Pre-U CAIE Mathematics: Core Topic Review | Pre-U CAIE 数学：核心知识点梳理

📚 Pre-U CAIE Mathematics: Core Topic Review | Pre-U CAIE 数学：核心知识点梳理

The Cambridge Pre-U Mathematics syllabus (9794) builds a deep and interconnected understanding of pure mathematics, probability, statistics and mechanics. This article systematically reviews the core knowledge points, serving as a concise yet comprehensive checklist for students preparing for examinations. We present essential concepts, key notations and typical applications across all major areas of the course.

剑桥 Pre-U 数学课程（9794）旨在使学生对纯数学、概率、统计和力学形成深入且贯通的理解。本文系统梳理核心知识点，为备考学生提供一份简明而全面的检查清单。我们将展示课程各主要领域中的基本概念、关键符号以及典型应用。

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1. Algebra and Functions | 代数与函数

Manipulation of polynomials: factor theorem, remainder theorem, division algorithm and partial fractions. Rational functions, their asymptotes and graph sketching are central to later analysis. The modulus function |x| is treated geometrically and algebraically; solving equations and inequalities involving |f(x)| requires splitting into cases or squaring under conditions.

多项式的操作：因式定理、余项定理、除法算式及部分分式。有理函数及其渐近线与图像绘制是后续分析的基础。绝对值函数 |x| 从几何与代数两个层面处理；解含有 |f(x)| 的方程和不等式需要分情况讨论或在适当条件下平方。

Exponential and logarithmic functions: y = eˣ and y = ln x, the laws of logarithms, change of base and solving exponential equations. Understanding the effect of transformations such as f(x + a), f(ax), |f(x)| and f(|x|) on graphs is essential for function analysis.

指数与对数函数：y = eˣ 和 y = ln x，对数的运算法则、换底公式以及指数方程的求解。理解函数变换，如 f(x + a)、f(ax)、|f(x)| 和 f(|x|) 对图形的影响，对函数分析至关重要。

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2. Trigonometry | 三角学

Radian measure, arc length s = rθ and sector area A = ½ r²θ. Fundamental identities: sin²θ + cos²θ ≡ 1, 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ. Solving trigonometric equations, including those involving compound angles, double angles (sin 2θ, cos 2θ, tan 2θ) and the forms a sin θ ± b cos θ expressed as R sin(θ ± α) or R cos(θ ± α).

弧度制，弧长 s = rθ 与扇形面积 A = ½ r²θ。基本恒等式：sin²θ + cos²θ ≡ 1，1 + tan²θ ≡ sec²θ，1 + cot²θ ≡ cosec²θ。求解三角方程，包括涉及复合角、倍角（sin 2θ，cos 2θ，tan 2θ）的方程，以及将 a sin θ ± b cos θ 表示为 R sin(θ ± α) 或 R cos(θ ± α) 的形式。

The graphs of sine, cosine and tangent functions, their symmetries and periodicities. Inverse trigonometric functions arcsin, arccos and arctan, their domains, ranges and principal values are applied in solving equations and in calculus.

正弦、余弦和正切函数的图像及其对称性与周期性。反三角函数 arcsin、arccos 和 arctan 的定义域、值域和主值，在解方程和微积分中均有应用。

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3. Coordinate Geometry and Curves | 坐标几何与曲线

Straight lines: gradient, intercepts, distance between two points, midpoints. Equation of a circle in standard and general forms; finding tangents and normals to a circle. Parametric equations of curves, converting between parametric and Cartesian forms, and finding gradients using parametric differentiation.

直线：斜率、截距、两点间距离、中点。圆的标准方程与一般方程；求圆的切线与法线。曲线的参数方程，在参数形式与笛卡儿形式之间转换，并使用参数微分法求导数。

Loci problems that lead to equations of circles, lines or parabolas. Understanding the geometry of conic sections is explored through coordinate methods and later linked to vector descriptions.

由轨迹问题得出圆、直线或抛物线的方程。通过坐标方法理解圆锥曲线的几何，后期这些内容会与向量描述联系起来。

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4. Sequences and Series | 数列与级数

Arithmetic and geometric sequences: formulas for the n-th term and sum of the first n terms. Convergence of a geometric series when the common ratio |r| < 1; sum to infinity S∞ = a / (1 − r). The binomial expansion for (1 + x)ⁿ, where n is rational, and the range of validity |x| < 1. Expansion of (a + b)ⁿ using the general binomial theorem.

