📚 Year 13 OCR Maths: Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This concise handbook brings together the essential formulae, theorems and key results you need for the Year 13 OCR A‑level Mathematics course. Use it as a portable revision tool to reinforce your understanding of Pure, Statistics and Mechanics topics.
这本速查手册汇集了 Year 13 OCR A‑level 数学课程中必备的公式、定理与关键结论。你可以将其作为便携复习工具，用于巩固纯数、统计和力学的知识。
1. Binomial Theorem for Rational Powers | 有理指数二项式定理
For any rational index n and |x| < 1, the binomial expansion is (1+x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … . This series is infinite when n is not a positive integer, and you must state the range of validity.
对任意有理指数 n 且 |x| < 1，二项展开式为 (1+x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … 。当 n 不是正整数时，该级数无限延伸，且必须写明其适用范围。
To expand (a + bx)ⁿ, first rewrite as aⁿ (1 + (b/a)x)ⁿ and then apply the standard expansion, as long as |(b/a)x| < 1.
展开 (a + bx)ⁿ 时，先改写为 aⁿ (1 + (b/a)x)ⁿ，然后套用标准展开，前提是 |(b/a)x| < 1。
(a + bx)ⁿ = aⁿ [1 + n(b/a)x + n(n−1)/2! (b/a)² x² + …]
2. Trigonometric Formulae & Equations | 三角公式与方程
The reciprocal and quotient identities form the foundation: sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ.
倒数商数恒等式是基础：sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = 1/tan θ = cos θ/sin θ。
Pythagorean identities: sin²θ + cos²θ ≡ 1, 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ.
毕达哥拉斯恒等式：sin²θ + cos²θ ≡ 1，1 + tan²θ ≡ sec²θ，1 + cot²θ ≡ cosec²θ。
Double-angle formulae: sin 2θ ≡ 2 sin θ cos θ, cos 2θ ≡ cos²θ − sin²θ ≡ 2 cos²θ − 1 ≡ 1 − 2 sin²θ.
倍角公式：sin 2θ ≡ 2 sin θ cos θ，cos 2θ ≡ cos²θ − sin²θ ≡ 2 cos²θ − 1 ≡ 1 − 2 sin²θ。
For harmonic form, a sin θ + b cos θ can be written as R sin(θ ± α) or R cos(θ ∓ α), where R = √(a² + b²) and tan α = b/a.
对于谐波形式，a sin θ + b cos θ 可写作 R sin(θ ± α) 或 R cos(θ ∓ α)，其中 R = √(a² + b²) 且 tan α = b/a。
R sin(θ + α) = R sin θ cos α + R cos θ sin α
3. Differentiation Techniques | 微分技巧
The chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx).
链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。
Product rule: d/dx [u(x)v(x)] = u′v + uv′.
乘法法则：d/dx [u(x)v(x)] = u′v + uv′。
Quotient rule: d/dx [u/v] = (v u′ − u v′)/v².
除法法则：d/dx [u/v] = (v u′ − u v′)/v²。
Standard derivatives: d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec² x; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x.
标准导数：d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (tan x) = sec² x；d/dx (eˣ) = eˣ；d/dx (ln x) = 1/x。
d/dx [f(g(x))] = f'(g(x)) · g'(x)
4. Integration & Area | 积分与面积
Integration is the reverse of differentiation. The indefinite integral ∫ f(x) dx gives a family of functions differing by a constant.
积分是微分的逆运算。不定积分 ∫ f(x) dx 给出相差一个常数的函数族。
Standard integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1); ∫ 1/x dx = ln|x| + C; ∫ eˣ dx = eˣ + C; ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C.
标准积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)；∫ 1/x dx = ln|x| + C；∫ eˣ dx = eˣ + C；∫ sin x dx = −cos x + C；∫ cos x dx = sin x + C；∫ sec² x dx = tan x + C。
The definite integral ∫ab f(x) dx gives the signed area between the curve and the x‑axis. To find the area between a curve and the y‑axis, use ∫ x dy.
定积分 ∫ab f(x) dx 给出曲线与 x 轴之间的带符号面积。求曲线与 y 轴之间的面积，使用 ∫ x dy。
Integration by substitution: ∫ f(g(x)) g‘(x) dx = ∫ f(u) du, with u = g(x). For definite integrals, remember to change the limits.
换元积分法：∫ f(g(x)) g‘(x) dx = ∫ f(u) du，其中 u = g(x)。对于定积分，记住要变换积分限。
∫ u dv = uv − ∫ v du (integration by parts)
5. Exponentials & Logarithms | 指数与对数
The natural exponential and logarithm are inverses: eln x = x for x > 0, and ln(eˣ) = x for all real x.
自然指数与自然对数互为反函数：当 x > 0 时，eln x = x，且对全体实数 ln(eˣ) = x。
Laws of logs: ln(ab) = ln a + ln b; ln(a/b) = ln a − ln b; ln(aᵏ) = k ln a. The same rules apply to logs with any base.
