📚 Year 13 Edexcel Maths Formula & Theorem Quick Reference Handbook | Year 13 Edexcel 数学公式定理速查手册
This handbook provides a concise summary of key formulas and theorems for the Edexcel A Level Mathematics (Year 13) course, covering Pure Mathematics, Statistics, and Mechanics. Use it for quick revision and exam preparation.
本手册简要总结了适用于 Edexcel A Level 数学（Year 13）课程的核心公式与定理，涵盖纯数、统计与力学，供快速复习与备考使用。
1. Algebra & Functions | 代数与函数
For the quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula: x = [-b ± √(b² – 4ac)] / (2a). The discriminant Δ = b² – 4ac determines the nature of the roots; Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
对于二次方程 ax² + bx + c = 0，求根公式为 x = [-b ± √(b² – 4ac)] / (2a)。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。
The Factor Theorem states that (x – a) is a factor of polynomial p(x) if and only if p(a) = 0. The Remainder Theorem states that when p(x) is divided by (x – a), the remainder is p(a).
因式定理：若多项式 p(a) = 0，则 (x – a) 是 p(x) 的因式。余式定理：p(x) 除以 (x – a) 的余数为 p(a)。
Composite functions are defined as fg(x) = f(g(x)); the inverse function f⁻¹(x) satisfies f(f⁻¹(x)) = x. The domain of f⁻¹ is the range of f.
复合函数定义为 fg(x) = f(g(x))；反函数 f⁻¹(x) 满足 f(f⁻¹(x)) = x。f⁻¹ 的定义域是 f 的值域。
Modulus function |x| yields the absolute value. Equations like |ax + b| = c lead to ax + b = ±c, while inequalities require careful consideration of intervals, e.g. |x – a| > b ⇔ x – a < -b or x - a > b.
模函数 |x| 表示绝对值。方程 |ax + b| = c 变为 ax + b = ±c；不等式需分段处理，例如 |x – a| > b ⇔ x – a < -b 或 x - a > b。
Graph transformations: y = f(x) + a translates vertically by a; y = f(x + a) translates horizontally by -a; y = a f(x) stretches vertically by scale factor a; y = f(ax) stretches horizontally by scale factor 1/a.
函数图像变换：y = f(x) + a 竖直平移 a；y = f(x + a) 水平平移 -a；y = a f(x) 竖直伸缩 a 倍；y = f(ax) 水平伸缩 1/a 倍。
2. Coordinate Geometry & Circles | 坐标几何与圆
The gradient of a line through (x₁, y₁) and (x₂, y₂) is m = (y₂ – y₁) / (x₂ – x₁); the equation in point-slope form is y – y₁ = m(x – x₁). Two lines are parallel if m₁ = m₂, perpendicular if m₁ m₂ = -1.
过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m = (y₂ – y₁) / (x₂ – x₁)；点斜式方程为 y – y₁ = m(x – x₁)。两直线平行则 m₁ = m₂，垂直则 m₁ m₂ = -1。
The equation of a circle with centre (a, b) and radius r is (x – a)² + (y – b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (-g, -f) and radius √(g² + f² – c).
以 (a, b) 为圆心、r 为半径的圆方程为 (x – a)² + (y – b)² = r²。一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (-g, -f)，半径为 √(g² + f² – c)。
Tangents and chords: the gradient of the radius to a point on the circle is m; the tangent there has gradient -1/m (perpendicular). The midpoint of a chord lies on the line from the centre perpendicular to the chord.
切线与弦：圆上某点的半径斜率为 m，则切线斜率为 -1/m（垂直）。弦的中点在过圆心且垂直于弦的直线上。
3. Sequences & Series | 数列与级数
An arithmetic sequence has nth term uₙ = a + (n – 1)d, sum to n terms Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + L), where L is the last term.
等差数列的通项为 uₙ = a + (n – 1)d，前 n 项和 Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + L)，其中 L 为末项。
A geometric sequence has nth term uₙ = arⁿ⁻¹, sum Sₙ = a (1 – rⁿ) / (1 – r) for |r| < 1, and sum to infinity S∞ = a / (1 - r) valid when |r| < 1.
等比数列的通项为 uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a (1 – rⁿ) / (1 – r)（|r| < 1 时收敛），无穷和 S∞ = a / (1 - r) 仅当 |r| < 1 成立。
Binomial expansion for integer n: (a + b)ⁿ = Σ (n choose r) aⁿ⁻ʳ bʳ. For negative or fractional n, (1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + … valid for |x| < 1. Use partial fractions to expand rational functions if needed.
