20 Hard-Level
Practice Problems
Official ACT-style questions verified by expert instructors across all major math domains. Study concepts first, then take the timed quiz.
Review these six domains before the quiz. Every concept below maps directly to at least one of the 20 questions.
Algebra & Functions
Systems of equations, quadratics, absolute value inequalities, and function composition are all heavily tested on the ACT.
b²−4ac = 0 → 1 real root
b²−4ac < 0 → no real roots
- Sum of roots = −b/a • Product of roots = c/a
- f(g(x)): evaluate g(x) first, then substitute into f
- |ax + b| ≤ c → −c ≤ ax + b ≤ c
- Vertex x-coordinate of parabola: x = −b / (2a)
Geometry & Measurement
Circles, special right triangles, coordinate geometry, and area or volume appear on every ACT exam.
45-45-90: sides 1 : 1 : √2
- Complete the square to find center and radius of a circle
- Altitude from right angle to hypotenuse: h = leg&sub1; × leg&sub2; / hypotenuse
- Distance: √((x&sub2;−x&sub1;)² + (y&sub2;−y&sub1;)²)
- Perpendicular lines: slopes multiply to −1
Trigonometry
SOH-CAH-TOA, unit circle values, trig identities, and Law of Cosines all appear on the ACT.
tanθ = sinθ / cosθ
c² = a²+b²−2ab·cos(C) [Law of Cosines]
sin45°=cos45°=√2/2
sin60°=√3/2, cos60°=1/2
- secθ = 1/cosθ, cscθ = 1/sinθ, cotθ = 1/tanθ
- Period of y = A·sin(Bx+C)+D is 2π/|B|
- ASTC rule: All (Q1), Sin (Q2), Tan (Q3), Cos (Q4) positive
Statistics & Probability
Mean, median, combinations, permutations, and probability rules all appear frequently on the ACT.
nPr = n! / (n−r)!
P(A∪B) = P(A)+P(B)−P(A∩B)
- Mean = sum / count; find missing value: sum = mean × n
- P(complement) = 1 − P(event)
- Without replacement: each draw changes the denominator
- Order matters → Permutation; Order doesn’t → Combination
Pre-Calculus & Advanced Algebra
Logarithms, exponential equations, polynomial division, and rational expressions are the hardest ACT topics.
log(a/b) = log a − log b
log(a^n) = n·log a • log_b(b^n) = n
Remainder Theorem: remainder of f(x)÷(x−a) = f(a)
- To factor x²−9: difference of squares → (x+3)(x−3)
- To factor x²−x−6: find two numbers that multiply to −6 and add to −1
- log_b(1) = 0 for any valid base b
Complex Numbers & Sequences
Arithmetic and geometric sequences, complex number arithmetic, and series sums are the final ACT hard-problem category.
Arithmetic sum: S_n = (n/2)(2a&sub1; + (n−1)d)
Geometric n-th term: a_n = a&sub1; · r^(n−1)
(a+bi)(a−bi) = a²+b²
(3+2i)(3−2i) = 9+4 = 13
- Always substitute i² = −1 when expanding complex products
- Conjugate of (a+bi) is (a−bi); useful for division
- Sum of geo series (|r|<1): S = a&sub1;/(1−r)
Choose the best answer. Explanation appears immediately after answering. Navigate with the dots below.
Fully worked solutions for all 20 questions. Every step verified. Best reviewed after completing the quiz.