Year 13 AQA Formula & Theorem Quick Reference Handbook | Year 13 AQA 公式定理速查手册

📚 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

e 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。

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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°。

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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。

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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。

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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。

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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。

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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。

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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。

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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。

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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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