整数次幂的二项展开：(a + b)ⁿ = Σ C(n,r) aⁿ⁻ʳ bʳ。对于负指数或分数指数，(1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + …，要求 |x| < 1。若含有理式可先用部分分式分解再展开。
Sigma notation: Σ uᵣ from r=1 to n. Standard results: Σ r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6, Σ r³ = n²(n+1)²/4.
求和符号 Σ：Σ r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6, Σ r³ = n²(n+1)²/4。
4. Trigonometry | 三角学
Radians: 180° = π rad, arc length s = rθ, sector area A = ½ r²θ, segment area = ½ r²(θ – sin θ).
弧度制：180° = π rad，弧长 s = rθ，扇形面积 A = ½ r²θ，弓形面积 = ½ r²(θ – sin θ)。
Fundamental identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ. For any triangle: sine rule a/sin A = b/sin B = c/sin C, cosine rule a² = b² + c² – 2bc cos A.
基本恒等式：sin²θ + cos²θ = 1；1 + tan²θ = sec²θ；1 + cot²θ = cosec²θ。正弦定理 a/sin A = b/sin B = c/sin C，余弦定理 a² = b² + c² – 2bc cos A。
Compound angle formulas: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).
和差角公式：sin(A ± B) = sin A cos B ± cos A sin B；cos(A ± B) = cos A cos B ∓ sin A sin B；tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)。
Double angle: sin 2A = 2 sin A cos A; cos 2A = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A; tan 2A = 2 tan A / (1 – tan²A).
倍角公式：sin 2A = 2 sin A cos A；cos 2A = cos²A – sin²A = 2cos²A – 1 = 1 – 2sin²A；tan 2A = 2 tan A / (1 – tan²A)。
R-formula: a sin θ ± b cos θ = R sin(θ ± α), a cos θ ± b sin θ = R cos(θ ∓ α) where R = √(a² + b²) and tan α = b/a (choose α appropriately).
R 公式：a sin θ ± b cos θ = R sin(θ ± α)，a cos θ ± b sin θ = R cos(θ ∓ α)，其中 R = √(a² + b²)，tan α = b/a（适当选取 α）。
5. Exponentials & Logarithms | 指数与对数
The natural exponential function y = eˣ has derivative eˣ and inverse ln x. eˣ and ln x are related by e^(ln x) = x, ln(eˣ) = x. Key limits: eˣ → 0 as x → -∞, eˣ → ∞ as x → ∞.
自然指数函数 y = eˣ 的导数为 eˣ，其反函数为 ln x。满足 e^(ln x) = x，ln(eˣ) = x。极限：x → -∞ 时 eˣ → 0，x → ∞ 时 eˣ → ∞。
Logarithm laws: ln(ab) = ln a + ln b; ln(a/b) = ln a – ln b; ln(aᵏ) = k ln a. Change of base: logₐ b = ln b / ln a or log₁₀ b / log₁₀ a.
对数运算法则：ln(ab) = ln a + ln b；ln(a/b) = ln a – ln b；ln(aᵏ) = k ln a。换底公式：logₐ b = ln b / ln a。
Exponential growth/decay: y = A e^(kt), where k > 0 models growth, k < 0 models decay. The half-life for decay is given by T₁/₂ = (ln 2)/|k|.
指数增长/衰减：y = A e^(kt)，k > 0 为增长，k < 0 为衰减。衰减的半衰期 T₁/₂ = (ln 2)/|k|。
Solving equations: to solve a e^(bx) = c, take ln: bx = ln(c/a). For logs, use exponentiation: ln(x) = t ⇒ x = eᵗ. Watch domain: ln x is defined only for x > 0.
求解方程：a e^(bx) = c 可先取对数 bx = ln(c/a)；解 ln(x) = t 则 x = eᵗ。注意 ln x 仅当 x > 0 有定义。
6. Differentiation | 微分
Basic derivatives: d/dx (xⁿ) = n xⁿ⁻¹, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec² x.
基本导数：d/dx (xⁿ) = n xⁿ⁻¹，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (tan x) = sec² x。
Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x). Product rule: (uv)’ = u’v + uv’. Quotient rule: (u/v)’ = (u’v – uv’)/v².
链式法则：y = f(g(x)) 则 dy/dx = f'(g(x))
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