📚 Year 12 Cambridge Mathematics: Vocabulary and Terminology Quick-Memorisation Guide | 剑桥 Year 12 数学：词汇术语速记指南
Mastering the precise vocabulary of Cambridge Year 12 Mathematics is essential for interpreting exam questions correctly and delivering clear solutions. This guide breaks down key terminology across Pure, Statistics, and Mechanics into simple, memorable explanations. Each term is paired with its Chinese equivalent to strengthen bilingual understanding, helping you build confidence in both languages while revising efficiently.
掌握剑桥 Year 12 数学的精准词汇是准确理解考题、提供清晰解答的关键。这份指南将纯数学、统计和力学中的核心术语拆解成简单、易记的解释。每一个术语都与对应的中文解释配对，强化双语理解，帮助你在高效复习的同时，在两种语言环境中都建立信心。
1. Algebraic Expressions | 代数表达式
Variable: A symbol (usually a letter) that represents a number we do not yet know. In 2x + 5, the variable is x. Think of it as a ‘varying’ placeholder that can take different values.
变量： 表示未知数的符号（通常是字母）。在 2x + 5 中，变量是 x。可以把它想象成一个可以取不同值的“可变”占位符。
Constant: A fixed number that does not change. In the expression 3x + 7, the constant is 7. It stays ‘constant’, unaffected by the variable.
常量： 一个不变的固定数字。在表达式 3x + 7 中，常量是 7。它保持不变，不受变量影响。
Coefficient: The number multiplying the variable. In 4x², the coefficient is 4. It tells us the ‘co-pilot’ that scales the variable term.
系数： 乘以变量的数。在 4x² 中，系数是 4。它告诉我们“协同”缩放变量项的数字。
Term: A single mathematical quantity that can be a number, a variable, or a product of numbers and variables. For instance, 5x²y, −3, and z are all individual terms.
项： 单个数学量，可以是一个数、一个变量，或数及变量的乘积。例如，5x²y、−3 和 z 都是单独的项。
Expression: A combination of terms using operations such as addition or subtraction, without an equals sign. 2x² + 3x − 5 is an expression.
表达式： 通过加法或减法等运算组合在一起的项构成，没有等号。2x² + 3x − 5 就是一个表达式。
Equation: A statement that two expressions are equal, containing an equals sign. 2x + 3 = 11 is an equation; solving it finds the value of the variable.
方程： 声明两个表达式相等的陈述，含有等号。2x + 3 = 11 是一个方程；解方程可以求出变量的值。
Identity: An equation that holds true for all values of the variable. The symbol ≡ is often used. For example, x² − 1 ≡ (x − 1)(x + 1) is an identity.
恒等式： 对所有变量值都成立的方程。常用符号 ≡ 表示。例如，x² − 1 ≡ (x − 1)(x + 1) 是一个恒等式。
Factor Theorem: If f(a) = 0 for a polynomial f(x), then (x − a) is a factor of f(x). It is a quick tool to break down cubics and quartics.
因式定理： 对于多项式 f(x)，若 f(a) = 0，则 (x − a) 是 f(x) 的一个因式。它是分解三次、四次多项式的快捷工具。
Remainder Theorem: When a polynomial f(x) is divided by (x − a), the remainder is f(a). This allows rapid evaluation without long division.
余式定理： 当多项式 f(x) 除以 (x − a) 时，余式为 f(a)。这可以避开长除法，快速求值。
2. Functions and Graphs | 函数与图像
Function: A rule that assigns each input exactly one output. Written as f(x) = …, it must pass the vertical line test. Think of a function as a machine that produces a unique output for every input.
函数： 将每个输入恰好对应一个输出的规则。写作 f(x) = …，必须通过垂直线检验。可以把函数想象成一台机器，为每个输入产生唯一的输出。
Domain: The set of all possible input values (x-values) for which the function is defined. For f(x) = √x, the domain is x ≥ 0.
定义域： 函数有定义的所有可能输入值（x 值）的集合。对于 f(x) = √x，定义域是 x ≥ 0。
Range: The set of all possible output values (y-values) the function can produce. The range of f(x) = x² is y ≥ 0.
值域： 函数可以产生的所有可能输出值（y 值）的集合。f(x) = x² 的值域是 y ≥ 0。
One-to-One Function: A function where each output corresponds to exactly one input. It passes both vertical and horizontal line tests and has an inverse that is also a function.
一一对应函数： 每个输出恰好对应一个输入的函数。既能通过垂直线检验，也能通过水平线检验，其反函数也是一个函数。
Inverse Function: Reverses the effect of the original function, swapping x and y. Written as f⁻¹(x). Only one‑to‑one functions have true inverses over their whole domain.
