Pre-U CIE Mathematics Vocabulary Guide u2014 Pre-U CIE u6570u5b66uff1au8bcdu6c47u672fu8bedu901fu8bb0u6307u5357

Pre-U CIE Mathematics: Essential Vocabulary & Terminology Quick Guide | Pre-U CIE 数学：词汇术语速记指南

The Cambridge Pre-U Mathematics (9794) syllabus is a rigorous qualification designed to prepare students for university-level study in mathematics, engineering, physics, and related disciplines. One of the biggest challenges for students – especially those studying in a bilingual environment – is mastering the precise mathematical vocabulary required to understand exam questions, write coherent proofs, and communicate mathematical ideas effectively. This guide provides a curated collection of essential terms across all major topic areas: Pure Mathematics, Probability & Statistics, and Mechanics.

剑桥 Pre-U 数学（9794）课程大纲是一项严格的资格考试，旨在为学生进入大学阶段的数学、工程、物理及相关学科做好准备。对学生而言 – 尤其是双语环境下的学习者 – 最大的挑战之一，就是掌握精确的数学词汇，以理解考题、写出条理清晰的证明，并有效传达数学思想。本指南汇总了各主要领域的核心术语：纯数学、概率与统计，以及力学。

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1. Pure Mathematics — Core Terminology | 纯数学—核心术语

1.1 Algebra & Functions | 代数与函数

English Term	中文术语	Explanation / 解释
Function	函数	A mapping from a domain to a codomain: each input has exactly one output. / 从定义域到值域的映射：每个输入有且仅有一个输出。
Domain	定义域	The set of all possible input values for a function. / 函数所有可能输入的集合。
Range / Codomain	值域	The set of all possible output values. / 所有可能输出值的集合。
One-to-one (Injective)	单射	A function where each output corresponds to exactly one input. / 每个输出值仅对应一个输入值的函数。
Onto (Surjective)	满射	A function whose range equals its entire codomain. / 值域等于整个陪域的函数。
Bijection	双射	A function that is both injective and surjective; has an inverse. / 既是单射又是满射的函数；存在逆函数。
Inverse function	反函数	Denoted f⁻¹(x); reverses the mapping. Only exists for bijections. / 记为 f⁻¹(x)；逆转映射。仅双射存在反函数。
Composite function	复合函数	(f∘g)(x) = f(g(x)); applying one function after another. / 将一个函数应用于另一个函数的结果。
Polynomial	多项式	Expression of the form aₙxⁿ + … + a₁x + a₀. / 形如 aₙxⁿ + … + a₁x + a₀ 的表达式。
Root (of an equation)	根	A value that satisfies f(x) = 0. / 满足 f(x) = 0 的值。
Factor theorem	因式定理	If f(a) = 0, then (x−a) is a factor of f(x). / 若 f(a)=0，则 (x−a) 是 f(x) 的因式。
Remainder theorem	余式定理	When f(x) is divided by (x−a), the remainder is f(a). / 当 f(x) 除以 (x−a) 时，余数为 f(a)。
Modulus function	绝对值函数	|x| gives the distance of x from zero; always non-negative. / |x| 表示 x 到零的距离；恒为非负。
Asymptote	渐近线	A line that a curve approaches but never reaches. / 曲线无限接近但永不相交的直线。
Rational function	有理函数	A ratio of two polynomials; careful about undefined points. / 两个多项式之比；需注意无定义点。
Partial fractions	部分分式	Decomposing a rational expression into simpler fractions. / 将有理表达式分解为更简单的分式之和。
Discriminant	判别式	For ax²+bx+c, Δ = b²−4ac; determines the nature of roots. / 对于 ax²+bx+c，Δ=b²−4ac；决定根的性质。

When approaching a question on functions in the Pre-U exam, always begin by stating the domain and codomain. Questions frequently test the distinction between “for all x” (∀) and “there exists x” (∃), as well as the conditions under which an inverse function exists. A classic pitfall is forgetting to restrict the domain when finding an inverse of a non-bijective function.

