📚 Year 13 AQA Formula & Theorem Quick Reference Handbook | Year 13 AQA 公式定理速查手册
In Year 13 AQA examinations, mastering essential formulas and theorems across Mathematics and Sciences is crucial. This bilingual quick-reference handbook consolidates the key results needed for success, presented in both English and Chinese for clarity. Use it as a daily study companion to sharpen your recall and deepen understanding.
在 AQA 13 年级考试中，掌握数学和科学的核心公式与定理至关重要。这本双语速查手册汇集了应考所需的关键结论，并用中英文清晰呈现，方便随时查阅。将其作为日常学习伙伴，可帮助强化记忆、加深理解。
1. Algebraic Foundations | 代数基础
The quadratic formula solves ax² + bx + c = 0 for real or complex roots.
二次公式用于求解方程 ax² + bx + c = 0 的实根或复根。
x = (-b ± √(b² – 4ac)) / 2a
Completing the square transforms a quadratic into vertex form: a(x – h)² + k.
配方法将二次式写成顶点式：a(x – h)² + k。
x² + bx = (x + b/2)² – (b/2)²
The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 two real roots, Δ = 0 one repeated root, Δ < 0 no real roots.
判别式 Δ = b² – 4ac 判定根的性质：Δ > 0 两个不等实根，Δ = 0 一个重根，Δ < 0 无实根。
For simultaneous linear equations, the elimination method systematically removes variables.
对于联立线性方程，消元法系统地消去变量。
ax + by = e, cx + dy = f → x = (ed – bf)/(ad – bc), y = (af – ec)/(ad – bc)
Binomial expansion for a positive integer n: (1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + … + xⁿ.
正整数 n 的二项展开式：(1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + … + xⁿ。
2. Trigonometry & Identities | 三角恒等式
The sine rule relates sides and opposite angles in any triangle.
正弦定理适用于任意三角形，边长与对角的正弦成比例。
a/sin A = b/sin B = c/sin C
The cosine rule generalises Pythagoras’ theorem for non-right triangles.
余弦定理是勾股定理在非直角三角形中的推广。
a² = b² + c² – 2bc cos A
Pythagorean identity is the foundation of trigonometric relationships.
勾股恒等式是三角函数关系的基础。
sin²θ + cos²θ ≡ 1
Double-angle formulas expand trigonometric expressions.
倍角公式用于展开三角函数表达式。
sin 2θ ≡ 2 sinθ cosθ, cos 2θ ≡ cos²θ – sin²θ ≡ 2cos²θ – 1 ≡ 1 – 2sin²θ
Radians measure angles; 180° = π rad, so 1 rad ≈ 57.3°.
弧度制计量角度：180° = π 弧度，因此 1 弧度 ≈ 57.3°。
3. Exponentials & Logarithms | 指数与对数
The natural exponential function eˣ is its own derivative and integral.
自然指数函数 eˣ 的导数与积分均为其自身。
d(eˣ)/dx = eˣ, ∫eˣ dx = eˣ + C
Logarithm laws simplify multiplication, division and exponentiation.
对数运算法则简化乘、除和乘方运算。
logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx – logₐy, logₐ(xⁿ) = n logₐx
The natural logarithm ln is the inverse of eˣ: e^(ln x) = x for x > 0.
自然对数 ln 是 eˣ 的反函数：对于 x > 0，e^(ln x) = x。
ln(e) = 1, ln(1) = 0
Exponential growth and decay follow N = N₀e^(kt), where k > 0 for growth, k < 0 for decay.
指数增长与衰减遵循 N = N₀e^(kt)，其中 k > 0 增长，k < 0 衰减。
4. Differentiation | 微分
The power rule is the most basic derivative: d(xⁿ)/dx = nxⁿ⁻¹ for any real n.
幂函数求导法则：d(xⁿ)/dx = nxⁿ⁻¹，对任意实数 n 成立。
The chain rule differentiates composite functions.
链式法则用于复合函数求导。
dy/dx = (dy/du) × (du/dx)
The product rule: d(uv)/dx = u dv/dx + v du/dx.
乘法法则：d(uv)/dx = u dv/dx + v du/dx。
The quotient rule: d(u/v)/dx = (v du/dx – u dv/dx)/v².
除法法则：d(u/v)/dx = (v du/dx – u dv/dx)/v²。
Derivatives of trigonometric functions must be memorised.
三角函数的导数需要熟记。
d(sin x)/dx = cos x, d(cos x)/dx = -sin x, d(tan x)/dx = sec² x
Stationary points occur where dy/dx = 0; use the second derivative to test nature.
驻点满足 dy/dx = 0；利用二阶导数判别极大或极小。
5. Integration | 积分
Integration reverses differentiation: ∫ f'(x) dx = f(x) + C.
积分是微分的逆运算：∫ f'(x) dx = f(x) + C。
The fundamental power rule for integration (n ≠ -1).
