📚 Year 12 OCR Maths: Vocabulary and Key Terms Quick Memorisation Guide | OCR 数学关键术语速记指南
In Year 12 OCR Mathematics, a firm grip on terminology turns abstract ideas into usable tools. This guide pairs every must-know term with a memory shortcut, covering Pure, Statistics and Mechanics.
在OCR Year 12数学中，掌握术语能将抽象概念转化为解题利器。本指南为纯数学、统计与力学中的必背术语逐一配上记忆捷径。
1. Core Notation & Operators | 核心符号与运算符
The summation symbol Σ (capital sigma) tells you to add a sequence of terms. For instance, Σ xᵢ from i=1 to n means x₁ + x₂ + … + xₙ. Think of the stretched M shape as ‘Many terms Merged’.
求和符号 Σ 表示把一串项加起来，如 Σ xᵢ (i=1 到 n) 就是 x₁ + x₂ + … + xₙ。把 Σ 想象成拉长的 M，代表“Many Merged”。
The implication arrow ⇒ reads as ‘implies’ or ‘leads to’. If P ⇒ Q, then whenever P is true, Q must be true. The converse Q ⇒ P is a separate statement and often false.
蕴含箭头 ⇒ 读作“蕴含”。P ⇒ Q 表示 P 真时 Q 必真。其逆命题 Q ⇒ P 是独立断言，经常不成立。单箭头像单向的推理路径，记住不要擅自回头。
The factorial n! multiplies all whole numbers from n down to 1. 5! = 5 × 4 × 3 × 2 × 1 = 120. The exclamation mark hints at how fast the product grows.
阶乘 n! 是从 n 乘到 1 的连乘积，如 5! = 120。感叹号暗示结果会“惊叹般”变大。
The identity symbol ≡ means two expressions are equivalent for all values of the variable, e.g. (x+1)² ≡ x² + 2x + 1. The three bars remind you of ‘exactly equal, always’.
恒等号 ≡ 表示两边对所有变量取值都相等，如 (x+1)² ≡ x²+2x+1。三道横杠可记作“永远完全相等”。
2. Algebra & Polynomials | 代数与多项式
A polynomial is an expression like 3x⁴ − 2x² + 5. Its degree is the highest power of x, here 4. The word ‘degree’ shares its root with ‘de-gree’ – the top step of the power ladder.
多项式是如 3x⁴ − 2x² + 5 的式子，其次数degree是最高次幂 4。把“degree”联想成阶梯的最高一级。
The factor theorem states that (x − a) is a factor of polynomial f(x) exactly when f(a) = 0. You test a value; if the function outputs zero, you have captured a root. Link: ‘factor’ and ‘zero’ are two sides of the same coin.
因式定理：(x − a) 是 f(x) 的因式当且仅当 f(a)=0。代入 a 得 0 就找到根和因式，记牢“因式为零，零即是因”。
The discriminant Δ = b² − 4ac from the quadratic ax² + bx + c tells you about roots: Δ > 0 gives two real distinct roots, Δ = 0 one repeated root, Δ < 0 no real roots. Visualise Δ as a 'detective' that decides how many times the graph crosses the x-axis.
判别式 Δ = b² − 4ac（来自二次式 ax²+bx+c）判断根的性质：Δ > 0 两个不等实根，Δ = 0 重根，Δ < 0 无实根。把 Δ 想成 “Detective” 侦探，它调查曲线与 x 轴的交点数量。
The remainder theorem: when f(x) is divided by (x − a), the remainder is f(a). If you forget, link ‘remainder’ to ‘evaluation’: the remainder is what’s left when you plug in a.
余数定理：f(x) 除以 (x − a) 的余数等于 f(a)。记忆口诀：“代入 a，余数抓”。
3. Functions & Graphs | 函数与图像
A function f : x ↦ f(x) maps each input to exactly one output. If an x-value could produce two y-values, it is not a function. Think of a vending machine: one button gives one snack.
函数 f : x ↦ f(x) 把每个输入唯一地映射到一个输出。一个 x 给多个 y 就不是函数。可以类比自动售货机：一个按钮只掉出一种零食。
The domain is the set of allowed inputs, and the range is the set of all outputs that can appear. Domain comes before range just as D comes before R in the alphabet.
定义域 domain 是允许的输入集，值域 range 是可能的所有输出集。字母表里 D 在 R 前，对应先输入后输出。
An asymptote is a line that a graph approaches but never touches. Visualise a hyperbola chasing its asymptotes forever. Associate ‘asymptote’ with ‘asymptomatic’ – both have something that never fully appears.
渐近线 asymptote 是图形无限接近但永不相交的直线。想象双曲线永远追着渐近线跑。把 asymptote 与 asymptomatic (无症状) 联系——都是“永远不现身”的感觉。
Graph transformations: f(x + a) shifts left by a, f(x) + a lifts up by a, f(ax) squeezes horizontally by factor 1/a. The ‘inside the bracket’ change always affects x oppositely. Sing ‘inside opposite, outside same’.