在 Pre-U 考试中遇到函数题时，始终先阐明定义域和值域。考题经常检验”对所有 x”（∀）与”存在 x”（∃）的区别，以及反函数存在的条件。一个经典的陷阱是在求非双射函数的反函数时，忘记限制定义域。

1.2 Sequences & Series | 数列与级数

English Term	中文术语	Explanation / 解释
Sequence	数列	An ordered list of numbers following a rule. / 按某种规律排列的一列数。
Series	级数	The sum of the terms of a sequence. / 数列各项之和。
Arithmetic progression	等差数列	a, a+d, a+2d, …; constant difference d. / 公差 d 恒定的数列。
Geometric progression	等比数列	a, ar, ar², …; constant ratio r. / 公比 r 恒定的数列。
Common difference	公差	The fixed value added to get the next term. / 相邻两项之间的固定差值。
Common ratio	公比	The fixed multiplier to get the next term. / 相邻两项之间的固定比值。
Convergence	收敛	A series approaches a finite limit as n→∞. / 当 n→∞ 时级数趋近于有限值。
Divergence	发散	A series does not approach a finite limit. / 级数不趋近于有限值。
Sum to infinity	无穷项和	For |r|<1, S∞ = a/(1−r) for a geometric series. / 当 |r|<1 时，等比级数的无穷项和为 a/(1−r)。
Binomial expansion	二项展开	(1+x)ⁿ = 1 + nx + n(n−1)x²/2! + …; valid for |x|<1 when n is not a positive integer. / 当 n 不是正整数时，展开在 |x|<1 时有效。
Sigma notation (Σ)	求和符号	Compact representation of a series sum. / 级数求和的紧凑表示。

The binomial expansion for non-integer powers is a hallmark of Pre-U difficulty. Students must carefully note the radius of convergence (|x| < 1) and be prepared to manipulate expressions into the form (1 + x)ⁿ before expanding. A common exam technique involves expanding a product of two binomial series and collecting like terms up to a specified power.

非整数次幂的二项展开是 Pre-U 难度的典型标志。学生必须注意收敛半径（|x| < 1），并准备先将表达式化为 (1 + x)ⁿ 的形式再展开。常见的一种考试技巧是展开两个二项级数的乘积，并合并到指定次幂的同类项。

1.3 Coordinate Geometry | 坐标几何

English Term	中文术语	Explanation / 解释
Gradient / Slope	斜率 / 梯度	Measure of steepness; Δy/Δx. / 陡峭程度的度量；Δy/Δx。
Intercept	截距	Where a line or curve crosses an axis. / 直线或曲线与坐标轴的交点。
Parametric equation	参数方程	Variables expressed in terms of a parameter t. / 用参数 t 表示变量。
Normal (line)	法线	A line perpendicular to the tangent at a point. / 在某点与切线垂直的直线。
Locus	轨迹	The set of all points satisfying a given condition. / 满足给定条件的所有点的集合。
Conic sections	圆锥曲线	Circles, ellipses, parabolas, hyperbolas from intersecting a plane with a cone. / 平面与圆锥相交所得的圆、椭圆、抛物线和双曲线。
Ellipse	椭圆	x²/a² + y²/b² = 1; sum of distances to two foci is constant. / 到两焦点距离之和为常数的点的轨迹。
Hyperbola	双曲线	x²/a² − y²/b² = 1; difference of distances to two foci is constant. / 到两焦点距离之差为常数的点的轨迹。
Eccentricity	离心率	Measures how elongated a conic section is. / 度量圆锥曲线拉伸程度的参数。
Directrix	准线	A fixed line used with the focus to define a conic. / 与焦点一起定义圆锥曲线的固定直线。

Parametric differentiation is a key skill: dy/dx = (dy/dt) ÷ (dx/dt). Pre-U exams love to ask for the equation of a tangent or normal at a point given in parametric form, so memorize this chain. For conic sections, be prepared to convert between Cartesian and parametric forms fluently.

参数微分是一项关键技能：dy/dx = (dy/dt) ÷ (dx/dt)。Pre-U 考试热衷于要求求出参数形式下某点的切线或法线方程，请牢记此链式法则。对于圆锥曲线，要准备好熟练地在直角坐标形式和参数形式之间转换。

