📚 Pre-U OCR Mathematics: Vocabulary and Terminology Quick-Memorisation Guide | Pre-U OCR 数学：词汇术语速记指南
Mastering mathematical terminology is crucial for answering Pre-U OCR questions accurately and efficiently. This guide provides effective memory hooks and bilingual explanations for key terms across pure mathematics, mechanics, and statistics. It is designed to help you recall precise definitions, avoid common misconceptions, and interpret exam questions correctly under time pressure.
掌握数学术语对于准确、高效地解答 Pre-U OCR 试题至关重要。本指南为纯数学、力学和统计学中的关键术语提供有效的记忆技巧和双语解释，旨在帮助你快速回忆精确定义、避开常见误区，并在时间压力下正确解读考题。
1. Core Vocabulary in Pure Mathematics: Sets, Indices and Surds | 纯数学核心词汇：集合、指数与根式
Natural numbers (positive whole numbers) are often denoted by ℕ. Remember ‘nature starts from 1’, so ℕ = {1, 2, 3, …}. To avoid confusion with the inclusion of 0, use the phrase ‘No zero in nature’ to recall that 0 is excluded in most OCR contexts.
自然数（正整数）通常用 ℕ 表示。记住“自然从1开始”，因此 ℕ = {1, 2, 3, …}。为避免与包含0的定义混淆，用口诀“自然无0”来记住0不在ℕ中。
The symbol ∈ means ‘is an element of’, while ∉ means ‘is not an element of’. Think of the letter ‘e’ for ‘element’. For subsets, ⊆ reminds you of a ‘containment’ with the line beneath representing equality possibility. Imagine the line as a floor allowing equality.
符号 ∈ 表示“属于”，而 ∉ 表示“不属于”。可以联想字母“e”代表“element（元素）”。对于子集，⊆ 像是一个包含符号，下方的横线暗示可能相等，把横线想象成允许相等的地面。
A surd is an irrational root, such as √2 or √3. The word sounds like ‘absurd’, which helps you remember that surds cannot be expressed exactly as fractions — their decimal forms go on forever without repeating.
根式（surd）是无理根式，如 √2 或 √3。这个词发音像“absurd（荒谬）”，帮你记住根式无法精确写成分数——它们的小数展开无限不循环。
The index laws state that aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ. Notice how multiplication of powers adds the little numbers, like stacking ‘m’ and ‘n’ together. For the power of a power, think of the exponents being ‘multiplied’ inside a nesting, so the m and n become multiplied.
指数法则指出 aᵐ × aⁿ = aᵐ⁺ⁿ 以及 (aᵐ)ⁿ = aᵐⁿ。注意幂相乘时小指数相加，就像把 m 和 n 叠在一起。对于幂的幂，把指数想象成嵌套的乘法，因此 m 与 n 相乘。
A logarithm logₐ x asks: ‘to what power must I raise a to get x?’ The term can be memorised as ‘log-a-x = power to which a must be raised’. Link ‘log’ with ‘logical question’ — it is the logical inverse of exponentiation. Always equate logₐ 1 = 0 because a⁰ = 1.
对数 logₐ x 问的是：“a 的多少次幂等于 x？” 术语可记作“log-a-x = a 的几次方”。把“log”与“逻辑提问（logical question）”联系起来——它是指数运算的逻辑逆运算。牢记 logₐ 1 = 0，因为 a⁰ = 1。
2. Functions, Graphs and Transformations | 函数、图像与变换
A function f: x → y assigns exactly one output y for each input x. The vital phrase ‘one-to-one or many-to-one’ distinguishes a function from a mere relation. Use the vertical line test: a graph represents a function if any vertical line intersects it at most once.
函数 f: x → y 为每个输入 x 分配唯一一个输出 y。关键口诀“一对一或多对一”将函数与一般关系区分开。使用垂线检验：如果任何垂直线与图形最多只有一个交点，则该图形表示一个函数。
The domain is the set of all possible input values, and the range is the set of all possible output values. Think ‘domain → driven in’ (inputs you drive into the function) and ‘range → results out’. Always state the domain when defining a function in Pre-U.