图象变换：f(x+a) 左移 a，f(x)+a 上移 a，f(ax) 水平缩放 1/a 倍。口诀：“括号内相反，括号外照常”。
4. Differentiation | 微分
The derivative f'(x) or dy/dx measures the gradient of a curve at a point. It is the limit of (f(x+h) − f(x))/h as h → 0. ‘Derivative’ sounds like ‘derived slope’ – the slope derived from the original curve.
导数 f'(x) 或 dy/dx 衡量曲线在某点的梯度，是差商的极限。Derivative 可联想为“derived slope”衍生出的斜率。
A stationary point occurs where f'(x) = 0. It can be a local maximum, minimum, or point of inflection. Use a sign table to classify. Recall ‘stationary’ = slope zero, like a stationary car.
驻点 stationary point 满足 f'(x)=0，可能是极大、极小或拐点。用导数符号表判断。把 stationary 记为“静止”——斜率为零，如同静止的车。
The chain rule: dy/dx = dy/du × du/dx. It lets you differentiate composite functions like sin(2x). Imagine peeling layers: ‘outer derivative times inner derivative’.
链式法则：dy/dx = dy/du × du/dx，用于 sin(2x) 等复合函数。像剥洋葱：“外导乘内导”。
The product rule: if y = u v, then dy/dx = u’ v + u v’. Memorise ‘first prime times second plus first times second prime’. Quotient rule follows a similar pattern.
积法则：乘积 u v 的导数是 u’ v + u v’。背诵口诀：“前导后不导加前不导后导”。商法则同样可记。
5. Integration | 积分
Integration is the reverse of differentiation. The indefinite integral ∫ f(x) dx yields a family of functions plus a constant C. Think ‘integrate = anti-derivative + C’.
积分是微分的逆运算。不定积分 ∫ f(x) dx 得到一族原函数加常数 C。记作“积分 = 反导数 + C”。
The definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve and the x‑axis from x = a to x = b. The limit notation reminds you of a sum ∑ that becomes continuous.
定积分 ∫ₐᵇ f(x) dx 表示曲线与 x 轴在区间 [a,b] 上的带号面积。其符号由求和的 Σ 拉长而来，说明积分是连续求和。
Finding the area under a curve: areas above the axis count positive, below negative. Use separate integrals when the curve crosses the axis. Always sketch first to avoid sign errors.
求曲线下方面积时，x轴上方面积为正，下方为负。曲线穿过轴时要分段积分。先画草图，避免符号错误。
The fundamental theorem of calculus links differentiation and integration: if F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a). This is the ‘evaluate and subtract’ shortcut.
微积分基本定理连接微分与积分：若 F'(x)=f(x)，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。它就是“代入相减”的捷径。
6. Exponentials & Logarithms | 指数与对数
The function eˣ is the exponential function where the gradient equals the value at every point. The constant e ≈ 2.718. ‘e’ is the unique base for a function that is its own derivative.
指数函数 eˣ 满足在每一点的导数等于函数值本身，e ≈ 2.718。e 可以记为“exponential eigenfunction (特征函数)”，其导数就是自己。
A logarithm logₐ x answers the power to which a must be raised to get x. The natural log ln x is logₑ x. The log undoes the exponential: ln(eˣ) = x.
对数 logₐ x 是问 a 的多少次幂等于 x。自然对数 ln x 即 logₑ x。对数和指数互为逆运算：ln(eˣ)=x。
Three log laws: log(ab) = log a + log b, log(a/b) = log a − log b, log(aᵏ) = k log a. Sing the jingle: ‘multiply in, add out; divide in, subtract out; power in, multiply out’.
三个对数律：乘变加、除变减、幂变乘。口诀：“积化加，商化减，指数提到前”。
Growth models P = P₀ e^(kt) describe continuous growth (k > 0) or decay (k < 0). 'k positive, plus people; k negative, leaving party'.
增长模型 P = P₀ e^(kt)，k>0 增长，k<0 衰减。联想：k 正，人口增多；k 负，聚会散场。
7. Trigonometry | 三角学
Angles in OCR are measured in radians: π rad = 180°. Radian is length of arc divided by radius. π rad = 180° – the half-circle link helps you convert.
OCR 中角度用弧度 radian：π rad = 180°。弧度是弧长除以半径。用半圆 π=180° 作为转换基点。
The CAST diagram shows which trig ratios are positive in each quadrant: Cosine positive, All positive, Sine positive, Tangent positive going anticlockwise from the fourth quadrant. Remember the rhyme ‘All Students Take Calculus’ or ‘CAST’ itself.
CAST 图表示各象限正的三角比：第四象限 Cos 正，第一 All 正，第二 Sin 正，第三 Tan 正。口诀“CAST”或“All Students Take Coffee”。
Key identities: sin²θ + cos²θ ≡ 1; tanθ ≡ sinθ/cosθ. The first is Pythagoras on the unit circle. See sin² + cos² = 1² – the radius squared equals 1.