1.4 Trigonometry | 三角学

English Term	中文术语	Explanation / 解释
Radian	弧度	Angle where arc length equals radius; π rad = 180°. / 弧长等于半径时所对的圆心角；π 弧度 = 180°。
Secant (sec)	正割	sec θ = 1/cos θ. / sec θ = 1/cos θ。
Cosecant (cosec)	余割	cosec θ = 1/sin θ. / cosec θ = 1/sin θ。
Cotangent (cot)	余切	cot θ = 1/tan θ = cos θ/sin θ. / cot θ = 1/tan θ = cos θ/sin θ。
Compound angle formula	复合角公式	sin(A±B) = sinAcosB ± cosAsinB; cos(A±B) = cosAcosB ∓ sinAsinB. / 用于展开两角和差的三角公式。
Double angle formula	倍角公式	sin2θ = 2sinθcosθ; cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
Harmonic form	调和形式	asinθ + bcosθ = Rsin(θ+α) where R = √(a²+b²). / asinθ + bcosθ = Rsin(θ+α)，其中 R = √(a²+b²)。
General solution	通解	For sinθ=k: θ=nπ+(−1)ⁿα; for cosθ=k: θ=2nπ±α; for tanθ=k: θ=nπ+α. / 三角方程的所有解的表达式。
Small angle approximations	小角近似	sinθ ≈ θ; cosθ ≈ 1−θ²/2; tanθ ≈ θ (θ in radians, small). / 当 θ 很小时（弧度），上述近似成立。
Arcsin / Arccos / Arctan	反正弦 / 反余弦 / 反正切	Inverse trigonometric functions; range restrictions apply. / 反三角函数；需注意值域限制。

The harmonic form Rsin(θ+α) or Rcos(θ−α) is crucial for solving equations of the type asinθ + bcosθ = c and for finding maximum/minimum values. In Pre-U, you are expected to derive R and α without a formula sheet – practice deriving R = √(a² + b²) and tanα = b/a from first principles.

调和形式 Rsin(θ+α) 或 Rcos(θ−α) 对于解 asinθ + bcosθ = c 类方程以及求极值至关重要。在 Pre-U 考试中，你需要在没有公式表的情况下推导 R 和 α – 练习从基本原理推导 R = √(a² + b²) 和 tanα = b/a。

1.5 Calculus | 微积分

English Term	中文术语	Explanation / 解释
Derivative	导数	f'(x) = lim[h→0] [f(x+h)−f(x)]/h; rate of change. / 函数的变化率，由极限定义。
Chain rule	链式法则	dy/dx = (dy/du) × (du/dx). / 复合函数求导法则。
Product rule	乘法法则	d(uv)/dx = u(dv/dx) + v(du/dx). / 两函数乘积的导数。
Quotient rule	除法法则	d(u/v)/dx = [v(du/dx) − u(dv/dx)]/v². / 两函数之前的导数。
Implicit differentiation	隐函数微分	Differentiate both sides of an equation w.r.t. x, treating y as a function of x. / 对等式两边关于 x 求导，将 y 视为 x 的函数。
Second derivative	二阶导数	d²y/dx²; gives concavity and nature of stationary points. / 用于判断凹性和驻点性质。
Stationary point	驻点	Where dy/dx = 0; can be maximum, minimum, or inflection. / dy/dx=0 处的点；可为极大、极小或拐点。
Point of inflection	拐点	Where concavity changes; f”(x) = 0 and changes sign. / 凹性改变的点；f”(x)=0 且符号改变。
Connected rates of change	相关变化率	Relating two rates via the chain rule. / 通过链式法则关联两个变化率。
Definite integral	定积分	∫ₐᵇ f(x)dx; gives the signed area under a curve. / 曲线下的带符号面积。
Indefinite integral	不定积分	∫ f(x)dx = F(x) + C; the antiderivative. / 反导数，需加积分常数 C。
Integration by substitution	换元积分	Let u = g(x), then ∫ f(g(x))g'(x)dx = ∫ f(u)du. / 通过变量替换简化积分。
Integration by parts	分部积分	∫ u(dv/dx)dx = uv − ∫ v(du/dx)dx. / 基于乘法法则的积分技巧。
Improper integral	反常积分	Integral with infinite limit(s) or an integrand with a vertical asymptote. / 积分限含无穷或被积函数含垂直渐近线。
Volume of revolution	旋转体体积	V = π∫ y² dx (about x-axis) or V = π∫ x² dy (about y-axis). / 绕 x 轴：V = π∫ y² dx；绕 y 轴：V = π∫ x² dy。
Differential equation	微分方程	An equation involving a function and its derivatives. / 含有函数及其导数的方程。
Separation of variables	分离变量	Rewriting dy/dx = f(x)g(y) as ∫ dy/g(y) = ∫ f(x)dx. / 将 dy/dx = f(x)g(y) 改写并分别积分。

Pre-U calculus demands fluency in all differentiation and integration techniques. A critical skill is choosing the right method for integration: substitution when you see a function and its derivative, parts when you have a product of functions from different families (e.g., x·eˣ), and partial fractions for rational functions. For differential equations, always check whether the equation is separable; if it is, the method of separation of variables gives a straightforward path to the general solution. Remember to include the constant of integration and use initial conditions to find particular solutions.

