2x+1, find f(f(3)).

f(3) = 7, f(7) = 15 ✓

5

Geometry: Lines, Angles & Polygons

Interior angle sum of n-gon = (n−2) × 180° Each interior angle of regular n-gon = (n−2)×180°/n Exterior angles always sum to 360°

Parallel lines cut by transversal: alternate interior angles = equal; co-interior = 180°

Triangle inequality: each side < sum of other two sides

Q: Each interior angle of a regular hexagon?

(6−2)×180/6 = 720/6 = 120° ✓

6

Circles & Area/Volume

Circle: A = πr², C = 2πr Arc length = (θ/360) × 2πr Sector area = (θ/360) × πr² Cylinder: V = πr²h; Sphere: V = (4/3)πr³ Cone: V = (1/3)πr²h

Inscribed angle = half the central angle subtending the same arc

Tangent to circle is perpendicular to the radius at the point of tangency

Q: Arc length of 60° sector with r=6?

(60/360)×2π×6 = 2π ✓

7

Coordinate Geometry

Distance: d = √[(x₂−x₁)² + (y₂−y₁)²] Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2) Circle eq: (x−h)² + (y−k)² = r² Parabola: y = a(x−h)² + k [vertex (h,k)]

Reflection over y=x: swap (x,y) → (y,x)

Slope of perpendicular bisector = −1/m of segment

8

Trigonometry

SOH-CAH-TOA: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj sin²θ + cos²θ = 1 Law of Sines: a/sin A = b/sin B = c/sin C Law of Cosines: c² = a² + b² − 2ab·cos C

Special angles: sin 30°=½, cos 30°=√3/2, tan 45°=1, sin 60°=√3/2

Radian ↔ Degree: π rad = 180°; multiply by 180/π to convert to degrees

Q: In right triangle, opp=3, hyp=5. Find sin θ.

sin θ = 3/5 = 0.6 ✓

9

Statistics & Probability

Mean = Σx / n Median = middle value (sorted) Mode = most frequent value P(A and B) = P(A)·P(B) [independent events] P(A or B) = P(A)+P(B)−P(A and B)

Counting: Permutation P(n,r) = n!/(n−r)!; Combination C(n,r) = n!/[r!(n−r)!]

Expected value = Σ(value × probability)

10

Logarithms & Exponents

aᵐ · aⁿ = aᵐ⁺ⁿ; aᵐ/aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ 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 Change of base: log_b a = ln a / ln b

log_b(b) = 1 always; log_b(1) = 0 always; b^(log_b x) = x

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