反函数： 逆转原函数的作用，交换 x 与 y。写作 f⁻¹(x)。只有一一对应的函数在其整个定义域上才有真正的反函数。
Composite Function: A function applied inside another, written as f(g(x)) or f∘g(x). Work from the inner function outward.
复合函数： 将一个函数套进另一个函数中，写作 f(g(x)) 或 f∘g(x)。应从内层函数向外层计算。
Modulus Function: Denotes absolute value, written as |x|. It makes any negative input positive, creating a V‑shaped graph with a sharp corner at the origin.
绝对值函数（模函数）： 表示绝对值，写作 |x|。它将任何负输入变为正数，产生一个原点处有尖锐拐角的 V 形图像。
Asymptote: A line that a graph approaches but never touches. Hyperbolas like y = 1/x have horizontal and vertical asymptotes. Think of it as an ‘invisible boundary’.
渐近线： 图像无限靠近但永不触及的直线。像 y = 1/x 这样的双曲线具有水平和垂直渐近线。可将其视为“隐形边界”。
3. Trigonometry | 三角函数
Radian: A unit of angle measurement based on arc length. π radians = 180°. Use radian mode on your calculator for all calculus involving trig functions.
弧度： 基于弧长的角度度量单位。π 弧度 = 180°。对所有涉及三角函数的微积分运算，计算器需设为弧度模式。
Exact Trigonometric Values: Special angles such as 30°, 45°, 60° have neat sine, cosine, and tangent values expressed with surds. Memorise the common triangles to recall them instantly.
精确三角值： 特殊角度如 30°、45°、60° 的正弦、余弦和正切值可用根式表示。熟记常用三角形，以便立即回忆。
CAST Diagram: A memory aid showing which trig ratios are positive in each quadrant. Starting from top‑right: Cosine (C), All (A), Sine (S), Tangent (T). Tells you signs of sin, cos, tan.
CAST 图： 一种辅助记忆各象限中哪些三角比为正的工具。从右上角开始逆时针：余弦 (C)、全正 (A)、正弦 (S)、正切 (T)。可判断 sin、cos、tan 的符号。
Trigonometric Identity (Pythagorean): sin²θ + cos²θ ≡ 1. This identity links sine and cosine, allowing you to convert between them. Also useful: tan θ ≡ sin θ / cos θ.
三角恒等式（毕达哥拉斯）： sin²θ + cos²θ ≡ 1。该恒等式联结了正弦与余弦，可实现相互转换。同时常用：tan θ ≡ sin θ / cos θ。
Inverse Trig Functions: arcsin, arccos, arctan (or sin⁻¹, cos⁻¹, tan⁻¹). They return an angle given a ratio, but beware of restricted ranges – they only give principal values.
反三角函数： arcsin、arccos、arctan（或 sin⁻¹、cos⁻¹、tan⁻¹）。给定比值时返回一个角度，但注意其受限的值域——它们只给出主值。
4. Exponentials and Logarithms | 指数与对数
Exponential Function: f(x) = eˣ, where e ≈ 2.71828. Its graph grows rapidly, and its derivative is itself. It models continuous growth and decay.
指数函数： f(x) = eˣ，其中 e ≈ 2.71828。其图像快速增长，导函数等于自身。它模拟了连续的增长与衰减。
Natural Logarithm (ln): The inverse of eˣ, written as ln x. So ln(eˣ) = x and eˡⁿˣ = x. The natural log answers: ‘to what power must e be raised to get x?’
自然对数 (ln)： eˣ 的反函数，写作 ln x。因此 ln(eˣ) = x 且 eˡⁿˣ = x。自然对数回答：“e 的多少次方等于 x？”
Laws of Logarithms: logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx − logₐy; logₐxⁿ = n logₐx. These allow you to break products into sums and pull down exponents.
对数运算法则： logₐ(xy) = logₐx + logₐy；logₐ(x/y) = logₐx − logₐy；logₐxⁿ = n logₐx。可将乘积拆为和，并将指数拉到前面。
Change of Base: logₐb = logₓb / logₓa. Use this to evaluate logs with unfamiliar bases on your calculator, often converting to base 10 or e.
换底公式： logₐb = logₓb / logₓa。用它将不熟悉底数的对数在计算器上求值，通常换成以 10 或 e 为底。
5. Introduction to Calculus | 微积分入门
Derivative: The rate at which a function changes, written as f'(x) or dy/dx. It gives the gradient of the tangent at any point and reveals instantaneous change.
导数： 函数变化的速度，写作 f'(x) 或 dy/dx。它给出了任意点处切线的斜率，揭示了瞬间变化率。
Differentiation from First Principles: Uses the limit definition f'(x) = limₕ→₀ [f(x+h)−f(x)]/h. It proves why the power rule works and is the foundation of calculus.
