📚 Year 13 CIE Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 CIE 数学：公式定理速查手册
This handbook brings together the essential formulas, identities, and theorems you need for Year 13 CIE Mathematics. Use it for last-minute revision or to check your recall of key results across Pure Mathematics, Mechanics, and Statistics.
本手册汇集了 Year 13 CIE 数学所需的关键公式、恒等式与定理，适合考前快速回顾，也可用于检查纯数学、力学与统计核心结论的掌握情况。
1. Algebraic Identities & Factorisation | 代数恒等式与因式分解
Quadratic expansion and factorisation form the backbone of algebraic manipulation. Knowing these identities by heart saves time in exams.
二次项展开与因式分解是代数运算的基础，牢记这些恒等式能在考试中节省大量时间。
(a ± b)² = a² ± 2ab + b²
a² − b² = (a + b)(a − b)
The cube of a binomial appears frequently in series expansion and polynomial manipulation.
二项式的立方在级数展开和多项式处理中经常出现。
(a ± b)³ = a³ ± 3a²b + 3ab² ± b³
For solving quadratic equations ax² + bx + c = 0, the quadratic formula is indispensable.
求解二次方程 ax² + bx + c = 0 时，求根公式必不可少。
x = [−b ± √b² − 4ac] / (2a)
Δ = b² − 4ac determines the nature of the roots: two real, one repeated, or no real roots.
记住判别式 Δ = b² − 4ac 决定根的性质：两个不等实根、一个重根或无实根。
2. Exponentials & Logarithms | 指数与对数
The laws of exponents and logarithms are vital for solving equations and modelling growth processes.
指数与对数的运算法则是解方程和模拟增长过程的关键。
aˣ × aʸ = aˣ⁺ʸ
aˣ ÷ aʸ = aˣ⁻ʸ
(aˣ)ʸ = aˣʸ
The logarithmic identities mirror the exponential ones and are often tested in CIE P3.
对数恒等式与指数法则互相呼应，在 CIE P3 考卷中经常出现。
logₐ(xy) = logₐ x + logₐ y
logₐ(x/y) = logₐ x − logₐ y
logₐ(xⁿ) = n logₐ x
Change of base formula is essential: logₐ b = (log_c b)/(log_c a).
换底公式非常重要：logₐ b = (log_c b)/(log_c a)。
The natural logarithm ln x = logₑ x and its relationship with eˣ are central to calculus.
自然对数 ln x = logₑ x 及其与 eˣ 的关系是微积分的核心。
eln x = x and ln(eˣ) = x
3. Trigonometry | 三角学
Trigonometric identities are divided into Pythagorean, compound angle, double angle, and factor formulae. CIE P3 requires fluent use of all.
三角恒等式分为勾股恒等式、和角公式、倍角公式与和差化积。CIE P3 要求能熟练运用全部公式。
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Compound angle formulae:
和角公式：
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 identities reduce the degree of trigonometric equations.
倍角公式常用来降低三角方程的次数。
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
tan 2θ = (2 tan θ) / (1 − tan²θ)
The factor formulae are useful for integrating products of sine and cosine.
和差化积公式在积分正弦与余弦的乘积时非常有用。
sin P + sin Q = 2 sin[(P+Q)/2] cos[(P−Q)/2]
sin P − sin Q = 2 cos[(P+Q)/2] sin[(P−Q)/2]
cos P + cos Q = 2 cos[(P+Q)/2] cos[(P−Q)/2]
cos P − cos Q = −2 sin[(P+Q)/2] sin[(P−Q)/2]
4. Differentiation | 微分
Differentiation rules enable you to find gradients, rates of change, and stationary points. The following standard forms must be memorised.
微分法则用于求梯度、变化率和驻点。以下标准形式必须熟记。
| f(x) | f'(x) |
| xⁿ | n xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x).
链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。
Product rule: d/dx (u v) = u' v + u v'.
乘积法则：d/dx (u v) = u' v + u v'。
Quotient rule: d/dx (u/v) = (u' v − u v') / v².
商法则：d/dx (u/v) = (u' v − u v') / v²。
Implicit differentiation is used when y is not explicitly expressed in terms of x.
隐函数微分适用于 y 没有明确表示成 x 的函数的情况。
5. Integration | 积分
Integration is the reverse of differentiation and is used for finding areas, volumes, and solving differential equations.
积分是微分的逆运算，用于求面积、体积以及解微分方程。
| f(x) | ∫ f(x) dx |
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln |x| + C |
| eˣ | eˣ + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| sec² x | tan x + C |
area between a curve and the x-axis is given by ∫ₐᵇ y dx, using the limits of integration.
对于定积分，曲线与 x 轴之间的面积通过 ∫ₐᵇ y dx 给出，使用积分上下限。
Integration by substitution and by parts are key techniques in CIE P3.
换元积分法与分部积分法是 CIE P3 的核心技巧。
∫ u (dv/dx) dx = u v − ∫ v (du/dx) dx
approximate integration): ∫ₐᵇ y dx ≈ ½ h [y₀ + 2(y₁+y₂+...+yₙ₋₁) + yₙ], where h = (b−a)/n.
梯形法则（近似积分）：∫ₐᵇ y dx ≈ ½ h [y₀ + 2(y₁+y₂+...+yₙ₋₁) + yₙ]，其中 h = (b−a)/n。
6. Differential Equations | 微分方程
Separating variables is the most common method for solving first-order differential equations.