积分的基本幂法则（n ≠ -1）。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
Definite integrals calculate area under a curve between limits a and b.
定积分计算曲线下 x = a 到 x = b 之间的面积。
∫ₐᵇ f(x) dx = F(b) – F(a)
Integration by substitution simplifies integrals by changing variables.
换元积分法通过变量代换简化积分。
∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x)
Integration by parts derives from the product rule.
分部积分法源自乘法法则。
∫ u dv = uv – ∫ v du
Area between two curves y = f(x) and y = g(x) from a to b is ∫ |f(x) – g(x)| dx.
两曲线间的面积：∫ₐᵇ |f(x) – g(x)| dx。
6. Sequences & Series | 数列与级数
Arithmetic sequence: uₙ = a + (n-1)d; sum Sₙ = (n/2)[2a + (n-1)d].
等差数列：uₙ = a + (n-1)d；前 n 项和 Sₙ = (n/2)[2a + (n-1)d]。
Geometric sequence: uₙ = arⁿ⁻¹; sum Sₙ = a(1 – rⁿ)/(1 – r) for |r| ≠ 1.
等比数列：uₙ = arⁿ⁻¹；前 n 项和 Sₙ = a(1 – rⁿ)/(1 – r)，|r| ≠ 1。
Sum to infinity for a convergent geometric series: S∞ = a/(1 – r), |r| < 1.
收敛等比级数的无穷和：S∞ = a/(1 – r)，要求 |r| < 1。
Binomial series extends to fractional and negative indices (1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + … valid for |x| < 1.
二项级数可推广至分数及负指数：(1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + …，|x| < 1 时收敛。
7. Vectors & Matrices | 向量与矩阵
A position vector describes a point relative to the origin: r = xi + yj + zk.
位置向量描述点相对于原点的方向：r = xi + yj + zk。
Magnitude of a vector: |v| = √(x² + y² + z²).
向量的模：|v| = √(x² + y² + z²)。
Dot product: a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃.
点积：a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃。
Cross product (3D): a × b = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k.
叉积（三维）：a × b = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k。
For matrices, determinant of 2×2: det([[a,b],[c,d]]) = ad – bc.
2×2 矩阵的行列式：det([[a,b],[c,d]]) = ad – bc。
8. Probability & Statistics | 概率与统计
Basic probability: P(A) = n(A)/n(S) for equally likely outcomes.
基本概率：等可能结果时 P(A) = n(A)/n(S)。
Addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
Conditional probability: P(A|B) = P(A ∩ B)/P(B).
条件概率：P(A|B) = P(A ∩ B)/P(B)。
Binomial distribution X ~ B(n, p): P(X = r) = ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ.
二项分布 X ~ B(n, p)：P(X = r) = ⁿCᵣ pʳ (1-p)ⁿ⁻ʳ。
Mean of binomial: μ = np; variance: σ² = np(1-p).
二项分布的均值 μ = np；方差 σ² = np(1-p)。
Normal distribution X ~ N(μ, σ²) and standardisation Z = (X – μ)/σ.
正态分布 X ~ N(μ, σ²) 及标准化 Z = (X – μ)/σ。
9. Mechanics Essentials | 力学要点
Constant acceleration (suvat) equations for motion in a straight line.
匀加速直线运动（suvat）方程。
v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t
Newton’s second law links force, mass and acceleration: F = ma.
牛顿第二定律将力、质量与加速度联系起来：F = ma。
Momentum: p = mv; impulse = change in momentum = FΔt.
动量 p = mv；冲量 = 动量变化 = FΔt。
Work done by a constant force: W = Fs cosθ.
恒力做功：W = Fs cosθ。
Kinetic energy: Ek = ½mv²; gravitational potential energy: Ep = mgh.
动能：Ek = ½mv²；重力势能：Ep = mgh。
10. Key Theorems & Results | 核心定理与结论
The Fundamental Theorem of Calculus joins differentiation and integration.
微积分基本定理将微分与积分联系起来。
d/dx [∫ₐˣ f(t) dt] = f(x)
Trapezium rule approximates area under a curve using trapezoids.
梯形法则用梯形近似计算曲线下面积。
∫ₐᵇ f(x) dx ≈ ½h[y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], h = (b – a)/n
De Moivre’s theorem for complex numbers: (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ.
棣莫弗定理： (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ。
Euler’s identity connects five fundamental constants: e^(iπ) + 1 = 0.
欧拉恒等式关联五个基本常数：e^(iπ) + 1 = 0。
Remainder theorem: when polynomial f(x) is divided by (x – a), remainder = f(a).
余式定理：多项式 f(x) 除以 (x – a)，余数为 f(a)。
Factor theorem: (x – a) is a factor of f(x) if and only if f(a) = 0.
因式定理：(x – a) 是 f(x) 的因式当且仅当 f(a) = 0。
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