📚 Year 13 CIE Maths: A Rapid Guide to Memorising Essential Terminology | Year 13 CIE 数学：词汇术语速记指南
In Year 13 CIE Mathematics, precise use of terminology is just as vital as mastering the techniques themselves. Examiners routinely award marks for correctly interpreting keywords in questions and using the right vocabulary in structured answers. This guide breaks down the most frequently tested terms across Pure Mathematics, Mechanics, and Probability & Statistics, pairing each with a concise definition and a clear Chinese equivalent, so you can build fluency and confidence in both languages. Use this as a daily revision checklist to secure those terminology marks.
在 Year 13 CIE 数学中，准确使用术语与掌握解题技巧同等重要。阅卷人经常将分数分配给那些能正确理解题目关键词并在结构化答案中使用恰当词汇的考生。本指南梳理了纯数学、力学和概率统计中最常考的术语，每个术语都配有简洁的定义和清晰的中文对应表达，帮助你建立中英双语的流利度和信心。将此作为每日复习清单，牢牢把握术语相关的分数。
1. Differentiation and Integration Terms | 微分与积分术语
The cornerstone of Pure Mathematics 3 is calculus, and a host of specific terms describe the operations, results, and properties of functions under differentiation and integration. Memorising these will help you follow the logic of ‘show that’ and ‘determine’ questions without hesitation.
纯数学 3 的基石是微积分，众多专门术语描述了函数在微分和积分下的运算、结果和性质。熟记这些术语能帮助你毫不迟疑地跟上“证明”和“求”这类问题的逻辑。
Derivative refers to the instantaneous rate of change of a function f(x) with respect to x, usually denoted as f'(x) or dy/dx. If f'(x) > 0 on an interval, the function is said to be increasing on that interval.
导数（Derivative）指的是函数 f(x) 关于 x 的瞬时变化率，通常记作 f'(x) 或 dy/dx。如果在某个区间上 f'(x) > 0，则称函数在该区间递增（increasing）。
Stationary point is a point on a curve where the derivative equals zero. A stationary point can be classified as a local maximum, a local minimum, or a point of inflection. The second derivative, d²y/dx², helps determine the nature: if it is negative, the point is a maximum; if positive, a minimum.
驻点（Stationary point）是曲线上导数等于零的点。驻点可分类为局部极大值（local maximum）、局部极小值（local minimum）或拐点（point of inflection）。二阶导数（second derivative） d²y/dx² 有助于判断其性质：若为负，则是极大值；若为正，则是极小值。
Integration is the reverse process of differentiation. The indefinite integral of a function f(x) includes an arbitrary constant, denoted as + c. The definite integral from a to b yields the signed area between the curve and the x-axis. The phrase integrate by parts uses the formula ∫ u dv = uv − ∫ v du, while integration by substitution changes the variable to simplify the integrand.
积分（Integration）是微分的逆过程。函数 f(x) 的不定积分（indefinite integral）包含一个任意常数，记作 + c。定积分（definite integral）从 a 到 b 给出曲线与 x 轴之间的符号面积。术语分部积分法（integrate by parts）使用公式 ∫ u dv = uv − ∫ v du，而换元积分法（integration by substitution）则通过变量替换简化被积函数。
When a differential equation involves a rate of change proportional to the current quantity, we write dy/dt = ky and obtain an exponential growth or decay model. The resulting general solution contains the constant eᵏᵗ.
当微分方程涉及变化率与当前量成正比时，我们记作 dy/dt = ky，并得到指数增长或衰减（exponential growth or decay）模型。由此得到的通解包含常数 eᵏᵗ。
2. Trigonometric Identities and Functions | 三角恒等式与函数
Trigonometry in Year 13 extends well beyond right‑angled triangles into compound angles, reciprocal functions, and harmonic form. The vocabulary here is dense, but mastering it allows you to decode questions about exact values, proofs, and solutions of equations within a given interval.
Year 13 的三角学远远超出直角三角形的范畴，涉及复角、倒数函数和谐波形式。此处的词汇十分密集，但掌握后能让你轻松解读关于精确值、证明和给定区间内方程求解的问题。
Secant (sec θ), cosecant (cosec θ), and cotangent (cot θ) are the three reciprocal trigonometric functions, defined as sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ. Their domains are restricted where the original functions are zero.
