📚 Year 12 AQA Mathematics: Formula & Theorem Quick Reference Handbook | AQA 数学公式定理速查手册
This quick reference handbook collates the essential formulae, theorems and standard results covered in Year 12 AQA Mathematics. It is designed as a convenient study companion for revision and during independent practice. Each section presents key content in English alongside a direct Chinese translation, ensuring accessibility for bilingual learners. Topics span pure mathematics, statistics and mechanics as outlined by the AQA specification for the first year of the A-level course.
本速查手册汇总了 AQA 数学 Year 12 阶段需要掌握的核心公式、定理与标准结果，旨在为复习和自主练习提供便捷的学习伴侣。每个小节均以英文陈述搭配中文译文，方便双语学习者对照理解。内容涵盖 AQA A-level 第一年课程要求的纯数学、统计与力学三大板块。
1. Algebraic Laws & Quadratic Formula | 代数法则与二次公式
The quadratic formula solves any equation of the form ax² + bx + c = 0 (a ≠ 0) and is given by the expression below. The discriminant Δ = b² − 4ac determines the nature of the roots: two distinct real roots when Δ > 0, one repeated real root when Δ = 0, and no real roots when Δ < 0.
二次公式可求解任何形如 ax² + bx + c = 0 (a ≠ 0) 的方程，见下方公式。判别式 Δ = b² − 4ac 决定根的性质：Δ > 0 有两个不等的实根，Δ = 0 有一个重根，Δ < 0 无实根。
x = (−b ± √(b² − 4ac)) / 2a
The factor theorem states that for a polynomial f(x), (x − a) is a factor if and only if f(a) = 0. Remainder theorem: when f(x) is divided by (x − a), the remainder is f(a).
因式定理：多项式 f(x) 有因式 (x − a) 当且仅当 f(a) = 0。余式定理：f(x) 除以 (x − a) 的余数为 f(a)。
Basic index laws include aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, and a⁻ⁿ = 1/aⁿ. The surd rule √a × √b = √(ab) holds for a, b ≥ 0.
基本指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。根式规则 √a × √b = √(ab) 在 a, b ≥ 0 时成立。
2. Coordinate Geometry: Line & Circle | 坐标几何：直线与圆
The distance between two points (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²]. The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). Gradient m of a line through these points is (y₂ − y₁)/(x₂ − x₁).
两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]。中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。过这两点的直线斜率（梯度）m = (y₂ − y₁)/(x₂ − x₁)。
The equation of a straight line can be written as y − y₁ = m(x − x₁), or y = mx + c, where c is the y-intercept. Perpendicular lines satisfy m₁ × m₂ = −1.
直线方程可写为点斜式 y − y₁ = m(x − x₁) 或斜截式 y = mx + c，c 为 y 轴截距。两直线垂直当且仅当斜率之积 m₁ × m₂ = −1。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).
圆心 (a, b)、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (−g, −f)，半径为 √(g² + f² − c)。
3. Sequences, Series & Binomial Expansion | 数列、级数与二项展开
For an arithmetic sequence, the n-th term is uₙ = a + (n − 1)d, and the sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l), where l is the last term.
等差数列的第 n 项为 uₙ = a + (n − 1)d，前 n 项和为 Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
For a geometric sequence, uₙ = arⁿ⁻¹, and the sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. The sum to infinity converges when |r| < 1: S∞ = a/(1 − r).
等比数列第 n 项 uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。当 |r| < 1 时，无穷和收敛：S∞ = a/(1 − r)。
The binomial expansion for rational n negative or fractional, valid for |x| < 1, is (1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + ... For positive integer n, the expansion is finite and coefficients are ⁿCᵣ.
对有理数 n（含负数和分数），在 |x| < 1 时有二项展开 (1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + ... 当 n 为正整数时展开有限，系数为 ⁿCᵣ。
(a + b)ⁿ = aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + … + bⁿ, where ⁿCᵣ = n! / [r!(n − r)!]