分离变量法是求解一阶微分方程最常用的方法。
If dy/dx = g(x)h(y), then ∫ (1/h(y)) dy = ∫ g(x) dx
Forming a differential equation from a word problem involves expressing a rate of change in terms of the variable quantities.
由应用题建立微分方程，需要将变化率用变量表示出来。
Exponential growth and decay models follow the form dN/dt = k N, leading to N = N₀ ekt.
指数增长与衰减模型遵循 dN/dt = k N，解得 N = N₀ ekt。
For logistic growth or limited growth models, you often encounter dN/dt = k(N₀ − N).
对于逻辑增长或有限增长模型，常遇到 dN/dt = k(N₀ − N)。
7. Vectors | 向量
Vector operations are heavily examined in Pure 3, covering dot product, angles, and lines.
向量运算在 P3 中是常考内容，涉及点积、夹角与直线方程。
The magnitude of a vector a = xi + yj + zk is |a| = √x² + y² + z².
向量 a = xi + yj + zk 的模为 |a| = √x² + y² + z²。
a ⋅ b = x₁x₂ + y₁y₂ + z₁z₂ = |a||b| cos θ
The angle between two vectors can be found using cos θ = (a ⋅ b) / (|a| |b|).
两向量之间的夹角可通过 cos θ = (a ⋅ b) / (|a| |b|) 求得。
in vector form: r = a + t b, where a is a point on the line and b is the direction vector.
直线方程的向量形式：r = a + t b，其中 a 是直线上一点，b 是方向向量。
Two lines intersect if there exists t and s such that a₁ + t b₁ = a₂ + s b₂.
两直线相交当存在 t 和 s 满足 a₁ + t b₁ = a₂ + s b₂。
To find the point of intersection of two lines, solve the parametric equations simultaneously.
为求两直线交点，需联立参数方程求解。
8. Complex Numbers | 复数
Complex numbers extend the real number system and are essential for solving certain polynomials and representing transformations.
复数扩展了实数系统，对于解某些多项式以及表示变换至关重要。
z = x + iy, i² = −1
The modulus |z| = √x² + y² and argument arg(z) = θ = tan⁻¹(y/x) (with quadrant adjustments).
模长 |z| = √x² + y²，辐角 arg(z) = θ = tan⁻¹(y/x)（需要考虑象限）。
Polar form: z = r(cos θ + i sin θ) = r eiθ.
极坐标形式：z = r(cos θ + i sin θ) = r eiθ。
De Moivre's theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ.
棣莫弗定理：(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。
This is particularly powerful for finding powers and roots of complex numbers.
该定理在求复数的幂与根时特别有效。
The nth roots of a complex number z₀ are given by z₀^(1/n) = r1/n cis((θ+2kπ)/n), k = 0,1,...,n−1.
复数 z₀ 的 n 次方根为 z₀^(1/n) = r1/n cis((θ+2kπ)/n)，k = 0,1,...,n−1。
9. Mechanics Formulas | 力学公式
For CIE Mechanics (typically M1 or M2), the following formulas form the foundation of kinematics and dynamics.
在 CIE 力学（通常涉及 M1 或 M2）中，以下公式是运动学与动力学的基础。
Constant acceleration equations (SUVAT):
匀加速运动方程 (SUVAT)：
- v = u + a t
- s = u t + ½ a t²
- s = ½ (u + v) t
- v² = u² + 2 a s
- s = v t − ½ a t²
Newton's second law: F = m a.
牛顿第二定律：F = m a。
ction: F_friction ≤ μ R, where R is the normal reaction and μ the coefficient of friction.
摩擦力：F_friction ≤ μ R，其中 R 为法向反作用力，μ 为摩擦系数。
e by a constant force: W = F d cos θ, where θ is the angle between force and displacement.
恒力做功：W = F d cos θ，θ 为力与位移的夹角。
Power: P = F v, average power = work / time.
功率：P = F v，平均功率 = 功/时间。
Momentum: p = m v, Impulse = change in momentum = F Δt.
动量：p = m v，冲量 = 动量的变化量 = F Δt。
10. Statistics Formulas | 统计公式
Key formulas from S1 (and S2) for CIE include measures of central tendency, probability, and the normal distribution.
CIE S1（及 S2）的关键公式涵盖集中趋势度量、概率和正态分布。
Mean for grouped data: x̄ = Σ f x / Σ f.
分组数据的均值：x̄ = Σ f x / Σ f。
Variance: σ² = Σ f (x − x̄)² / Σ f = (Σ f x² / Σ f) − x̄².
方差：σ² = Σ f (x − x̄)² / Σ f = (Σ f x² / Σ f) − x̄²。
Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
概率加法公式：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。
For independent events, P(A ∩ B) = P(A) P(B).
对于独立事件，P(A ∩ B) = P(A) P(B)。
Permutations: nPr = n! / (n−r)!, Combinations: nCr = n! / [r! (n−r)!].
排列：nPr = n! / (n−r)!，组合：nCr = n! / [r! (n−r)!]。
Normal distribution: Z = (X − μ) / σ ~ N(0,1).
正态分布：Z = (X − μ) / σ ~ N(0,1)。
For binomial distribution X ~ B(n, p), mean = n p, variance = n p (1−p).
二项分布 X ~ B(n, p)，均值 = n p，方差 = n p (1−p)。
Continuity correction: when approximating binomial with normal, add or subtract 0.5 appropriately.
连续性校正：用正态分布近似二项分布时，适当加减 0.5。
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