第一性原理求导： 使用极限定义 f'(x) = limₕ→₀ [f(x+h)−f(x)]/h。它证明了幂法则的原理，是微积分的基础。
Power Rule: If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. A quick shortcut that works for any real exponent n.
幂法则： 若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹。对任何实数指数 n 都适用的快捷公式。
Stationary Point: A point where f'(x) = 0, meaning the gradient is zero. Could be a local maximum, local minimum, or point of inflection. Use the second derivative or gradient sign test to classify.
驻点： f'(x) = 0 的点，意味着梯度为零。可能是局部极大值、局部极小值或拐点。使用二阶导数或梯度符号检验进行分类。
Second Derivative: Denoted f”(x) or d²y/dx². It tells you the rate of change of the gradient, describing concavity and helping determine the nature of stationary points.
二阶导数： 记作 f”(x) 或 d²y/dx²。它表明梯度的变化率，描述凹凸性，有助于确定驻点的性质。
Indefinite Integral: The antiderivative, written as ∫ f(x) dx = F(x) + C. It reverses differentiation, and the constant C accounts for the unknown initial value.
不定积分： 反导数，写作 ∫ f(x) dx = F(x) + C。它逆转求导过程，常数 C 用于表示未知的初始值。
Definite Integral: Evaluates the net area under a curve between limits a and b: ∫ₐᵇ f(x) dx = F(b) − F(a). It can be negative if the curve lies below the x‑axis.
定积分： 计算曲线下方在区间 a 到 b 的净面积：∫ₐᵇ f(x) dx = F(b) − F(a)。若曲线在 x 轴下方，结果可为负值。
6. Vectors | 向量
Vector: A quantity with both magnitude and direction. Represented by a directed line segment or column notation. Velocity and force are vectors; speed and mass are scalars.
向量： 既有大小又有方向的量。用有向线段或列向量表示。速度和力是向量；速率和质量是标量。
Magnitude: The length or size of a vector, found using Pythagoras’ theorem: |v| = √(x² + y²) in 2D. It is always non‑negative.
模（大小）： 向量的长度或大小，通过勾股定理求得：在二维中 |v| = √(x² + y²)。总是非负值。
Unit Vector: A vector with magnitude 1, often used to indicate direction. The standard basis vectors are i (along x‑axis) and j (along y‑axis). Any vector can be written as xi + yj.
单位向量： 模为 1 的向量，常用来指示方向。标准基向量是 i（沿 x 轴）和 j（沿 y 轴）。任何向量都可写为 xi + yj。
Position Vector: A vector that starts at the origin and ends at a given point. It pinpoints location. For point P(3,4), the position vector is 3i + 4j.
位置向量： 以原点为起点、给定点为终点的向量。它标定位置。对于点 P(3,4)，位置向量是 3i + 4j。
Dot Product: a·b = |a||b| cos θ. It multiplies vectors to give a scalar, useful for finding angles and checking perpendicularity. If a·b = 0, the vectors are orthogonal.
点积（数量积）： a·b = |a||b| cos θ。将向量相乘得到一个标量，用于求角度和检验垂直性。若 a·b = 0，则向量正交。
7. Sequences and Series | 序列与级数
Arithmetic Sequence: A list of numbers where each term is obtained by adding a constant difference d to the previous term. E.g., 3, 7, 11, 15,… with d = 4.
等差数列： 每一项通过在前一项上加一个常量公差 d 得到的数列。例如 3, 7, 11, 15,…，d = 4。
Geometric Sequence: A sequence where each term is multiplied by a constant ratio r. E.g., 2, 6, 18, 54,… with r = 3.
等比数列： 每一项乘以一个常量公比 r 得到的数列。例如 2, 6, 18, 54,…，r = 3。
nth Term (Arithmetic): uₙ = a + (n−1)d, where a is the first term. Quickly generates any term without listing all preceding ones.
第 n 项（等差）： uₙ = a + (n−1)d，其中 a 是首项。无需列出所有前面的项即可快速生成任一项。
Sum of Arithmetic Series: Sₙ = n/2 [2a + (n−1)d] or n/2 (a + l), where l is the last term. This formula averages the first and last term.
等差数列求和： Sₙ = n/2 [2a + (n−1)d] 或 n/2 (a + l)，其中 l 是末项。这个公式实际上是首项与末项的平均值乘以项数。
Sum of Geometric Series: Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1. It handles rapid growth or decay sequences efficiently.