4. Trigonometric Ratios & Identities | 三角比与恒等式
In a right-angled triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Radian measure: π rad = 180°. Arc length s = rθ, area of sector A = ½ r²θ for θ in radians.
在直角三角形中，sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。弧度制：π rad = 180°。弧长 s = rθ，扇形面积 A = ½ r²θ（θ 以弧度计）。
Fundamental identities: tan θ = sin θ / cos θ, and sin²θ + cos²θ = 1. Derived forms: sin²θ = 1 − cos²θ, cos²θ = 1 − sin²θ. The sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A.
基本恒等式：tan θ = sin θ / cos θ，sin²θ + cos²θ = 1。派生形式：sin²θ = 1 − cos²θ，cos²θ = 1 − sin²θ。正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A。
Area of a triangle using two sides and included angle: Area = ½ ab sin C. The exact values of sin, cos and tan for 0°, 30°, 45°, 60°, 90° (and their radian equivalents) must be known.
三角形面积（已知两边及其夹角）：面积 = ½ ab sin C。必须熟记 0°、30°、45°、60°、90°（及对应弧度）的 sin、cos、tan 精确值。
5. Exponential & Logarithmic Functions | 指数函数与对数函数
The function y = aⁿ satisfies logₐ y = x. Natural logarithm ln x uses base e. Key logarithm laws: logₐ(xy) = logₐ x + logₐ y; logₐ(x/y) = logₐ x − logₐ y; logₐ(xⁿ) = n logₐ x.
y = aⁿ 等价于 logₐ y = x。自然对数 ln x 以 e 为底。关键对数法则：logₐ(xy) = logₐ x + logₐ y；logₐ(x/y) = logₐ x − logₐ y；logₐ(xⁿ) = n logₐ x。
The derivative of eⁿ is itself, and the derivative of ln x is 1/x. Exponential growth and decay models take the form y = A eᵏⁿ or similar, where k is the continuous growth rate.
eⁿ 的导数仍为自身，ln x 的导数为 1/x。指数增长和衰减模型常表示为 y = A eᵏⁿ 等形式，其中 k 为连续增长率。
The change of base formula: logₐ x = log₆ x / log₆ a. Solving exponential equations often involves taking logs of both sides, e.g., aⁿ = b ⇒ x = logₐ b = ln b / ln a.
换底公式：logₐ x = log₆ x / log₆ a。求解指数方程常通过两边取对数：aⁿ = b ⇒ x = logₐ b = ln b / ln a。
6. Differentiation Rules & Standard Derivatives | 微分法则与标准导数
The derivative of xⁿ is nxⁿ⁻¹ for any real n. Notation: if y = f(x), then the derivative is dy/dx or f'(x). The gradient of a curve at a point equals the value of the derivative at that point. The second derivative d²y/dx² describes concavity.
xⁿ 的导数为 nxⁿ⁻¹（n 为任意实数）。记号：若 y = f(x)，则导数为 dy/dx 或 f'(x)。曲线在某点的切线斜率等于函数在该点的导数值。二阶导数 d²y/dx² 描述曲线的凹凸性。
Chain rule: if y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. Product rule: d/dx (uv) = u dv/dx + v du/dx. Quotient rule: d/dx (u/v) = (v du/dx − u dv/dx) / v².
链式法则：设 y = f(u), u = g(x)，则 dy/dx = dy/du × du/dx。积法则：d/dx (uv) = u dv/dx + v du/dx。商法则：d/dx (u/v) = (v du/dx − u dv/dx) / v²。
Standard derivatives include: d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x, d/dx (eⁿ) = eⁿ, d/dx (ln x) = 1/x. For aⁿ, d/dx (aⁿ) = aⁿ ln a.
常用标准导数：d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x，d/dx (eⁿ) = eⁿ，d/dx (ln x) = 1/x。d/dx (aⁿ) = aⁿ ln a。
7. Integration Basics & Area Under Curve | 积分基础与曲线下面积
Integration reverses differentiation. The indefinite integral of xⁿ (n ≠ −1) is ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, where C is the constant of integration. For n = −1, ∫ 1/x dx = ln|x| + C.