Pre-U 的微积分要求熟练掌握所有微分和积分技巧。一项关键技能是为积分选择正确的方法：当看到函数及其导数时用换元法，当遇到不同族函数的乘积（如 x·eˣ）时用分部积分，对有理函数用部分分式法。对于微分方程，始终检查方程是否可分离；如果可以，分离变量法提供了通向通解的直路。记住加上积分常数，并利用初始条件求特解。

1.6 Vectors | 向量

English Term	中文术语	Explanation / 解释
Magnitude	模 / 大小	|v| = √(x²+y²+z²); the length of a vector. / 向量的长度。
Unit vector	单位向量	A vector with magnitude 1; v̂ = v/|v|. / 模为 1 的向量。
Scalar (dot) product	标量积（点积）	a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃. / 结果是一个标量。
Vector (cross) product	向量积（叉积）	a×b = |a||b|sinθ n̂; gives a vector perpendicular to both. / 结果是一个垂直于两向量的向量。
Position vector	位置向量	The vector from the origin to a point. / 从原点到某一点的向量。
Vector equation of a line	直线的向量方程	r = a + λd; where a is a point on the line and d is the direction vector. / a 是直线上一点，d 是方向向量。
Vector equation of a plane	平面的向量方程	r·n = a·n (scalar product form) or r = a + λd₁ + μd₂ (parametric form). / 点法式：r·n = a·n；参数式：r = a + λd₁ + μd₂。
Angle between vectors	向量夹角	cosθ = (a·b)/(|a||b|). / 通过点积公式计算。
Angle between line and plane	直线与平面夹角	sinθ = |d·n|/(|d||n|), where d is the line direction and n is the plane normal. / d 为直线方向向量，n 为平面法向量。
Intersection of line and plane	直线与平面的交点	Substitute parametric line equation into plane equation, solve for λ. / 将直线参数方程代入平面方程，解出 λ。
Distance from point to plane	点到平面距离	|(r·n − a·n)|/|n|, where r is the point and n is the plane normal. / r 为点，n 为平面法向量。

Vector questions in Pre-U often combine geometry with algebraic manipulation. The cross product (a×b) is a three-dimensional concept that does not exist in two dimensions – its magnitude represents the area of the parallelogram spanned by a and b. A common exam question asks for the shortest distance from a point to a line or plane; the shortest distance from point P to line r = a + λd is |(a−p)×d|/|d|.

Pre-U 中的向量题常将几何与代数运算结合。叉积（a×b）是三维概念，在二维中不存在 – 其模表示由 a 和 b 张成的平行四边形的面积。常见的考题是求点到直线或平面的最短距离；点 P 到直线 r = a + λd 的最短距离为 |(a−p)×d|/|d|。

1.7 Complex Numbers | 复数

English Term	中文术语	Explanation / 解释
Imaginary unit (i)	虚数单位	i² = −1. / i² = −1。
Complex number	复数	z = x + iy; x is the real part, y the imaginary part. / x 为实部，y 为虚部。
Complex conjugate	共轭复数	z* = x − iy; used to rationalise denominators. / 用于分母有理化。
Modulus	模	|z| = √(x²+y²); the distance from the origin in the complex plane. / 复平面上到原点的距离。
Argument	辐角	arg(z) = θ where tanθ = y/x; the angle from the positive real axis. / 从正实轴量起的角度。
Polar form	极坐标形式	z = r(cosθ + isinθ) where r = |z|, θ = arg(z). / 用模和辐角表示复数。
Euler’s formula	欧拉公式	e^(iθ) = cosθ + isinθ. / 联系指数函数与三角函数的桥梁。
Exponential form	指数形式	z = re^(iθ). / 复数最简洁的表示形式。
De Moivre’s theorem	棣莫弗定理	(cosθ + isinθ)ⁿ = cos(nθ) + isin(nθ). / 复数幂运算的核心定理。
Roots of unity	单位根	The n solutions to zⁿ = 1; equally spaced on the unit circle. / zⁿ=1 的 n 个解，在单位圆上均匀分布。
Fundamental Theorem of Algebra	代数基本定理	Every non-constant polynomial has at least one complex root. / 每个非常数多项式至少有一个复根。

De Moivre’s theorem is one of the most powerful tools in Pre-U mathematics. It not only simplifies the computation of powers of complex numbers but also allows for the derivation of trigonometric identities (e.g., expressing cos5θ in terms of cosθ). The exponential form z = re^(iθ) makes multiplication and division trivial: z₁z₂ = r₁r₂e^(i(θ₁+θ₂)). For finding nth roots of a complex number, add multiples of 2π to the argument before dividing by n.

