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Formula Book

Every Mathematics topic's formulas in one place. Tap a chapter below to jump to it.

Syllabus note

Syllabus mapping should be validated against the official syllabus (csca.cn).

Sets & Inequalities

Set notation and operations, Venn diagrams, and solving linear, quadratic and absolute-value inequalities.

Sets & Set Operations

Lesson →
  • |A ∪ B| = |A| + |B| − |A ∩ B|
  • (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
  • (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ
  • A − B = A ∩ Bᶜ
  • If B ⊆ A then |A − B| = |A| − |B|
  • |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |B ∩ C| − |A ∩ C| + |A ∩ B ∩ C|
  • |A| + |Aᶜ| = |U|
  • |A ∩ B| = |A| + |B| − |A ∪ B|

Inequalities

Lesson →
  • |x| < a ⇔ −a < x < a (a > 0)
  • |x| > a ⇔ x < −a or x > a (a > 0)
  • |x − c| < r ⇔ x ∈ (c − r, c + r)
  • x² < a² ⇔ −a < x < a; x² > a² ⇔ x < −a or x > a
  • a/b > 0 ⇔ a and b have the same sign (b ≠ 0)
  • (x − a)(x − b) > 0 ⇔ x < min(a, b) or x > max(a, b)
  • (x − a)(x − b) < 0 ⇔ x strictly between a and b
  • √f(x) defined ⇔ f(x) ≥ 0; 1/g(x) defined ⇔ g(x) ≠ 0 — intersect with the solution set

Functions

Domain and range, monotonicity, parity, and the power, exponential, logarithmic and trigonometric families.

Domain & Range

Lesson →
  • dom(1/g) = {x | g(x) ≠ 0}
  • dom(√g) = {x | g(x) ≥ 0}
  • dom(log_a g) = {x | g(x) > 0}, a > 0, a ≠ 1
  • range of ax² + bx + c (a > 0): [−Δ/4a, ∞)
  • range of ax² + bx + c (a < 0): (−∞, −Δ/4a]
  • Vertex of ax² + bx + c at x = −b/2a; extremal value y = f(−b/2a)
  • dom(f + g) = dom(f) ∩ dom(g); dom(f ∘ g) requires g(x) ∈ dom(f)
  • range of eˣ: (0, ∞); range of ln x: ℝ; range of sin x: [−1, 1]

Monotonicity

Lesson →
  • f′(x) > 0 ⇒ f strictly increasing
  • f′(x) < 0 ⇒ f strictly decreasing
  • Local extremum candidate: f′(x₀) = 0
  • a > b and f increasing ⇒ f(a) > f(b)
  • Composition of two decreasing functions is increasing
  • Increasing + increasing = increasing; decreasing + decreasing = decreasing
  • f increasing ⇔ f⁻¹ exists and is increasing on the range
  • aˣ with a > 1 increasing; with 0 < a < 1 decreasing

Even & Odd Functions

Lesson →
  • Even: f(−x) = f(x)
  • Odd: f(−x) = −f(x)
  • ∫₋ₐᵃ even f = 2∫₀ᵃ f; ∫₋ₐᵃ odd f = 0
  • xⁿ is even if n even, odd if n odd
  • cos is even; sin and tan are odd
  • Even ± even = even; odd ± odd = odd; even × odd = odd; odd × odd = even
  • Odd f defined at 0 ⇒ f(0) = 0
  • Parity needs a symmetric domain: f(−x) must exist whenever f(x) does

Power Functions

Lesson →
  • xᵃ · xᵇ = xᵃ⁺ᵇ; (xᵃ)ᵇ = xᵃᵇ
  • x⁻ⁿ = 1/xⁿ; x^(m/n) = ⁿ√(xᵐ)
  • √(x²) = |x|
  • (xy)ⁿ = xⁿyⁿ
  • For x > 1, larger exponent ⇒ larger value
  • For 0 < x < 1, larger exponent ⇒ smaller value
  • (x/y)ⁿ = xⁿ/yⁿ (y ≠ 0); x^(1/2) = √x (x ≥ 0)
  • Large-x growth: exponential > power > logarithm