积分是微分的逆运算。xⁿ (n ≠ −1) 的不定积分为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 C 为积分常数。当 n = −1 时，∫ 1/x dx = ln|x| + C。
Definite integral from a to b, ∫ₐᵇ f(x) dx, gives the net area between the curve y = f(x) and the x-axis. Area between a curve and the x-axis is found by integrating the function and applying the limits, taking absolute values where the curve goes below the axis.
定积分 ∫ₐᵇ f(x) dx 给出曲线 y = f(x) 与 x 轴之间的净面积。曲线与 x 轴围成的面积通过对函数积分并代入上下限求得，曲线在 x 轴下方时需取绝对值。
Standard integrals: ∫ cos x dx = sin x + C, ∫ sin x dx = −cos x + C, ∫ sec² x dx = tan x + C, ∫ eⁿ dx = eⁿ + C, ∫ aⁿ dx = aⁿ/ln a + C.
常用标准积分：∫ cos x dx = sin x + C，∫ sin x dx = −cos x + C，∫ sec² x dx = tan x + C，∫ eⁿ dx = eⁿ + C，∫ aⁿ dx = aⁿ/ln a + C。
8. Vectors in Two Dimensions | 二维向量
A vector is a quantity with magnitude and direction. In component form, a = a₁ i + a₂ j. The magnitude (length) of a is |a| = √(a₁² + a₂²). A unit vector in the direction of a is â = a / |a|.
向量是具有大小和方向的量。分量形式：a = a₁ i + a₂ j。向量 a 的大小（模）为 |a| = √(a₁² + a₂²)。沿 a 方向的单位向量是 â = a / |a|。
Vector addition and scalar multiplication: a + b = (a₁ + b₁)i + (a₂ + b₂)j, and k a = k a₁ i + k a₂ j. The scalar (dot) product is a · b = a₁ b₁ + a₂ b₂ = |a||b| cos θ, where θ is the angle between vectors. Two vectors are perpendicular if a · b = 0.
向量加法与数乘：a + b = (a₁ + b₁)i + (a₂ + b₂)j，k a = k a₁ i + k a₂ j。数量积（点积）：a · b = a₁ b₁ + a₂ b₂ = |a||b| cos θ，其中 θ 为两向量夹角。若 a · b = 0，则两向量垂直。
Position vectors describe the location of a point relative to the origin. The vector from point A to point B is AB = b − a. Midpoint M of AB has position vector (a + b)/2.
位置向量描述点相对于原点的位置。由点 A 到点 B 的向量为 AB = b − a。线段 AB 中点 M 的位置向量为 (a + b)/2。
9. Statistical Measures & Probability Distributions | 统计测量与概率分布
Data summary: mean μ = Σx/n (population), sample mean x̄ = Σx/n. Variance σ² = Σ(x − μ)²/n or the formula σ² = Σx²/n − μ². Standard deviation is the square root of variance. Quartiles and interquartile range describe spread.
数据概括：总体均值 μ = Σx/n，样本均值 x̄ = Σx/n。方差 σ² = Σ(x − μ)²/n 或公式 σ² = Σx²/n − μ²。标准差为方差的平方根。四分位数和四分位距用于描述分布。
For a discrete random variable X, expectation E(X) = Σ x P(X = x), and Var(X) = E(X²) − [E(X)]². If X ~ B(n, p), the binomial distribution, then P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, E(X) = np, Var(X) = np(1 − p).
对离散随机变量 X，期望 E(X) = Σ x P(X = x)，方差 Var(X) = E(X²) − [E(X)]²。若 X ~ B(n, p) 服从二项分布，则 P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，E(X) = np，Var(X) = np(1 − p)。
The normal distribution X ~ N(μ, σ²) has bell-shaped curve. Standard normal Z = (X − μ)/σ ~ N(0, 1). Probabilities are found using tables or calculator for the standard normal distribution, and symmetry and total area = 1 are used.