正割（sec θ）、余割（cosec θ）和余切（cot θ）是三个倒数三角函数，定义为 sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = 1/tan θ。它们在其原函数为零的点处定义域受限。
The compound angle formulae include sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B. From these derive the double angle formulae, such as cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
复角公式（compound angle formulae）包括 sin(A ± B) = sin A cos B ± cos A sin B 和 cos(A ± B) = cos A cos B ∓ sin A sin B。由此可推导出倍角公式（double angle formulae），例如 cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。
Expressing a sin θ + b cos θ in the form R sin(θ ± α) or R cos(θ ± α) is called the harmonic form, where R = √(a² + b²) and α is a phase angle. This form is extremely useful for finding the maximum and minimum values of trigonometric expressions and for solving equations.
将 a sin θ + b cos θ 表示为 R sin(θ ± α) 或 R cos(θ ± α) 的形式称为谐波形式（harmonic form），其中 R = √(a² + b²)，α 是相位角。这种形式对于求三角函数表达式的最大值和最小值以及解方程极为有用。
3. Vectors in Three Dimensions | 三维向量
Moving from 2D to 3D vectors introduces new terminology for lines, planes, and spatial relationships. CIE questions frequently ask you to find the point of intersection, the shortest distance, or the equation of a plane in various forms.
从二维向量扩展到三维向量，引入了关于直线、平面和空间关系的新术语。CIE 试题经常要求你找出交点、最短距离或用各种形式表示平面方程。
A position vector of a point P is the vector OP → from the origin to P. The direction vector of a line is any vector parallel to that line. The vector equation of a line is written as r = a + tb, where a is a point on the line, b is a direction vector, and t is a scalar parameter.
点 P 的位置向量（position vector）是从原点指向 P 的向量 OP →。直线的方向向量（direction vector）是平行于该直线的任何向量。直线的向量方程（vector equation of a line）写作 r = a + tb，其中 a 是直线上的一点，b 是方向向量，t 是标量参数。
A plane can be expressed in vector form as r ⋅ n = a ⋅ n, where n is a normal vector perpendicular to the plane. The Cartesian equation of a plane is ax + by + cz = d, where the coefficients a, b, c are the components of a normal vector.
平面（plane）可以用向量形式（vector form）表示为 r ⋅ n = a ⋅ n，其中 n 是垂直于该平面的法向量（normal vector）。平面的笛卡尔方程（Cartesian equation of a plane）为 ax + by + cz = d，其中系数 a、b、c 是法向量的分量。
The angle between two lines is found using the dot product of their direction vectors. The angle between a line and a plane is the complement of the angle between the line’s direction vector and the plane’s normal vector. The shortest (perpendicular) distance from a point to a plane involves projecting the connecting vector onto the unit normal.
两条直线的夹角（angle between two lines）利用它们方向向量的点积求得。直线与平面的夹角（angle between a line and a plane）是直线方向向量与平面法向量夹角的余角。点到平面的最短（垂直）距离（shortest (perpendicular) distance）涉及将连接向量投影到单位法向量上。
4. Complex Numbers | 复数
Complex numbers are a rich topic where precise terminology separates sophisticated reasoning from guesswork. Knowing exactly what is meant by argument, modulus, and conjugate unlocks both algebraic manipulations and geometric interpretations on the Argand diagram.
复数是一个内涵丰富的主题，精确的术语能够区分严谨推理与胡乱猜测。准确理解辐角、模和共轭的含义，有助于打开代数运算和阿干特图上几何解释的大门。
A complex number is written as z = x + iy, where x is the real part Re(z) and y is the imaginary part Im(z). The complex conjugate of z is z* = x − iy. The modulus, |z|, is the distance from the origin on the Argand diagram, given by √(x² + y²). The argument, arg z, is the angle measured from the positive real axis to the line representing z.
一个复数写作 z = x + iy，其中 x 是实部 Re(z)，y 是虚部 Im(z)。z 的共轭复数（complex conjugate）是 z* = x − iy。模（modulus）|z| 是阿干特图上到原点的距离，由 √(x² + y²) 给出。辐角（argument）arg z 是从正实轴到代表 z 的直线所测得的角度。
A complex number can be expressed in modulus‑argument form: z = r(cos θ + i sin θ) = reⁱᶿ, where r = |z| and θ = arg z. De Moivre’s theorem states that for any integer n, (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, a powerful tool for finding powers and roots of complex numbers.