Exponential Functions

Lesson →
  • aˣ · aʸ = aˣ⁺ʸ; aˣ/aʸ = aˣ⁻ʸ
  • (aˣ)ʸ = aˣʸ
  • a^0 = 1; a⁻ˣ = 1/aˣ
  • a^f = a^g ⇔ f = g (a > 0, a ≠ 1)
  • e ≈ 2.718; lim(n→∞)(1 + 1/n)ⁿ = e
  • (ab)ˣ = aˣbˣ; aˣ > 0 for all real x
  • a > 1 ⇒ aˣ increasing; 0 < a < 1 ⇒ aˣ decreasing
  • aˣ = b ⇔ x = log_a b (b > 0)

Logarithmic Functions

Lesson →
  • log_a(xy) = log_a x + log_a y
  • log_a(x/y) = log_a x − log_a y
  • log_a(xⁿ) = n · log_a x
  • log_a x = ln x / ln a
  • a^(log_a x) = x (x > 0)
  • log_a a = 1; log_a 1 = 0; ln e = 1
  • log_a x + log_a y = log_a(xy) — condense before solving
  • log_a b = 1 / log_b a (a, b > 0, ≠ 1)

Trigonometric Functions

Lesson →
  • sin²θ + cos²θ = 1
  • sin(A ± B) = sin A cos B ± cos A sin B
  • cos(A ± B) = cos A cos B ∓ sin A sin B
  • sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ
  • tan θ = sin θ / cos θ
  • 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ
  • cos 2θ = 2cos²θ − 1 = 1 − 2sin²θ; tan 2θ = 2tan θ/(1 − tan²θ)
  • sin(−θ) = −sin θ; cos(−θ) = cos θ; tan(−θ) = −tan θ
  • Periods: sin/cos 2π, tan π; A·sin(Bx): amplitude |A|, period 2π/|B|
  • Special values: sin/cos of 0, π/6, π/4, π/3, π/2 — instant recall

Sequences

Arithmetic and geometric sequences and series: general terms, sums, and applied growth problems.

Arithmetic Sequences

Lesson →
  • aₙ = a₁ + (n − 1)d
  • Sₙ = n(a₁ + aₙ)/2
  • Sₙ = n(2a₁ + (n − 1)d)/2
  • aₙ = Sₙ − Sₙ₋₁ (n ≥ 2)
  • a, b, c in A.P. ⇔ 2b = a + c
  • d = aₙ − aₙ₋₁; aₖ = aₘ + (k − m)d
  • 1 + 2 + … + n = n(n + 1)/2
  • Sₙ is quadratic in n (no constant term); aₙ is linear in n

Geometric Sequences

Lesson →
  • aₙ = a₁ · qⁿ⁻¹
  • Sₙ = a₁(1 − qⁿ)/(1 − q), q ≠ 1
  • S = a₁/(1 − q), |q| < 1
  • a, b, c in G.P. ⇔ b² = ac
  • Compound growth: A = P(1 + r)ⁿ
  • q = aₙ/aₙ₋₁; aₖ = aₘ · q^(k−m)
  • Sₙ = a₁(qⁿ − 1)/(q − 1), q ≠ 1 (equivalent form)
  • |q| ≥ 1 ⇒ infinite series diverges (no finite sum)

Calculus

Limits, derivative rules, and basic applications: tangents, monotonicity, extrema and simple optimization.

Limits

Lesson →
  • lim(x→0) sin x / x = 1
  • lim(n→∞)(1 + 1/n)ⁿ = e
  • lim(x→c)(f ± g) = lim f ± lim g
  • lim(x→c)(f · g) = lim f · lim g
  • 0/0: factor-cancel or rationalize first
  • lim(x→0)(1 − cos x)/x = 0; lim(x→0)(eˣ − 1)/x = 1
  • Two-sided limit exists ⇔ left and right limits agree
  • Continuous at c ⇔ lim(x→c) f(x) = f(c)

Derivatives

Lesson →
  • (xⁿ)′ = n·xⁿ⁻¹
  • (uv)′ = u′v + uv′
  • (u/v)′ = (u′v − uv′)/v²
  • [f(g(x))]′ = f′(g(x)) · g′(x)
  • (eˣ)′ = eˣ; (ln x)′ = 1/x; (sin x)′ = cos x; (cos x)′ = −sin x
  • (c)′ = 0; (cu)′ = cu′; (u ± v)′ = u′ ± v′
  • (tan x)′ = sec²x; (aˣ)′ = aˣ ln a; (log_a x)′ = 1/(x ln a)
  • f′(a) = lim(h→0)(f(a+h) − f(a))/h — slope of tangent at a