正态分布 X ~ N(μ, σ²) 呈钟形曲线。标准正态分布 Z = (X − μ)/σ ~ N(0, 1)。利用标准正态分布表或计算器求概率，常用对称性和总面积 = 1。
10. Hypothesis Testing (Binomial) | 假设检验（二项分布）
In a binomial hypothesis test, the null hypothesis H₀ states a population parameter (e.g., p = 0.5). The alternative hypothesis H₁ can be one-tailed (p < 0.5 or p > 0.5) or two-tailed (p ≠ 0.5). The test statistic is the number of successes.
在二项假设检验中，原假设 H₀ 给出了总体参数（如 p = 0.5）。备择假设 H₁ 可以是单尾（p < 0.5 或 p > 0.5）或双尾（p ≠ 0.5）。检验统计量为成功次数。
The p-value is the probability of obtaining a result at least as extreme as the observed value, assuming H₀ is true. If p-value ≤ significance level α (commonly 0.05), reject H₀. The critical region is the set of values that lead to rejection.
p 值是在 H₀ 为真的条件下，得到至少与实际观测结果一样极端的结果的概率。若 p 值 ≤ 显著性水平 α（通常 0.05），则拒绝 H₀。拒绝域是导致拒绝 H₀ 的检验统计量的取值集合。
For a binomial test, use cumulative binomial probabilities to find the p-value or determine critical values. Always state the conclusion in context: there is/is not sufficient evidence to support H₁.
对于二项检验，使用累积二项概率求 p 值或确定临界值。务必在结论中结合背景陈述：是否有充分证据支持 H₁。
11. Kinematics: Constant Acceleration Formulae | 运动学：匀加速公式
For motion in a straight line with constant acceleration a, initial velocity u, final velocity v, displacement s, and time t, the SUVAT equations are used. These assume acceleration is constant.
对于恒定加速度 a 的直线运动，初速度为 u，末速度为 v，位移为 s，时间为 t，可使用 SUVAT 公式组。这些公式均假定加速度恒定。
v = u + at
s = ut + ½ at²
s = ½ (u + v)t
v² = u² + 2as
s = vt − ½ at²
Quantities are vectors: assign a positive direction and use negative values for motion in the opposite direction. Displacement is the area under a velocity-time graph; gradient of a displacement-time graph gives velocity; gradient of a velocity-time graph gives acceleration.
这些量是矢量：需设定正方向，反向运动取负值。速度-时间图像下的面积表示位移；位移-时间图像的斜率表示速度；速度-时间图像的斜率表示加速度。
12. Forces, Newton’s Laws & Equilibrium | 力、牛顿定律与平衡
Newton’s First Law: an object remains at rest or in uniform motion unless acted upon by a resultant force. Second Law: F = ma, where F is the resultant force, m the mass, a the acceleration. Third Law: for every action, there is an equal and opposite reaction.
牛顿第一定律：物体将保持静止或匀速直线运动状态，除非受到合外力作用。第二定律：F = ma，F 为合外力，m 为质量，a 为加速度。第三定律：作用力与反作用力大小相等、方向相反。
Weight = mg acts vertically downwards. Normal reaction acts perpendicular to the contact surface. Tension in a light inextensible string is the same throughout. Friction F ≤ μR, where μ is the coefficient of friction and R is the normal reaction. In limiting equilibrium, F = μR.
重力 = mg 竖直向下。法向反作用力垂直于接触面。轻质不可伸长绳中的张力处处相等。摩擦力 F ≤ μR，其中 μ 为摩擦系数，R 为法向反力。在极限平衡时 F = μR。
Resolving forces: split each force into components parallel and perpendicular to a chosen direction. Equilibrium occurs when the resultant force in any direction equals zero, i.e., ΣF = 0. For two perpendicular directions, ΣFₓ = 0 and ΣFᵧ = 0.
力的分解：将各力沿选定方向分解为分力。平衡时任意方向合外力为零，即 ΣF = 0。对两个互相垂直的方向，有 ΣFₓ = 0 和 ΣFᵧ = 0。
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