复数可以用模—辐角形式（modulus‑argument form）表示：z = r(cos θ + i sin θ) = reⁱᶿ，其中 r = |z|，θ = arg z。棣莫弗定理（De Moivre’s theorem）指出，对于任何整数 n，(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ，这是求复数乘方和根的有力工具。
Loci in the complex plane are described using conditions such as |z − a| = r (a circle) or |z − a| = |z − b| (a perpendicular bisector). The principal argument is usually taken in the interval (–π, π].
复平面上的轨迹（loci）使用诸如 |z − a| = r（圆）或 |z − a| = |z − b|（垂直平分线）的条件来描述。辐角主值（principal argument）通常取在区间 (–π, π] 内。
5. Differential Equations | 微分方程
Formulating and solving differential equations is a skill that bridges Pure Maths with Mechanics. The language used in questions reveals whether you are dealing with a separable equation, a first‑order linear equation, or a situation that requires an integrating factor.
建立并求解微分方程是连接纯数学与力学的一项技能。题目中使用的语言揭示了你是在处理可分离变量方程、一阶线性方程，还是需要用到积分因子的情况。
A separable first‑order differential equation can be written in the form dy/dx = g(x)h(y). Rearranging to ∫ (1/h(y)) dy = ∫ g(x) dx is the solution method. A general solution contains an arbitrary constant, while a particular solution satisfies a given initial condition, such as y = y₀ when x = 0.
可分离变量的一阶微分方程（separable first‑order differential equation）可以写成 dy/dx = g(x)h(y) 的形式。化简为 ∫ (1/h(y)) dy = ∫ g(x) dx 是求解方法。通解（general solution）包含一个任意常数，而特解（particular solution）满足给定的初始条件，例如当 x = 0 时 y = y₀。
An integrating factor is used for equations of the form dy/dx + P(x)y = Q(x). The factor is e^(∫ P(x) dx), and multiplying through by it makes the left side an exact derivative of the product of y and the factor.
对于形如 dy/dx + P(x)y = Q(x) 的方程，可以使用积分因子（integrating factor）。该因子为 e^(∫ P(x) dx)，乘以它后，左侧变为 y 与该因子乘积的精确导数。
6. Mechanics: Kinematics and Forces | 力学：运动学与力
Mechanics terminology must be precise because each term defines a distinct vector or scalar quantity, and confusing them can derail a whole solution. Terms like displacement, velocity, and acceleration are strictly defined with respect to time derivatives, and force interactions rely on Newton’s laws expressed in vector form.
力学术语必须精确，因为每个术语定义了不同的矢量或标量，混淆它们可能会毁掉整个解题过程。位移、速度和加速度等术语严格根据时间导数定义，而力的相互作用则依赖于以矢量形式表达的牛顿定律。
Displacement is a vector from the origin to the current position; distance is the total scalar length of the path travelled. Velocity is the rate of change of displacement, v = dr/dt, and acceleration is a = dv/dt = d²r/dt².
位移（Displacement）是从原点到当前位置的矢量；路程（distance）是经过路径的总标量长度。速度（Velocity）是位移的变化率，v = dr/dt，加速度（acceleration）为 a = dv/dt = d²r/dt²。
When acceleration is constant, we use the suvat equations: v = u + at, s = ut + ½at², v² = u² + 2as, and s = ½(u + v)t. Here s is displacement, u is initial velocity, v is final velocity, and t is time. An object moving under gravity alone experiences uniform acceleration of g = 9.8 m s⁻² downwards.
当加速度恒定时，我们使用匀加速运动方程（suvat equations）：v = u + at，s = ut + ½at²，v² = u² + 2as，s = ½(u + v)t。这里 s 是位移，u 是初速度，v 是末速度，t 是时间。仅在重力作用下运动的物体受到向下的大小为 g = 9.8 m s⁻² 的均匀加速度。
A force is a vector quantity measured in newtons (N). Newton’s second law states F = ma, where the resultant force F is the vector sum of all forces acting on a particle. Friction opposes motion or attempted motion; its maximum value is F ≤ μR, where μ is the coefficient of friction and R is the normal reaction force perpendicular to the contact surface.