Basic Applications

Lesson →
  • Tangent: y = f(x₀) + f′(x₀)(x − x₀)
  • Critical points: f′(x) = 0 or undefined
  • Max on [a, b]: max{f(a), f(b), f(critical)}
  • f′: − → + ⇒ local min; + → − ⇒ local max
  • Second-derivative check: f′′(x₀) > 0 ⇒ local min
  • f′ > 0 increasing; f′ < 0 decreasing — sign chart decides
  • Optimization: single-variable f → f′ = 0 → verify → check endpoints/domain
  • Second-derivative check: f′′(x₀) < 0 ⇒ local max; = 0 inconclusive

Geometry

Coordinate geometry: lines, circles, and the conic sections — ellipses, hyperbolas and parabolas.

Analytic Geometry Basics

Lesson →
  • d = √((x₂−x₁)² + (y₂−y₁)²)
  • Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
  • Slope: m = (y₂−y₁)/(x₂−x₁)
  • m₁·m₂ = −1 ⇔ perpendicular (non-vertical)
  • Point-to-origin: d = √(x² + y²)
  • Section formula: internal division (λ:μ) → ((μx₁ + λx₂)/(λ+μ), (μy₁ + λy₂)/(λ+μ))
  • Centroid of triangle: ((x₁+x₂+x₃)/3, (y₁+y₂)/3)
  • Area via coordinates (shoelace) for polygon vertices in order

Lines

Lesson →
  • y = mx + b; y − y₀ = m(x − x₀)
  • Ax + By + C = 0
  • Parallel: m₁ = m₂; perpendicular: m₁m₂ = −1
  • Distance point→line: |Ax₀ + By₀ + C|/√(A² + B²)
  • Distance between parallel lines: |C₁ − C₂|/√(A² + B²)
  • Slope from general form: m = −A/B (B ≠ 0); x-intercept −C/A; y-intercept −C/B
  • Intersection = simultaneous solution of the two line equations
  • Vertical line x = c (undefined slope); horizontal line y = c (m = 0)

Circles

Lesson →
  • (x − a)² + (y − b)² = r²
  • Center (−D/2, −E/2), r = √(D² + E² − 4F)/2
  • d > r separate; d = r tangent; d < r secant
  • Tangent ⊥ radius at point of tangency
  • |r₁ − r₂| < d < r₁ + r₂ ⇔ two intersection points
  • Diameter endpoints (x₁,y₁),(x₂,y₂): (x−x₁)(x−x₂) + (y−y₁)(y−y₂) = 0
  • Tangent at (x₀,y₀) on x² + y² = r²: xx₀ + yy₀ = r²
  • Real circle needs D² + E² − 4F > 0; = 0 is a point, < 0 is empty

Ellipses

Lesson →
  • x²/a² + y²/b² = 1 (a > b > 0)
  • a² = b² + c²
  • Foci (±c, 0); vertices (±a, 0); co-vertices (0, ±b)
  • e = c/a, 0 < e < 1
  • Focal radii sum: PF₁ + PF₂ = 2a
  • Vertical major axis: y²/a² + x²/b² = 1; foci (0, ±c), vertices (0, ±a)
  • Minor-axis endpoints: (0, ±b); focal distance from center: c = ae
  • Directrix: x = ±a/e (major axis horizontal); PF = e · (distance to directrix)

Hyperbolas

Lesson →
  • x²/a² − y²/b² = 1
  • c² = a² + b²
  • Asymptotes: y = ±(b/a)x
  • Foci (±c, 0); vertices (±a, 0)
  • e = c/a > 1; |PF₁ − PF₂| = 2a
  • Vertical transverse axis: y²/a² − x²/b² = 1; foci (0, ±c); asymptotes y = ±(a/b)x
  • Asymptote rectangle: corners (±a, ±b) — diagonals are the asymptotes
  • Conjugate axis length 2b; transverse axis length 2a; c = ae