定义域是所有可能输入值的集合，值域是所有可能输出值的集合。联想“domain → 输入驱动入内”，而“range → 结果范围向外”。在 Pre-U 中定义函数时务必写明定义域。
Composite function fg(x) means apply g first, then f. The order matters: fg is ‘f following g’. Remember ‘fog’ as ‘first operate g’. Whenever you see f²(x), it usually means f(f(x)), not the square of f(x) — unless otherwise specified.
复合函数 fg(x) 表示先施加 g，再施加 f。顺序很重要：fg 就是“f 跟在 g 后”。可把“fog”记作“先操作 g”。看到 f²(x) 时，除非另有说明，通常表示 f(f(x))，而非 f(x) 的平方。
Inverse function f⁻¹(x) undoes the original function. Its graph is a reflection of y = f(x) in the line y = x. The mnemonic: ‘swap x and y and solve for y’. Ensure the function is one-to-one (or restrict the domain) before finding an inverse.
反函数 f⁻¹(x) 撤销原函数的操作。它的图像是 y = f(x) 关于直线 y = x 的反射。口诀：“交换 x 和 y，然后解出 y”。求反函数前必须确保函数是一一映射（或限制定义域）。
Transformations: f(x + a) shifts left by a (‘left for plus’ — counter-intuitive, remember ‘inside bracket, opposite sign’). af(x) stretches vertically by scale factor a; f(ax) compresses horizontally by 1/a (‘x multiplied by a makes graph squeeze’).
变换：f(x + a) 向左平移 a 单位（“加号向左”——反直觉，可记作“括号内，符号相反”）。af(x) 是垂直方向拉伸 a 倍；f(ax) 是将水平方向压缩为 1/a (“x 乘以 a 使图像挤窄”)。
3. Trigonometric Terminology | 三角学术语
The three primary ratios: sine (sin), cosine (cos), and tangent (tan). SOHCAHTOA is universally used: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. This memory aid works for right-angled triangle definitions.
三个基本比值：正弦（sin）、余弦（cos）和正切（tan）。通用口诀 SOHCAHTOA：正弦 = 对边/斜边，余弦 = 邻边/斜边，正切 = 对边/邻边。这个记忆法适用于直角三角形定义。
Radians are the natural measure for angles, where π rad = 180°. To convert degrees to radians, multiply by π/180. Picture a circle’s arc length equal to the radius — that defines 1 radian. Therefore, 2π radians correspond to a full circle.
弧度是角度的自然度量，其中 π 弧度 = 180°。将度数转换为弧度，乘以 π/180。想象一段弧长等于半径——这就是 1 弧度的定义。因此，整个圆对应 2π 弧度。
Secant (sec θ = 1/cos θ), cosecant (cosec θ = 1/sin θ), and cotangent (cot θ = 1/tan θ) are reciprocal trig functions. Notice the third letter in ‘sec’ points to ‘c’ (cosine); ‘cot’ has a ‘t’ hinting at ‘tan’. That can help you pair them correctly.
正割（sec θ = 1/cos θ）、余割（cosec θ = 1/sin θ）和余切（cot θ = 1/tan θ）是倒数三角函数。注意“sec”第三个字母指向“c”（余弦）；“cot”中有“t”，暗示与“tan”配对。这样就可正确配组。
Exact values: memorise the special triangles. The 45° right triangle gives sin 45° = 1/√2. The 30°-60° triangle yields sin 30° = 1/2, cos 30° = √3/2, and tan 60° = √3. Link the angles 30 and 60 — their sine and cosine values swap; the square root of 3 appears with 60 and 30 tangents.
精确值：记住特殊三角形。45° 直角三角形给出 sin 45° = 1/√2。30°-60° 三角形得出 sin 30° = 1/2，cos 30° = √3/2，tan 60° = √3。把 30 和 60 度联系起来——它们的正弦和余弦值互换；√3 出现在 60 和 30 的正切中。
Trigonometric identities like sin²θ + cos²θ ≡ 1 are fundamental. The symbol ≡ means identically equal. Use the mnemonic ‘Pythagoras on the unit circle’ because this identity comes from x² + y² = 1 on a circle of radius 1.