力（force）是一个以牛顿 (N) 为单位的矢量。牛顿第二定律指出 F = ma，其中合力 F 是作用在质点上的所有力的矢量和。摩擦力（friction）阻碍运动或运动趋势；其最大值为 F ≤ μR，其中 μ 是摩擦系数（coefficient of friction），R 是垂直于接触面的法向反力（normal reaction）。
Connected particles moving in a straight line share the same acceleration and tension in a light inextensible string. A pulley problem typically involves resolving forces for each particle and equating tensions.
相连质点（Connected particles）沿直线运动时，具有相同的加速度，且轻质不可伸长绳中的张力相同。滑轮（pulley）问题通常涉及对每个质点进行力的分解并令张力相等。
7. Moments and Equilibrium | 力矩与平衡
The terms around moments and equilibrium are highly specific and often appear in structured ‘state the conditions’ problems. A clear grasp of what it means for a body to be in both translational and rotational equilibrium is fundamental to rigid‑body statics.
围绕力矩与平衡的术语非常具体，经常出现在结构化的“陈述条件”问题中。清晰理解一个物体同时处于平移平衡和转动平衡意味着什么，是刚体静力学的基础。
The moment of a force about a point is the product of the force and the perpendicular distance from the point to the line of action of the force. The principle of moments states that for a body in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any point.
力对某点的力矩（moment）是力与该点到力作用线的垂直距离的乘积。力矩原理指出，对于处于转动平衡（rotational equilibrium）的物体，绕任何点的顺时针力矩之和等于逆时针力矩之和。
A rigid body is in equilibrium when both the resultant force and the resultant moment are zero. Conditions: ΣFₓ = 0, ΣFᵧ = 0, and ΣM = 0. A couple is a pair of equal, parallel, but oppositely directed forces whose lines of action do not coincide, producing rotation without translation; the moment of a couple is constant about any point.
当合力和合力矩都为零时，刚体处于平衡（equilibrium）状态。条件为：ΣFₓ = 0，ΣFᵧ = 0，ΣM = 0。力偶（couple）是一对大小相等、平行但方向相反的力，其作用线不重合，产生转动而无平动；力偶的力矩对任何点都是常数。
A centroid or centre of mass is the point where the whole weight of a body is considered to act. In uniform laminas, the centroid is at the geometric centre, and in more complex shapes it is found by taking moments of composite parts.
质心（centroid）或重心是物体全部重量被认为集中作用的点。在均匀薄片中，质心位于几何中心，对于更复杂的形状，则通过对各组成部分取矩来求得。
8. Probability Distributions | 概率分布
Statistics 2 demands a fluent understanding of distribution shapes, parameters, and related concepts such as expectation and variance. The language is statistical, yet the reasoning is highly mathematical, blending algebra with probabilistic interpretation.
统计学 2 要求对分布形状、参数以及期望和方差等相关概念有流利的理解。其语言是统计学的，但推理过程高度数学化，将代数与概率解释融为一体。
A discrete random variable takes a countable number of values, while a continuous random variable takes any value within an interval. The probability density function (pdf), f(x), for a continuous variable must satisfy f(x) ≥ 0 and ∫ f(x) dx = 1 over the domain. The cumulative distribution function F(x) = P(X ≤ x) is obtained by integration.
离散随机变量（discrete random variable）取可数个值，而连续随机变量（continuous random variable）取某个区间内的任何值。连续变量的概率密度函数（probability density function, pdf） f(x) 必须满足 f(x) ≥ 0 且在其定义域上 ∫ f(x) dx = 1。累积分布函数 F(x) = P(X ≤ x) 通过积分得到。
Expectation E(X) is the mean or long‑term average, and variance Var(X) = E[(X − μ)²] = E(X²) − [E(X)]². The binomial distribution, X ~ B(n, p), models the number of successes in n independent trials each with probability p. The Poisson distribution, X ~ Po(λ), describes the number of events occurring in a fixed interval when events are independent and the mean rate is λ.