核心恒等式 sin²θ + cos²θ ≡ 1 和 tanθ ≡ sinθ/cosθ。前者是单位圆上的勾股定理：对边平方加邻边平方等于 1²。
Reciprocal functions: secθ = 1/cosθ, cosecθ = 1/sinθ, cotθ = 1/tanθ. The third letter hints: Sec comes from Cos (but reciprocal), Cosec from Sin, Cot from Tan. Notice the ‘co’ swapping rule.
倒数三角函数：secθ = 1/cosθ, cosecθ = 1/sinθ, cotθ = 1/tanθ。第三个字母对应原函数：sec 对应 cos，cosec 对应 sin，cot 对应 tan。
8. Vectors | 向量
A vector has both magnitude and direction, written as column (x y) or using i, j notation. Displacement, velocity and force are vectors. ‘Vector’ sounds like ‘director’ – both point somewhere.
向量 vector 有大小和方向，可用列向量 (x y) 或 i, j 表示。位移、速度和力都是向量。把 vector 联想成“导演 direct-or”——都有指向。
The magnitude |v| of vector v = a i + b j is √(a² + b²). It is the length of the arrow, found by Pythagoras. Think ‘magnitude = distance from origin’.
向量 v = a i + b j 的模 magnitude |v| = √(a²+b²)，即箭头的长度，由勾股定理得出。记住“模就是原点到点的距离”。
A unit vector has magnitude 1 and shows direction only. Divide a vector by its magnitude to obtain a unit vector: û = v/|v|. The hat symbol (û) reads ‘hat’, like a direction cap.
单位向量 unit vector 的模为 1，只表示方向。任一向量除以其模即得单位向量 û = v/|v|。û 上面的小帽子提示“方向帽”。
The scalar (dot) product v · w = |v||w| cos θ. It gives a number, not a vector. When dot product is zero, vectors are perpendicular. ‘Dot gives a number, cross is for later’.
标量积（点积）v · w = |v||w| cos θ，结果是一个数。点积为零则向量垂直。记牢：“点积得数，垂直可判”。
9. Sequences & Series | 数列与级数
An arithmetic sequence adds a common difference d each time. The nth term is a + (n−1)d. ‘Arithmetic’ has ‘rithmetic’ inside – steadily adding, like counting.
等差数列 arithmetic sequence 每次加公差 d，通项 a + (n−1)d。Arithmetic 中有“rithmetic”，像匀速增加。
A geometric sequence multiplies by a common ratio r. The nth term is a rⁿ⁻¹. ‘Geo-metric’ suggests exponential growth, like populations.
等比数列 geometric sequence 每次乘公比 r，通项 a rⁿ⁻¹。Geometric 联想到几何级增长，如人口爆炸。
The sum of a geometric series Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. The sum to infinity exists when |r| < 1 and equals a/(1 − r). Imagine an endless staircase that eventually reaches a limit.
等比级数和 Sₙ = a(1−rⁿ)/(1−r)，且当 |r|<1 时无穷和 S∞ = a/(1−r)。想象无穷级台阶，最终逼近一个极限高度。
Sigma notation Σ aᵢ is a compact way to write a series. The index i tells you where to start and stop. ‘Sigma sums up the scene’.
Σ 记号是级数的紧凑写法，下标和上标标明起止。Σ 就像在“总结”总和。
10. Statistics: Key Terms | 统计学关键术语
The mean (average) is Σx/n. The median is the middle value; the mode is the most frequent. ‘Mean is mean to outliers, median stays in the middle’.
平均数 mean = Σx/n。中位数 median 是中间值，众数 mode 是频次最高的。记忆：“平均数怕极端，中位数坐中间”。
Standard deviation σ measures spread. A small σ means data cluster near the mean; large σ means they are spread out. ‘σ’ looks like a snake, wiggling away from the centre.
标准差 σ 衡量离散程度，σ 小数据聚集，σ 大数据分散。把 σ 想象成一条扭动的蛇，远离中心表示离散。
A binomial distribution X ~ B(n, p) models number of successes in n independent trials. Conditions: fixed n, two outcomes, constant probability p. ‘Bi-nomial’ = two outcomes.
二项分布 X ~ B(n,p) 用于 n 次独立试验的成功次数。条件：固定 n、两种结果、概率 p 恒定。Bi-nomial 中的“bi”提示两种可能。
The normal distribution N(µ, σ²) is the bell-shaped curve. 68% of data lie within µ ± σ, 95% within µ ± 2σ. The ‘bell’ rings a friendly shape.
正态分布 N(µ,σ²) 是钟形曲线，68% 数据在 µ±σ 内，95% 在 µ±2σ 内。钟形轮廓像一个微笑的钟。
11. Mechanics: Kinematics & Forces | 力学：运动学与力
Displacement s is the vector from start to end; distance is the scalar path length. Displacement can be negative if you go backwards. ‘Displacement’ = ‘place’ changed.
位移 displacement s 是从起点到终点的向量，路程 distance 是标量路径总长。位移可为负，指示反向。Displacement 中有“place”，位置变了。
The SUVAT equations link s, u, v, a,
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