等比数列求和： 当 r ≠ 1 时，Sₙ = a(1−rⁿ)/(1−r)。能高效处理快速增长或衰减的数列。
Sum to Infinity: For a convergent geometric series with |r| < 1, S∞ = a/(1−r). The series approaches a finite limit even though infinitely many terms are added.
无穷级数和： 对于收敛的等比级数，|r| < 1 时，S∞ = a/(1−r)。尽管有无穷多项相加，级数会趋近于一个有限极限。
8. Statistical Terminology | 统计学术语
Mean: The arithmetic average, found by summing all data values and dividing by the total number of items. Sensitive to outliers.
平均数（均值）： 算术平均数，通过所有数据值之和除以数据总个数求得。易受极端值影响。
Median: The middle value when data are ordered. It splits the dataset into two equal halves and is resistant to outliers.
中位数： 数据按顺序排列后的中间值。将数据集平分为两半，不受极端值影响。
Mode: The value that occurs most frequently. A dataset can be bimodal or have no mode. Particularly useful for categorical data.
众数： 出现频率最高的值。数据集可以是双众数或无众数。对分类数据尤其有用。
Standard Deviation: Measures the spread of data around the mean. A small standard deviation means data points are tightly clustered; a large one indicates wide dispersion. Square of it is variance.
标准差： 衡量数据在均值周围的分散程度。标准差小意味着数据点紧密聚集；大标准差则表明分布宽广。它的平方就是方差。
Interquartile Range (IQR): IQR = Q₃ − Q₁, the range of the middle 50% of data. It is robust against outliers and often used to identify them (e.g. values below Q₁ − 1.5×IQR).
四分位距 (IQR)： IQR = Q₃ − Q₁，即中间 50% 数据的范围。对异常值稳健，常用来识别异常值（如低于 Q₁ − 1.5×IQR 的值）。
Histogram & Frequency Density: In a histogram, area represents frequency. Frequency density = frequency ÷ class width. This corrects for unequal class widths when drawing bars.
直方图与频率密度： 在直方图中，面积代表频数。频率密度 = 频数 ÷ 组距。当组距不相等时，用此调整柱高绘制直方图。
9. Probability | 概率
Sample Space: The set of all possible outcomes of an experiment. Usually denoted S. For a fair coin, S = {Heads, Tails}.
样本空间： 试验所有可能结果的集合。通常记作 S。对于一枚均匀硬币，S = {正面, 反面}。
Event: A subset of the sample space – a specific outcome or group of outcomes. ‘Rolling an even number’ on a die is an event with outcomes {2,4,6}.
事件： 样本空间的一个子集——一个或一组特定结果。掷骰子时“掷出偶数”是一个事件，对应结果 {2,4,6}。
Independent Events: Two events where one occurring does not affect the probability of the other. For independent A and B, P(A and B) = P(A) × P(B).
独立事件： 两个事件中一个的发生不影响另一个的概率。若 A 和 B 独立，则 P(A 且 B) = P(A) × P(B)。
Mutually Exclusive: Events that cannot happen at the same time. P(A or B) = P(A) + P(B). There is no overlap in a Venn diagram.
互斥事件： 不可能同时发生的事件。P(A 或 B) = P(A) + P(B)。在文氏图中没有交集。
Conditional Probability: The probability of event A given that B has already occurred: P(A|B) = P(A and B) / P(B). It narrows the sample space to B.
条件概率： 在事件 B 已发生的条件下事件 A 的概率：P(A|B) = P(A 且 B) / P(B)。它将样本空间缩小到 B。
Tree Diagram: A branching diagram showing all possible outcomes and their probabilities. Multiply along branches for successive events; add terminal paths for combined probabilities.
树状图： 展示所有可能结果及其概率的分支图。沿分支相乘可得连续事件概率；将终端路径相加可得组合概率。
10. Basic Mechanics | 基础力学
Displacement: A vector describing the change in position from a starting point. It has direction and magnitude; distance, on the other hand, is scalar.
位移： 描述从起点出发位置变化的向量。具有方向和大小；而路程是标量。
Velocity: The rate of change of displacement with respect to time. It is a vector. Speed is the magnitude of velocity.
速度： 位移随时间的变化率。它是向量。速率是速度的大小。
Acceleration: The rate of change of velocity with respect to time. Constant acceleration leads to the SUVAT equations. Measured in m s⁻².
加速度： 速度随时间的变化率。恒定加速度导出 SUVAT 方程。单位为 m s⁻²。
SUVAT Equations: Five equations linking s (displacement), u (initial velocity), v (final velocity), a (acceleration), t (time). e.g. v = u + at, s = ut + ½at². Must be used when acceleration is constant.
SUVAT 方程： 连接 s（位移）、u（初速度）、v（末速度）、a（加速度）、t（时间）的五个方程
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