期望（Expectation） E(X) 是均值或长期平均值，方差（variance） Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²。二项分布（binomial distribution） X ~ B(n, p) 模拟了 n 次独立试验中每次成功概率为 p 时的成功次数。泊松分布（Poisson distribution） X ~ Po(λ) 描述了在固定区间内事件发生且独立时，其平均发生率为 λ 的事件次数。
For a continuous distribution like the normal distribution, we use the standard normal variable Z = (X − μ)/σ to find probabilities from tables. A normal approximation to the binomial or Poisson distribution requires a continuity correction because a discrete distribution is being approximated by a continuous one.
对于像正态分布（normal distribution）这样的连续分布，我们使用标准正态变量（standard normal variable） Z = (X − μ)/σ 来从表中查找概率。对二项分布或泊松分布的正态近似（normal approximation）需要连续性校正（continuity correction），因为离散分布正被连续分布所近似。
9. Sampling and Estimation | 抽样与估计
Statistical inference uses sample data to draw conclusions about a population. The terminology here is precise and prone to confusion: students often mix up ‘sample mean’ and ‘population mean’, or ‘unbiased estimator’ and ‘standard error’. Clarity here ensures you interpret hypothesis‑testing scenarios correctly.
统计推断利用样本数据对总体得出结论。此处的术语精确且容易混淆：学生经常将“样本均值”与“总体均值”，或“无偏估计量”与“标准误”混为一谈。清晰把握这些术语，能确保你正确解读假设检验情境。
A population is the entire set of items of interest, while a sample is a subset selected for study. A statistic is a quantity calculated from the sample, such as the sample mean x̄, used to estimate a population parameter, such as the population mean μ.
总体（population）是研究的全部对象集合，而样本（sample）是选取用于研究的一个子集。统计量（statistic）是由样本计算出的量，例如样本均值 x̄，用于估计总体参数（population parameter），例如总体均值 μ。
An estimator is a rule for calculating an estimate from a sample. It is unbiased if its expected value equals the true parameter. The standard error is the standard deviation of the sampling distribution of a statistic, e.g., for the sample mean it is σ/√n.
估计量（estimator）是从样本计算估计值的规则。如果其期望值等于真实参数，则它是无偏（unbiased）的。标准误（standard error）是统计量抽样分布的标准差，例如，样本均值的标准误为 σ/√n。
A confidence interval gives a range of values within which the population parameter is expected to lie with a certain probability, say 95%. The central limit theorem (CLT) states that the sample mean of a sufficiently large sample is approximately normally distributed regardless of the population distribution.
置信区间（confidence interval）给出了总体参数以一定概率（比如 95%）落在其中的一个取值范围。中心极限定理（CLT）指出，不论总体分布如何，足够大样本的样本均值近似服从正态分布。
10. Hypothesis Testing | 假设检验
Hypothesis testing is the formal framework for making decisions based on data. Every step has a specific technical name, from the null hypothesis to the critical region, and mislabelling them can lose marks even when calculations are correct.
假设检验是基于数据做出决策的正式框架。从原假设到拒绝域，每一步都有特定的技术名称，即使计算正确，标注错误也会丢分。
The null hypothesis H₀ is a statement of no effect or no difference, while the alternative hypothesis H₁ is what you are trying to prove. A one‑tailed test looks for a change in one direction; a two‑tailed test looks for any change. The significance level α is the probability of rejecting H₀ when it is true (Type I error).
原假设（null hypothesis） H₀ 是一个无效或没有差异的陈述，而备择假设（alternative hypothesis） H₁ 是你试图证明的陈述。单尾检验（one‑tailed test）考察单一方向的变化；双尾检验（two‑tailed test）考察任何方向的变化。显著性水平（significance level） α 是当 H₀ 为真时拒绝它的概率（第一类错误）。
The test statistic is calculated from the sample and compared to a critical value from tables. The critical region (or rejection region) is the set of values of the test statistic that leads to rejection of H₀. The p‑value is the probability, under H₀, of obtaining a result at least as extreme as the observed one. If p < α, reject H₀.
检验统计量（test statistic）由样本计算得出，并与查表得到的临界值（critical value）进行比较。拒绝域（critical region）是导致拒绝 H₀ 的检验统计量取值集合。p 值（p‑value）是在 H₀ 下，获得至少与观测值一样极端的结果的概率。若 p < α，则拒绝 H₀。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导