Parabolas

Lesson →
  • y² = 2px: focus (p/2, 0), directrix x = −p/2
  • x² = 2py: focus (0, p/2), directrix y = −p/2
  • PF = distance from P to directrix
  • y = a(x − h)² + k: vertex (h, k)
  • Focal length = |p|/2
  • y² = −2px (p > 0): opens left, focus (−p/2, 0), directrix x = p/2
  • x² = −2py (p > 0): opens down, focus (0, −p/2), directrix y = p/2
  • Axis of y² = 2px is the x-axis; axis of x² = 2py is the y-axis

Vectors

Vector coordinates, magnitude, dot product, parallelism/perpendicularity, and basic geometric applications.

Vectors

Lesson →
  • |a| = √(x² + y²)
  • a·b = x₁x₂ + y₁y₂ = |a||b|cos θ
  • cos θ = (a·b)/(|a||b|)
  • a ⊥ b ⇔ a·b = 0
  • a ∥ b ⇔ x₁y₂ = x₂y₁ (2D)
  • |a|² = a·a; unit vector û = a/|a| (a ≠ 0)
  • Midpoint of AB: (a + b)/2; vector AB = b − a (position-vector difference)
  • Projection of a onto b: ((a·b)/|b|²) b; scalar projection (a·b)/|b|

Complex Numbers

The complex plane, modulus and argument, the four operations, conjugates, and quadratic roots.

Complex Numbers

Lesson →
  • i² = −1
  • (a + bi)(a − bi) = a² + b²
  • |a + bi| = √(a² + b²)
  • 1/i = −i
  • Δ < 0 ⇒ x = (−b ± i√|Δ|)/2a
  • |z|² = z·z̄; z̄ = a − bi for z = a + bi
  • i-cycle: i¹ = i, i² = −1, i³ = −i, i⁴ = 1 (period 4)
  • z₁ + z₂ = (a+c) + (b+d)i; z₁z₂ via FOIL with i² = −1

Solid Geometry

Space lines and planes, and surface area / volume of prisms, pyramids, cylinders, cones and spheres.

Solid Geometry

Lesson →
  • Prism/cylinder: V = Bh
  • Pyramid/cone: V = Bh/3
  • Sphere: V = 4πr³/3; S = 4πr²
  • Cylinder lateral area: 2πrh; cone lateral area: πrl
  • a² + b² + c² = space diagonal² of a box
  • Pyramid/cone total surface = base area + lateral area; prism/cylinder total = 2B + lateral
  • Cone slant height: l = √(r² + h²)
  • Cross-section through the axis reduces cone/cylinder to an isosceles triangle/rectangle

Probability

Classical probability, counting (permutations/combinations), conditional probability and independence.

Probability

Lesson →
  • P(A) = |A|/|Ω|
  • A(n,k) = n!/(n−k)!; C(n,k) = n!/(k!(n−k)!)
  • P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
  • P(A|B) = P(A ∩ B)/P(B)
  • Independent: P(A ∩ B) = P(A) · P(B)
  • P(Aᶜ) = 1 − P(A); P(at least one) = 1 − P(none)
  • Bayes: P(B|A) = P(A|B)P(B)/P(A); total probability: P(A) = Σ P(A|Bᵢ)P(Bᵢ)
  • C(n,k) = C(n,n−k); C(n,0) = C(n,n) = 1; P(∅) = 0, P(Ω) = 1
  • Mutually exclusive: P(A ∩ B) = 0 ⇒ P(A ∪ B) = P(A) + P(B)
  • Order matters ⇒ A(n,k); order irrelevant ⇒ C(n,k)

Statistics

Mean, median, mode, variance and standard deviation, plus reading frequency tables and basic charts.

Statistics

Lesson →
  • mean x̄ = (Σx)/n
  • variance σ² = Σ(x − x̄)²/n
  • σ = √variance
  • Median position (odd n): (n+1)/2-th ordered value
  • Range = max − min
  • Median (even n): average of the two middle ordered values
  • Weighted/grouped mean: x̄ = (Σ fᵢxᵢ)/(Σ fᵢ)
  • σ = 0 ⇔ all values identical; larger σ ⇔ more spread (same units as data)