SAT Math Elite Practice | 20 Hard Questions

Topic 1 · Heart of Algebra

Linear Equations & Systems

Slope-intercept: y = mx + b (m = slope, b = y-intercept)
Point-slope: y − y₁ = m(x − x₁)
Parallel lines share the same slope; perpendicular slopes multiply to −1
Systems: substitute or eliminate to find (x, y) intersection
No solution ↔ parallel lines; infinite solutions ↔ same line

Slope = (y₂ − y₁) / (x₂ − x₁)

If 2x + 3y = 12 and x − y = 1, what is x?

→ From x = y+1: 2(y+1)+3y=12 → 5y=10 → y=2, x=3. Answer: 3

Topic 2 · Advanced Math

Quadratic Equations & Parabolas

Standard: ax² + bx + c = 0; vertex form: a(x−h)² + k
Quadratic formula: x = [−b ± √(b²−4ac)] / 2a
Discriminant b²−4ac: >0 two real roots, =0 one root, <0 no real roots
Vertex x-coordinate: x = −b / 2a
Sum of roots = −b/a; Product of roots = c/a

x = (−b ± √(b²−4ac)) / 2a

For x² − 5x + 6 = 0, find the roots.

→ (x−2)(x−3) = 0 → x = 2 or x = 3

Topic 3 · Advanced Math

Polynomials & Rational Expressions

Remainder Theorem: f(a) = remainder when f(x) ÷ (x−a)
Factor Theorem: (x−a) is a factor ↔ f(a) = 0
FOIL and difference of squares: (a−b)(a+b) = a²−b²
Long / synthetic division to simplify rational expressions

(a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b²

If f(x) = x³ − 4x + 3, what is f(1)?

→ 1 − 4 + 3 = 0. So (x−1) is a factor.

Topic 4 · Advanced Math

Exponents, Radicals & Exponential Growth

aᵐ · aⁿ = aᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ
a^(1/n) = ⁿ√a; a^(m/n) = (ⁿ√a)ᵐ
Exponential growth: y = A · bˣ; decay when 0 < b < 1
Percent change: multiply by (1 ± r%) each period

Growth: A(t) = A₀(1 + r)ᵗ Decay: A(t) = A₀(1 − r)ᵗ

Simplify: (8^(2/3)) / (2^(−1))

→ 8^(2/3) = (2³)^(2/3) = 2² = 4; 2^(−1) = 1/2; 4 ÷ (1/2) = 8

Topic 5 · Advanced Math

Functions — Composition, Inverse & Transformations

f(g(x)): evaluate inner function first, then outer
Inverse f⁻¹: swap x and y, solve for y
f(x)+k shifts up k; f(x+k) shifts left k
−f(x) reflects over x-axis; f(−x) reflects over y-axis

If f(x) = 2x+1 and g(x) = x², find f(g(3)).

→ g(3)=9; f(9)=19. Answer: 19

Topic 6 · Heart of Algebra

Inequalities & Absolute Value

Flip inequality sign when multiplying / dividing by a negative number
|x| < a ↔ −a < x < a
|x| > a ↔ x < −a OR x > a
Graph linear inequalities: solid line (≤,≥), dashed (‹,›)

Solve |2x − 3| ≤ 5

→ −5 ≤ 2x−3 ≤ 5 → −1 ≤ x ≤ 4

Topic 7 · Geometry

Geometry — Area, Volume & Angles

Circle: Area = πr²; Circumference = 2πr; Arc = (θ/360)·2πr
Triangle area = ½bh; Pythagorean: a²+b²=c²
Special right triangles: 30-60-90 (1:√3:2) and 45-45-90 (1:1:√2)
Cylinder V = πr²h; Cone V = ⅓πr²h; Sphere V = (4/3)πr³
Similar triangles: ratios of corresponding sides are equal

Pythagorean Triples: 3-4-5, 5-12-13, 8-15-17

Topic 8 · Geometry

Trigonometry & Radian Measure

SOH-CAH-TOA: sin=opp/hyp, cos=adj/hyp, tan=opp/adj
Radians: 180° = π rad; arc length s = rθ
sin²θ + cos²θ = 1 (Pythagorean identity)
Co-function: sin θ = cos(90°−θ)

In a right triangle, opposite = 5, hypotenuse = 13. Find sin θ.

→ sin θ = 5/13

Topic 9 · Problem Solving & Data

Statistics, Probability & Data Interpretation

Mean = sum ÷ count; Median = middle value when sorted; Mode = most frequent
Range = max − min; Standard deviation measures spread
Probability = favorable outcomes / total outcomes
P(A and B) = P(A)·P(B) if independent
P(A or B) = P(A) + P(B) − P(A and B)

Topic 10 · Problem Solving

Ratios, Rates, Percentages & Word Problems

Rate × Time = Distance; Work = Rate × Time
Percent increase: (new−old)/old × 100%
Unit rate: reduce ratio to "per 1" unit
Proportion: a/b = c/d → ad = bc (cross-multiply)

A car travels 120 miles in 2 hours. How far in 5 hours at the same rate?

→ Rate = 60 mph; 60 × 5 = 300 miles