三角恒等式如 sin²θ + cos²θ ≡ 1 是基础。符号 ≡ 表示恒等。借用口诀“单位圆上的毕达哥拉斯”，因为该恒等式源于半径为 1 的圆上 x² + y² = 1。
4. Sequences and Series | 数列与级数
A sequence is an ordered list of terms; a series is the sum of the terms. Think ‘sequence’ as ‘s’ for ‘string’, and ‘series’ as ‘s’ for ‘sum’. Arithmetic sequences have a common difference d; use the formula uₙ = a + (n − 1)d, where a is the first term.
数列是一组有序的项；级数是这些项的和。可联想“sequence”（数列）的“s”代表“string（串）”，“series”（级数）的“s”代表“sum（和）”。等差数列有公差 d；用公式 uₙ = a + (n − 1)d，其中 a 为首项。
Geometric sequences multiply by a constant ratio r each time. The nth term is a r ⁿ⁻¹. To remember, note that the index is (n−1) because the first term has no multiplication of r yet. The finite sum Sₙ = a(1 − rⁿ)/(1 − r), valid when r ≠ 1.
等比数列每次乘以常数比 r。第 n 项为 a r ⁿ⁻¹。为记忆，注意指数为 (n−1)，因为首项尚未乘以 r。有限求和 Sₙ = a(1 − rⁿ)/(1 − r)，当 r ≠ 1 时成立。
Convergent geometric series: if |r| < 1, the sum to infinity S∞ = a/(1 − r). Visualise 'infinite sum' as the series settling down to a single number, like adding smaller and smaller pieces that never exceed that limit.
收敛等比级数：若 |r| < 1，无穷和 S∞ = a/(1 − r)。将“无穷和”形象化为级数趋近于一个定数，如同不断添加越来越小的碎片，但总和不会超过该极限。
Sigma notation Σ (capital sigma) indicates summation. For example, Σ (from i=1 to n) uᵢ means u₁ + u₂ + … + uₙ. Remember that Σ is the Greek ‘S’, standing for sum. Always check the lower and upper limits to know where the sum starts and ends.
Σ 记号（大写 sigma）表示求和。例如，Σᵢ₌₁ⁿ uᵢ 表示 u₁ + u₂ + … + uₙ。记住 Σ 是希腊语的“S”，代表求和。务必检查上下限，以明确求和从何处开始、到何处结束。
The binomial expansion (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + … is valid for |x| < 1 when n is not a positive integer. Use the phrase 'n choose k' for the coefficient nCk. Mind the alternating signs when n is negative or fractional.
二项展开式 (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + … 在 n 不是正整数且 |x| < 1 时有效。用“n 选 k”来记忆系数 nCk。当 n 为负数或分数时，注意符号交替变化。
5. Calculus Language: Differentiation and Integration | 微积分语言：微分与积分
Differentiation finds the gradient of a curve. The derivative dy/dx means rate of change of y with respect to x. Think ‘difference quotient’ shrinking to zero. The notation f'(x) is Lagrange’s notation; d/dx is Leibniz’s. Pre-U uses both — be comfortable switching between them.
微分求曲线的梯度。导数 dy/dx 表示 y 关于 x 的变化率。把它想象成差商缩至零。f'(x) 是拉格朗日记号；d/dx 是莱布尼茨记号。Pre-U 两者都采用——要能熟练切换。
Chain rule: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x). Memorise as ‘derivative of the outside, times derivative of the inside’. Use the ‘peeling an onion’ image: differentiate the outer function first, then multiply by the inner derivative.
链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)。记为“外面求导，乘以内面求导”。使用“剥洋葱”形象：先对外层函数求导，再乘以内部函数的导数。
Product rule: (uv)’ = u’v + uv’. Think ‘first prime times second, plus first times second prime’. The mnemonic ‘udvee add vduu’ can help. For the quotient rule, remember ‘low d high minus high d low over low squared’ — the denominator is differentiated last and squared.