棣莫弗定理是 Pre-U 数学中最强大的工具之一。它不仅能简化复数幂的计算，还能推导三角恒等式（例如，将 cos5θ 用 cosθ 表示）。指数形式 z = re^(iθ) 使乘除运算变得简单：z₁z₂ = r₁r₂e^(i(θ₁+θ₂))。求复数的 n 次方根时，需在辐角上加 2π 的整数倍后再除以 n。

1.8 Matrices & Transformations | 矩阵与变换

English Term	中文术语	Explanation / 解释
Matrix	矩阵	A rectangular array of numbers with m rows and n columns. / m 行 n 列的数字阵列。
Determinant	行列式	det(M) for 2×2: ad−bc; zero means singular (non-invertible). / 若为零，则矩阵为奇异矩阵（不可逆）。
Inverse matrix	逆矩阵	M⁻¹ such that MM⁻¹ = I; for 2×2: (1/det)[d −b; −c a]. / 满足 MM⁻¹=I 的矩阵。
Identity matrix (I)	单位矩阵	Diagonal entries are 1, all others 0; like “1” for matrices. / 对角线为 1，其余为 0。
Eigenvalue	特征值	λ satisfying det(M−λI) = 0. / 满足 det(M−λI) = 0 的 λ。
Eigenvector	特征向量	Non-zero v such that Mv = λv. / 满足 Mv=λv 的非零向量。
Characteristic equation	特征方程	det(M−λI) = 0; gives eigenvalues. / 用于求特征值的方程。
Diagonalisation	对角化	M = PDP⁻¹ where D is a diagonal matrix of eigenvalues. / D 为特征值构成的对角矩阵。
Linear transformation	线性变换	A mapping from one vector space to another preserving addition and scalar multiplication. / 保持加法和标量乘法的向量空间映射。
Invariant line	不变直线	A line mapped to itself under a transformation (points may move along it). / 在变换下映射到自身的直线。
Invariant point	不动点	A point mapped to itself under a transformation. / 在变换下映射到自身的点。

Matrix algebra in Pre-U extends beyond simple 2×2 transformations. Students must be comfortable finding eigenvalues and eigenvectors for 2×2 and 3×3 matrices, then using them to diagonalise matrices. A key application is solving systems of coupled differential equations: if dx/dt = Mx, diagonalising M yields uncoupled equations that are easily solved. The distinction between invariant points (Mx = x) and invariant lines (Mx = λx + c, where the line as a set is preserved) is a subtle point often examined.

Pre-U 中的矩阵代数超出了简单的 2×2 变换。学生需能求解 2×2 和 3×3 矩阵的特征值和特征向量，并利用它们对角化矩阵。一个关键应用是求解耦合微分方程组：若 dx/dt = Mx，对角化 M 得到解耦的方程，易于求解。不动点（Mx=x）与不变直线（直线整体被保留）之间的微妙区别，经常成为考点。

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2. Probability & Statistics | 概率与统计

