Year 13 OCR Mathematics: Formula & Theorem Quick Reference Guide | Year 13 OCR 数学：公式定理速查手册

📚 Year 13 OCR Mathematics: Formula & Theorem Quick Reference Guide | Year 13 OCR 数学：公式定理速查手册

Welcome to your compact revision companion for Year 13 OCR Mathematics. This quick reference guide brings together essential formulas, theorems and statistical mechanics results from the OCR A Level specification. Use it as a bilingual checklist to reinforce your memory, clarify tricky points, and walk into the exam with confidence.

欢迎使用 Year 13 OCR 数学公式定理速查手册。本手册浓缩了 OCR A Level 考试大纲中纯数学、统计学和力学的核心公式与定理，以中英双语形式呈现，帮助你巩固记忆、扫清疑点，自信应考。

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1. Algebraic Laws & Sequences | 代数法则与数列

Indices rules form the backbone of algebraic manipulation. Recall that multiplying powers adds exponents, while raising to a power multiplies them. Negative and fractional indices represent reciprocals and roots respectively.

aᵐ × aⁿ = a^(m+n) , (aᵐ)ⁿ = a^(mn) , a⁻ⁿ = 1/aⁿ , a^(1/n) = ⁿ√a

指数律是代数运算的基础。幂相乘指数相加，幂的幂指数相乘；负指数表示倒数，分数指数表示开方。

The logarithm laws are indispensable for solving equations involving exponentials. The change-of-base formula lets you switch between bases.

logₐ(xy) = logₐx + logₐy , logₐ(x/y) = logₐx – logₐy , logₐ(xⁿ) = n logₐx
logₐb = log_c b / log_c a

对数律在解指数方程时不可或缺。换底公式可以转换任意底的对数。

Arithmetic and geometric sequences appear regularly. Know the nth term and sum formulas, and remember the infinite sum condition for convergent geometric series.

Arithmetic: aₙ = a + (n-1)d , Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l)
Geometric: aₙ = arⁿ⁻¹ , Sₙ = a(1 – rⁿ)/(1 – r) (r≠1), S∞ = a/(1 – r) for |r| < 1

等差与等比数列是常见考点，需熟记通项与求和公式。等比级数当公比绝对值小于 1 时收敛，无限和为 S∞ = a/(1 – r)。

The quadratic formula and discriminant determine the nature of roots instantly.

x = (-b ± √(b² – 4ac)) / (2a) , Δ = b² – 4ac

二次公式与判别式可快速判断根的性质：Δ > 0 两个不同实根，Δ = 0 重根，Δ < 0 无实根。

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2. Trigonometry | 三角学

Fundamental identities must be at your fingertips. They are the gateway to simplifying expressions and proving other relations.

sin²θ + cos²θ = 1
tanθ = sinθ / cosθ

基本恒等式是化简三角表达式和证明其他关系的基础。

Double angle formulas are heavily tested; recognise their three forms for cos2θ.

sin2θ = 2sinθ cosθ
cos2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ
tan2θ = 2tanθ / (1 – tan²θ)

倍角公式考试高频出现；注意余弦的三种等价形式，可根据需要选用。

The R-formula rewrites a linear combination of sine and cosine as a single sine or cosine wave, vital for solving equations and finding extremes.

a sinθ ± b cosθ = R sin(θ ± α)
a cosθ ± b sinθ = R cos(θ ∓ α)
R = √(a² + b²), tanα = b/a (for sinθ ± cosθ forms)

R 公式将正弦余弦的线性组合化为单一的正弦或余弦函数，用于解方程和求极值。

Core triangle laws and radian measures link geometry with calculus.

Sine rule: a/sinA = b/sinB = c/sinC = 2R
Cosine rule: a² = b² + c² – 2bc cosA
Area: ½ab sinC
Arc length: s = rθ , sector area: ½r²θ

正、余弦定理与弧度制下的弧长和扇形面积是连接几何与微积分的桥梁。

Small-angle approximations simplify limits and oscillatory models.

sinθ ≈ θ , cosθ ≈ 1 – ½θ² , tanθ ≈ θ (θ in radians, small)

小角近似用于求极限和简化振动模型，前提是弧度制且角度很小。

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3. Differentiation | 微分

Standard derivatives must be memorised, especially for exponentials, logs and trigonometric functions.

d/dx (xⁿ) = nxⁿ⁻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

基本导数公式必须牢记，尤其是指数、对数和三角函数的导数。

The chain, product and quotient rules handle composite and mixed functions.

Chain: dy/dx = dy/du × du/dx
Product: d/dx(uv) = u’v + uv’
Quotient: d/dx(u/v) = (u’v – uv’) / v²

链式法则、乘法法则和商法则分别处理复合函数与乘除形式的导数。

Parametric and implicit differentiation allow you to differentiate curves not given as y = f(x).

Parametric: dy/dx = (dy/dt) / (dx/dt)
Implicit: differentiate both sides w.r.t. x, treating y as a function, then collect dy/dx terms.

参数方程与隐函数微分法可以处理不以 y = f(x) 形式给出的曲线。

Second derivative and rates of change deepen the analysis of curvature and motion.

d²y/dx² = d/dx(dy/dx)
Connected rates: dA/dt = dA/dr × dr/dt

二阶导数描述凹凸性，相关变化率用于应用题中多个变量的瞬时关系。

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4. Integration | 积分

Indefinite integration is the reverse of differentiation; always add the constant of integration.

∫ 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

不定积分是微分的逆运算，不要忘记加积分常数 C。

Integration by substitution and by parts are the two principal techniques for non-standard integrals.

Substitution: let u = g(x), du = g'(x) dx, transform and integrate.
Parts: ∫ u dv = uv – ∫ v du (guideline: LIATE for choice of u)

换元积分法与分部积分法是非标准积分的两大主要方法。分部积分中通常按 ‘LIATE’ 选 u。

Definite integrals calculate areas and volumes of revolution. Remember to subtract lower limit from upper limit.

Area between curve and x-axis: A = ∫ₐᵇ |y| dx
Volume of revolution about x-axis: V = π ∫ₐᵇ y² dx

定积分计算面积和旋转体体积。绕 x 轴体积公式为 V = π ∫ y² dx。

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