乘法法则：(uv)’ = u’v + uv’。记作“第一求导乘第二，加第一乘第二求导”。口诀“u 导 v 加 u v 导”也有帮助。对于除法法则，记“下导上减上导下除以下平方”——分母最后求导并平方。
Integration is the reverse process of differentiation. The indefinite integral ∫ f(x) dx gives a family of functions plus the constant of integration +c. Always add ‘+c’ unless evaluating a definite integral. Use the mnemonic ‘integral includes the invisible constant’.
积分是微分的逆过程。不定积分 ∫ f(x) dx 给出一个函数族加上积分常数 +c。除非计算定积分，永远要加上“+c”。用口诀“积分包含隐常数”来提醒自己。
Definite integral ∫ₐᵇ f(x) dx represents the signed area under the curve between x = a and x = b. The numbers a and b are called limits of integration. Visualise summing infinitely many thin rectangles (Riemann sums) to remember that integration gives accumulated quantity.
定积分 ∫ₐᵇ f(x) dx 表示曲线下 x = a 到 x = b 之间的带符号面积。a 和 b 称为积分限。想象对无数薄矩形求和（黎曼和），从而记住积分给出累积量。
Separation of variables is a technique for solving differential equations: rewrite dy/dx = g(x)h(y) as (1/h(y)) dy = g(x) dx and integrate both sides. Think ‘split the dy and dx across the equals sign, then integrate’.
分离变量法是解微分方程的一种技巧：将 dy/dx = g(x)h(y) 改写为 (1/h(y)) dy = g(x) dx 并两边积分。想象“把 dy 和 dx 分别挪到等号两边，然后积分”。
6. Vectors: Direction and Magnitude | 向量：方向与大小
A vector has both magnitude and direction, unlike a scalar which has only size. Denote a vector as a bold letter or with an arrow, e.g., v or v⃗. In column form, it is written as (x y) or as xi + yj. Remember ‘vector = velocity’ — velocity needs direction.
向量既有大小又有方向，不像标量只有大小。向量可用粗体字母或加箭头表示，如 v 或 v⃗。列向量形式写作 (x y) 或 xi + yj。记住“向量 = 速度” —— 速度需要方向。
The magnitude or modulus of vector v = (x y) is given by |v| = √(x² + y²). This is the length of the vector. Think of Pythagoras: the components form a right triangle’s legs, and the magnitude is the hypotenuse.
向量 v = (x y) 的大小（模）为 |v| = √(x² + y²)。这就是向量的长度。想到毕达哥拉斯定理：分量构成直角三角形的两直角边，模就是斜边。
A unit vector has magnitude 1. The unit vector in the direction of v is v/|v|. The standard basis vectors i and j are unit vectors along the x- and y-axes. For 3D, add k along the z-axis. Use the mnemonic ‘i, j, k — in alphabetical order along axes’.
单位向量的模为1。沿 v 方向的单位向量为 v/|v|。标准基向量 i 和 j 是沿 x 轴和 y 轴的单位向量。对于三维，加上沿 z 轴的 k。用口诀“i, j, k —— 沿坐标轴按字母顺序”。
The scalar (dot) product a·b = |a||b| cos θ = a₁b₁ + a₂b₂. It gives a scalar. If a·b = 0, the vectors are perpendicular. The mnemonic ‘dot product is directional projection’ helps recall that it measures how much one vector extends in the direction of another.
标量积（点积）a·b = |a||b| cos θ = a₁b₁ + a₂b₂。结果为一标量。若 a·b = 0，则向量垂直。口诀“点积是方向上的投影”有助于回忆它衡量一个向量在另一个向量方向上的延伸量。
The vector equation of a line: r = a + t d, where a is a position vector on the line and d is a direction vector. Parameter t varies over ℝ. To achieve the Cartesian form, eliminate t. Think ‘a for anchor, d for direction’.