English Term	中文术语	Explanation / 解释
Sample space	样本空间	The set of all possible outcomes of an experiment. / 试验所有可能结果的集合。
Event	事件	A subset of the sample space. / 样本空间的子集。
Mutually exclusive	互斥	Events that cannot happen simultaneously; P(A∩B)=0. / 不能同时发生的事件。
Independent events	独立事件	P(A∩B)=P(A)×P(B); the occurrence of one does not affect the other. / 一个事件的发生不影响另一个事件的概率。
Conditional probability	条件概率	P(A|B) = P(A∩B)/P(B); probability of A given B has occurred. / 在 B 已发生的条件下 A 发生的概率。
Bayes’ theorem	贝叶斯定理	P(A|B) = P(B|A)·P(A)/P(B). / 用于反转条件概率。
Random variable	随机变量	A variable whose value depends on the outcome of a random experiment. / 取值取决于随机试验结果的变量。
Discrete distribution	离散分布	A distribution where the random variable takes countable values. / 随机变量可取值的集合是可数的。
Continuous distribution	连续分布	A distribution where the random variable takes any value in an interval. / 随机变量可取区间内任意值。
Probability density function (pdf)	概率密度函数	f(x) ≥ 0; ∫f(x)dx = 1; P(a<X<b) = ∫ₐᵇ f(x)dx. / 曲线下面积代表概率。
Cumulative distribution function (cdf)	累积分布函数	F(x) = P(X ≤ x) = ∫₋∞ˣ f(t)dt. / 到 x 为止的累积概率。
Expectation / Mean	期望 / 均值	E(X) = Σxp(x) (discrete) or ∫xf(x)dx (continuous). / 随机变量的加权平均。
Variance	方差	Var(X) = E[(X−μ)²] = E(X²)−[E(X)]². / 度量离散程度。
Standard deviation	标准差	σ = √Var(X). / 方差的平方根。
Binomial distribution	二项分布	X~B(n,p): P(X=r) = ⁿCᵣ pʳ(1−p)ⁿ⁻ʳ; n fixed trials, p constant. / n 次固定试验，每次成功概率为 p。
Poisson distribution	泊松分布	X~Po(λ): P(X=r) = e^(−λ)·λʳ/r!; models rare events. / 用于建模稀有事件。
Normal distribution	正态分布	X~N(μ,σ²); bell-shaped, symmetric. Standardise: Z=(X−μ)/σ ~ N(0,1). / 钟形对称分布。标准化：Z=(X−μ)/σ。
Central Limit Theorem	中心极限定理	Sample mean approximates normal as n→∞ regardless of the population distribution. / 无论总体分布如何，样本均值在大样本下近似正态。
Hypothesis test	假设检验	A procedure to decide whether to reject a null hypothesis H₀. / 决定是否拒绝原假设 H₀ 的程序。
Null hypothesis (H₀)	原假设	The default assumption being tested. / 被检验的默认假设。
Alternative hypothesis (H₁)	备择假设	The claim we accept if H₀ is rejected. / 当 H₀ 被拒绝时接受的假设。
Significance level (α)	显著性水平	The probability of a Type I error (rejecting H₀ when true); typically 5% or 1%. / 第一类错误的概率；通常取 5% 或 1%。
p-value	p 值	The probability of obtaining a result at least as extreme as observed, assuming H₀ is true. / 在 H₀ 成立时观测到至少与当前结果同等极端的概率。
Type I error	第一类错误	Rejecting H₀ when it is true (false positive). / 原假设为真时拒绝它（假阳性）。
Type II error	第二类错误	Failing to reject H₀ when it is false (false negative). / 原假设为假时未能拒绝它（假阴性）。
Correlation coefficient (r)	相关系数	Measures linear association between two variables; −1 ≤ r ≤ 1. / 度量两变量之间的线性关联程度。
Regression	回归	Finding the line of best fit; y = a + bx. / 寻找最佳拟合直线。
Least squares	最小二乘法	Minimising Σ(yᵢ − ŷᵢ)² to find regression coefficients. / 最小化残差平方和以求回归系数。

Statistics in Pre-U demands both computational fluency and conceptual understanding. Hypothesis testing questions typically follow a structured approach: (1) State H₀ and H₁ clearly, (2) Identify the test statistic and its distribution, (3) Calculate the test statistic from the data, (4) Find the p-value or critical region, (5) Compare with the significance level and draw a conclusion in context. Never forget to interpret your conclusion in the context of the problem – marks are routinely awarded for contextual interpretation. For the normal distribution, always sketch a bell curve and shade the relevant area; this reduces careless sign errors.

Pre-U 的统计部分要求计算熟练度和概念理解并重。假设检验题目通常遵循结构化步骤：(1) 明确表述 H₀ 和 H₁；(2) 确定检验统计量及其分布；(3) 从数据计算检验统计量；(4) 求 p 值或临界区域；(5) 与显著性水平比较，并在上下文语境中得出结论。永远不要忘记在问题语境中解释你的结论 – 评分标准通常会为语境解释赋分。对于正态分布，始终画出钟形曲线并标记相应区域；这可以减少因符号疏忽导致的错误。