对数运算法则：ln(ab) = ln a + ln b；ln(a/b) = ln a − ln b；ln(aᵏ) = k ln a。这些法则适用于任何底数的对数。
The rate of change of aekx is proportional to the function itself: d/dx (aekx) = k aekx. Exponential models of growth and decay often use y = Aekt.
aekx 的变化率与函数本身成正比：d/dx (aekx) = k aekx。指数增长与衰减模型常采用 y = Aekt。
Solving ax = b: take logs of both sides, x ln a = ln b ⇒ x = ln b / ln a.
求解 ax = b：两边取对数，x ln a = ln b ⇒ x = ln b / ln a。
y = ln x ⇔ x = eʸ
6. Parametric & Implicit Differentiation | 参数与隐函数微分
For parametric equations x = f(t), y = g(t), the gradient is dy/dx = (dy/dt) / (dx/dt). The second derivative is d²y/dx² = d/dt (dy/dx) / (dx/dt).
对于参数方程 x = f(t)、y = g(t)，梯度为 dy/dx = (dy/dt) / (dx/dt)。二阶导数为 d²y/dx² = d/dt (dy/dx) / (dx/dt)。
Implicit differentiation: when an equation defines y implicitly as a function of x, differentiate each term with respect to x, applying the chain rule to terms containing y. For example, d/dx (y²) = 2y dy/dx.
隐函数求导：当方程以隐式定义 y 为 x 的函数时，对每一项关于 x 求导，对含 y 的项施用链式法则。例如，d/dx (y²) = 2y dy/dx。
From implicit differentiation you can find dy/dx and then the gradient of tangents and normals at a point.
通过隐函数求导可求得 dy/dx，进而得到某点的切线与法线梯度。
dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0
7. Vectors in 3D | 三维向量
A vector a = a₁i + a₂j + a₃k has magnitude |a| = √(a₁² + a₂² + a₃²). The unit vector in the direction of a is â = a/|a|.
向量 a = a₁i + a₂j + a₃k 的模长为 |a| = √(a₁² + a₂² + a₃²)。沿 a 方向的单位向量为 â = a/|a|。
Scalar (dot) product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. Two vectors are perpendicular if their dot product is zero.
数量积（点积）：a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。若点积为零，则两向量垂直。
The vector equation of a line is r = p + λd, where p is a point on the line and d is the direction vector. The shortest distance from a point to a line can be found using the cross product or dot product formula: d = |(a−p) × d|/|d| (cross product is not required in OCR Pure, but the distance formula with dot product is used).
直线的向量方程为 r = p + λd，其中 p 为直线上一点，d 为方向向量。点到直线的最短距离可通过点积公式求出：distance = |(a−p) − ((a−p)·d/|d|²) d|。
a · b = |a||b| cos θ
8. Numerical Methods for Roots | 求根数值方法
To locate a root of f(x) = 0, use a sign-change check: if f(a) and f(b) have opposite signs and f is continuous on [a, b], then a root lies between a and b.
定位方程 f(x) = 0 的根时，可用符号变化检验：若 f(a) 与 f(b) 异号且 f 在 [a, b] 上连续，则在 a 与 b 之间必有一根。
The Newton–Raphson method: xₙ₊₁ = xₙ − f(xₙ)/f‘(xₙ). It converges quickly if the initial guess is close enough and f‘(x) is not small.
牛顿—拉弗森法：xₙ₊₁ = xₙ − f(xₙ)/f‘(xₙ)。若初值足够接近且 f‘(x) 不为零，该方法可快速收敛。
Numerical integration using the trapezium rule: ∫ab f(x) dx ≈ ½ h [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b−a)/n. Increasing n improves accuracy.
梯形法则数值积分：∫ab f(x) dx ≈ ½ h [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 h = (b−a)/n。增大 n 可提高精度。
xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)
9. Discrete Probability Distributions | 离散概率分布
The binomial distribution X ~ B(n, p) models the number of successes in n independent trials, each with success probability p: P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ.
二项分布 X ~ B(n, p) 描述 n 次独立试验中成功的次数，每次成功概率为 p：P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ。
The Poisson distribution X ~ Po(λ) models rare events occurring randomly at a constant average rate λ: P(X = r) = e−λ λʳ / r! . The mean and variance both equal λ.
泊松分布 X ~ Po(λ) 模拟以恒定平均率 λ 随机发生的稀有事件：P(X = r) = e−λ λʳ / r! 。其均值与方差均等于 λ。
For sums: if X ~ Po(λ₁) and Y ~ Po(λ₂) are independent, then X + Y ~ Po(λ₁ + λ₂). Binomial also adds under constant p: B(n₁, p) + B(n₂, p) ~ B(n₁+n₂, p).
关于和的性质：若 X ~ Po(λ₁)
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