直线的向量方程：r = a + t d，其中 a 是线上一个点的位置向量，d 是方向向量。参数 t 遍历所有实数。要得到笛卡儿形式，消去 t。记忆为“a 为锚点，d 为方向”。
7. Mechanics: Force, Motion and Equilibrium | 力学：力、运动与平衡
A force is a push or a pull, measured in newtons (N). It is a vector quantity. Weight is the force due to gravity: W = mg, where g ≈ 9.8 m s⁻². Tension is the pulling force in a string; thrust is a pushing force in a rod.
力是推或拉，单位是牛顿 (N)。它是向量。重量是由重力产生的力：W = mg，其中 g ≈ 9.8 m s⁻²。张力是绳子中的拉力；推力是杆中的推力。
Free-body diagrams isolate a single object and show all forces acting on it. Drawing them accurately is essential. Label forces with their types: reaction (R), friction (Fᵣ), tension (T), etc. The mnemonic ‘RFTR’ (reaction, friction, tension, weight) covers common forces.
受力图将单个物体隔离，并标出所有作用力。准确绘制受力图至关重要。用类型标注力：反作用力 (R)、摩擦力 (Fᵣ)、张力 (T)、等等。口诀“RFTW”覆盖了常见力。
Equilibrium means the net force and net moment are zero. For a particle, ΣF = 0. For a rigid body, both ΣF = 0 and ΣM = 0 must hold. Resolve forces horizontally and vertically. If three forces act, they can form a closed triangle when in equilibrium.
平衡意味着合力为零且合力矩为零。对于质点，ΣF = 0。对于刚体，必须同时满足 ΣF = 0 和 ΣM = 0。分解力至水平与竖直方向。若三力作用且平衡，它们可构成封闭三角形。
Moments measure the turning effect: moment = force × perpendicular distance from pivot. Units are N m. The principle of moments states that for equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments. Use ‘F × d’ perpendicular.
力矩衡量转动效应：力矩 = 力 × 到支点的垂直距离。单位是 N m。力矩原理指出，平衡时顺时针力矩之和等于逆时针力矩之和。用“F × d”且务必垂直。
Kinematics equations for constant acceleration a: v = u + at, s = ut + ½ at², v² = u² + 2as, s = ½(u + v)t. The mnemonic SUVAT helps recall the five variables: s (displacement), u (initial velocity), v (final velocity), a (acceleration), t (time). Choose the equation that omits the unknown you are not interested in.
匀加速运动学方程：v = u + at, s = ut + ½ at², v² = u² + 2as, s = ½(u + v)t。口诀 SUVAT 帮助回忆五个变量：s（位移）、u（初速度）、v（末速度）、a（加速度）、t（时间）。选择不含所关心未知量的方程。
8. Probability and Statistical Terms | 概率与统计术语
A random variable X represents the outcome of a random phenomenon. A discrete random variable takes distinct values, while a continuous one takes any value in an interval. The probability distribution lists probabilities for discrete variables; for continuous, use a probability density function (PDF).
随机变量 X 表示随机现象的结果。离散随机变量取孤立值，而连续随机变量可取区间内任意值。概率分布列出离散变量的取值及其概率；对于连续变量，用概率密度函数 (PDF)。
Expected value E(X) is the long-run average μ. For discrete: E(X) = Σ x·P(X=x). Think ‘expectation is the weighted average’. Variance Var(X) = E(X²) − [E(X)]² measures spread. Remember ‘mean of squares minus square of mean’ to calculate it quickly.
期望值 E(X) 是长期平均值 μ。离散：E(X) = Σ x·P(X=x)。可联想“期望即加权平均”。方差 Var(X) = E(X²) − [E(X)]² 衡量离散程度。记住“平方的均值减均值的平方”以便快速计算。
The binomial distribution B(n, p) models the number of successes in n independent trials, each with probability p. Denote X ~ B(n, p). The probability formula is P(X = r) = nCr · p ʳ · (1−p)ⁿ⁻ʳ. The coefficient nCr is read ‘n choose r’. Use ‘binomial => bi-nomial: two outcomes per trial’.
二项分布 B(n, p) 表示在 n
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