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3. Mechanics | 力学

English Term	中文术语	Explanation / 解释
Displacement	位移	Vector from initial to final position; not the same as distance. / 从初始位置到最终位置的向量；不同于路程。
Velocity	速度	Rate of change of displacement; v = ds/dt. / 位移的变化率；是向量。
Speed	速率	Magnitude of velocity; a scalar. / 速度的大小；是标量。
Acceleration	加速度	Rate of change of velocity; a = dv/dt = d²s/dt². / 速度的变化率。
Suvat equations	Suvat 方程	v = u+at; s = ut+½at²; v² = u²+2as; s = ½(u+v)t; s = vt−½at². For constant acceleration only. / 仅适用于匀加速运动。
Free fall	自由落体	Motion under gravity alone; a = g ≈ 9.8 ms⁻² downward. / 仅在重力作用下的运动。
Projectile motion	抛体运动	Horizontal: constant velocity; Vertical: constant acceleration g. / 水平方向：匀速；竖直方向：匀加速 g。
Range	射程	Horizontal distance travelled by a projectile before landing. / 抛体落地前水平方向行进的距离。
Time of flight	飞行时间	Total time from launch to landing. / 从发射到落地的总时间。
Newton’s First Law	牛顿第一定律	A body remains at rest or in uniform motion unless acted upon by a net force. / 物体在不受外力作用时保持静止或匀速直线运动。
Newton’s Second Law	牛顿第二定律	F = ma; net force equals mass times acceleration. / 合力等于质量乘以加速度。
Newton’s Third Law	牛顿第三定律	For every action there is an equal and opposite reaction. / 作用力与反作用力大小相等、方向相反。
Friction	摩擦力	F ≤ μR; opposes motion, where μ is the coefficient of friction and R is the normal reaction. / 阻碍运动，μ 为摩擦系数，R 为正压力。
Limiting friction	极限摩擦力	F = μR; maximum value of friction just before motion begins. / 运动开始前的最大摩擦力。
Tension	张力	Force transmitted through a string, cable, or rod when pulled. / 通过绳、缆或杆传递的拉力。
Thrust / Compression	推力 / 压力	Force transmitted through a rod when pushed; opposite of tension. / 通过杆传递的推力；与张力方向相反。
Resolving forces	力的分解	Splitting a force into perpendicular components (typically horizontal and vertical). / 将力分解为垂直分量（通常为水平和竖直）。
Equilibrium	平衡	Net force = 0 and net moment = 0; body is at rest or constant velocity. / 合力为零且合力矩为零。
Moment (of a force)	力矩	Force × perpendicular distance from pivot; tendency to rotate. / 力乘以到支点的垂直距离；产生转动的趋势。
Centre of mass	质心	The point where the entire mass of a body can be considered concentrated for translational motion. / 物体质量可视为集中的点。
Impulse	冲量	Force × time = change in momentum; J = Ft = mv − mu. / 力乘以作用时间等于动量变化。
Momentum	动量	p = mv; conserved in the absence of external forces. / 无外力时动量守恒。
Conservation of momentum	动量守恒	m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ for collisions with no external forces. / 无外力碰撞中的总动量不变。
Elastic collision	弹性碰撞	Both momentum and kinetic energy are conserved. / 动量和动能均守恒。
Inelastic collision	非弹性碰撞	Momentum is conserved but kinetic energy is not. / 动量守恒但动能不守恒。
Coefficient of restitution (e)	恢复系数	e = (speed of separation)/(speed of approach); 0 ≤ e ≤ 1. / 分离速度与接近速度之比。
Work	功	Force × distance moved in the direction of the force. / 力乘以在力的方向上移动的距离。
Kinetic energy (KE)	动能	KE = ½mv². / 由于运动而具有的能量。
Gravitational potential energy (GPE)	重力势能	GPE = mgh (relative to a reference level). / 相对于参考水平面的重力势能。
Conservation of energy	能量守恒	Total energy is constant in the absence of external work. / 无外力做功时总能量不变。
Power	功率	Rate of doing work; P = Fv. / 做功的速率。

Mechanics in Pre-U is heavily applied: you will model real-world scenarios such as a car towing a caravan, a particle sliding down a rough inclined plane, or a projectile fired from a cliff. The key to success is drawing a clear, well-labelled force diagram before writing any equations. Resolve forces parallel and perpendicular to the plane of motion. For connected particles (e.g., via a light inextensible string over a smooth pulley), write F = ma for each particle separately and link them via the string constraint (same acceleration magnitude, same tension magnitude). When friction is involved, always check whether it is limiting (F = μR) or not; the phrase “on the point of slipping” signals limiting friction.

Pre-U 中的力学强调应用：你需要对现实场景建模，例如汽车牵引拖车、物块沿粗糙斜面下滑、或从悬崖上发射的抛体。成功的关键在于列方程之前画一个清晰、标注完整的力图。沿运动平面方向及垂直方向分解力。对于连接体问题（例如通过轻质不可伸长绳跨过光滑滑轮连接的两个物体），分别对每个物体写 F=ma，并通过绳的约束条件（加速度大小相同、张力大小相同）关联起来。涉及摩擦力时，始终检查是否为极限摩擦力（F = μR）；”即将滑动”这句话就是极限摩擦力的信号。

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4. Exam Tips & Common Pitfalls | 考试技巧与常见陷阱

4.1 Reading the Question | 仔细审题

Many marks are lost because students misread command words. “Find” means calculate; “Show that” or “Prove” means you must provide a logical argument – the given answer is a hint, not something you can assume; “Hence” means use the previous result; “Hence or otherwise” means you may use an alternative method but the “hence” route is usually quicker; “Deduce” means reason from given information; “State” means write down without working; “Verify” means check by substitution. These distinctions are especially important in the Pre-U where proof-based questions carry significant weight.

很多分数丢在学生对指令词的理解错误上。“Find” 意味着计算；“Show that” 或 “Prove” 意味着你必须提供逻辑论证 – 给出的答案是一个提示，而不是你可以直接假设的结论；“Hence” 意味着使用前面的结果；“Hence or otherwise” 意味着你可以使用替代方法，但”hence”路径通常更快捷；“Deduce” 意味着从给定信息进行推理；“State” 意味着直接写出答案，无需过程；“Verify” 意味着通过代入进行检查。这些区别在 Pre-U 考试中尤为重要，因为证明类题目占比较大的分值。

4.2 Precision in Notation | 符号的精确性

Using correct mathematical notation is essential in Pre-U. Some common errors: using “=” instead of “≈” for approximations; omitting the “+C” in indefinite integrals; writing “dy/dx = …” without specifying what dy/dx actually equals; confusing f⁻¹(x) (inverse function) with [f(x)]⁻¹ (reciprocal); using degrees when radians are required for calculus. Always check whether your calculator is in the correct mode before starting any trigonometric calculation.

在 Pre-U 中使用正确的数学符号至关重要。一些常见错误：用”=”替代”≈”表示近似值；不定积分中遗漏”+C”；写”dy/dx = …”却没有具体写出 dy/dx 等于什么；混淆 f⁻¹(x)（反函数）和 [f(x)]⁻¹（倒数）；在微积分需要弧度时使用角度制。在开始任何三角计算之前，务必检查计算器是否处于正确的模式。

4.3 Checking Your Work | 检查你的解答

Develop the habit of sanity-checking your answers. If you find that a probability is negative or greater than 1, you have made an error. If a length or magnitude is negative, check your sign conventions. If your answer to a mechanics problem implies a car accelerates at 500 ms⁻², reconsider your working. A useful technique is dimensional analysis: if you expect a length but your expression has dimensions of time, something is wrong. For pure mathematics, substituting your answer back into the original equation is the gold standard of verification.

养成对答案进行合理性检查的习惯。如果你算出的概率为负数或大于 1，那一定有错误。如果长度或模为负数，检查你的符号约定。如果你的力学答案暗示汽车的加速度为 500 ms⁻²，重新审视你的解题过程。一个有用的技巧是量纲分析：如果你期望得到长度，但表达式的量纲是时间，那一定有问题。对于纯数学，将答案代回原方程是验证的黄金标准。

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Conclusion | 结语

Mastering the vocabulary of Pre-U CIE Mathematics is a vital step toward examination success. Each term in this guide is a building block – when you encounter it in a question, you should instantly recognise what is being asked and which technique to apply. Build a systematic revision plan: create flashcards with the English term on one side and the Chinese equivalent plus a one-sentence explanation on the other; practise past paper questions while consciously using the correct terminology in your written solutions; and explain concepts aloud in both languages to strengthen your bilingual mathematical fluency.

掌握 Pre-U CIE 数学的词汇是通向考试成功的关键一步。本指南中的每个术语都是一块基石 – 当你在考题中遇到它时，你应当立刻识别出题目在问什么以及应该运用哪种技巧。制定系统的复习计划：制作闪卡，一面写英文术语，另一面写中文对应词及一句话解释；在做往年真题时有意识地在书面解答中使用正确的术语；并用两种语言口头解释概念，以加强你的双语数学表达能力。

Good luck with your studies! | 祝你学习顺利！