等差数列与等比数列：第 n 项及前 n 项和的公式。等比级数当公比 |r| < 1 时的收敛性；无穷和 S∞ = a / (1 − r)。二项展开式 (1 + x)ⁿ，其中 n 为有理数，适用范围为 |x| < 1。利用一般二项式定理展开 (a + b)ⁿ。

Sequences defined by recurrence relations, such as xₙ₊₁ = f(xₙ), and their behaviour. Use of sigma notation Σ for sums, basic properties and applications in proofs by induction for series sums.

由递推关系定义的数列，如 xₙ₊₁ = f(xₙ)，及其性态。应用求和符号 Σ 的基本性质，并在数列求和的归纳证明中使用。

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5. Complex Numbers | 复数

The imaginary unit i with i² = −1. Representation of a complex number z = a + ib in Cartesian form; conjugate z* = a − ib, modulus |z| = √(a² + b²) and argument arg(z). Operations: addition, subtraction, multiplication, division. Modulus-argument (polar) form z = r(cos θ + i sin θ) and the exponential form z = r e^(iθ).

虚数单位 i 满足 i² = −1。复数的表示：笛卡儿形式 z = a + ib；共轭复数 z* = a − ib，模 |z| = √(a² + b²) 与辐角 arg(z)。四则运算：加、减、乘、除。模–辐角（极）形式 z = r(cos θ + i sin θ) 以及指数形式 z = r e^(iθ)。

De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, used to find powers and roots of complex numbers. Loci in the Argand diagram, such as |z − a| = r (circle) and arg(z − a) = α (half-line). Solving polynomial equations with complex roots, including the fundamental theorem of algebra.

棣莫弗定理：(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ，用于求复数的幂与根。阿尔冈图上的轨迹，如 |z − a| = r（圆）和 arg(z − a) = α（射线）。求解具有复数根的多项式方程，包括代数基本定理的应用。

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6. Differentiation | 微分

The derivative as the limit of a difference quotient. Standard derivatives: d/dx (xⁿ) = n xⁿ⁻¹, derivatives of eˣ, ln x, sin x, cos x, tan x, and their inverses. The chain rule, product rule and quotient rule are used routinely.

导数作为差商的极限。标准导数：d/dx (xⁿ) = n xⁿ⁻¹，eˣ、ln x、sin x、cos x、tan x 及其反函数的导数。链式法则、乘积法则和商法则是常规工具。

d/dx [f(g(x))] = f'(g(x)) g'(x)

Implicit differentiation for relations where y cannot be easily isolated; parametric differentiation dy/dx = (dy/dt) / (dx/dt). Applications: finding gradients, equations of tangents and normals, stationary points and their classification via the second derivative test. Connected rates of change and small changes using differentials.

隐函数求导法，用于不易解出 y 的关系式；参数求导法 dy/dx = (dy/dt) / (dx/dt)。应用：求斜率、切线与法线方程、驻点，并通过二阶导数判别其类型。关联变化率以及利用微分求微小变化。

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7. Integration | 积分

Integration as the reverse process of differentiation. Standard integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ 1/x dx = ln|x| + C, integrals of eˣ, sin x, cos x, sec²x, etc. Definite integrals and the fundamental theorem of calculus: ∫ₐᵇ f(x) dx = F(b) − F(a).

积分作为微分的逆运算。标准积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1），∫ 1/x dx = ln|x| + C，eˣ、sin x、cos x、sec²x 等函数的积分。定积分与微积分基本定理：∫ₐᵇ f(x) dx = F(b) − F(a)。

Methods of integration: substitution, integration by parts (∫ u dv = uv − ∫ v du), and the use of partial fractions. Integration of rational functions, trigonometric integrals and use of trigonometric identities to simplify integrands. Applications: area under a curve, area between curves, volumes of revolution about the x- and y-axes.

积分方法：换元积分法、分部积分法（∫ u dv = uv − ∫ v du）以及部分分式法。有理函数积分、三角函数的积分，并利用三角恒等式简化被积函数。应用：曲线下的面积、曲线间的面积，以及绕 x 轴和 y 轴旋转的体积。

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8. Differential Equations | 微分方程

Formation of differential equations from physical contexts. First order separable equations: dy/dx = g(x)h(y) solved by separating variables. Differential equations of the form dy/dx = f(x) or dy/dx = f(y) and the interpretation of their slope fields and solution curves.

从物理情境建立微分方程。一阶可分离方程：dy/dx = g(x)h(y)，通过分离变量求解。形如 dy/dx = f(x) 或 dy/dx = f(y) 的微分方程，及其斜率场与解曲线的解释。

Modelling growth and decay (e.g., exponential growth dy/dt = ky). Second order homogeneous linear differential equations with constant coefficients: a d²y/dx² + b dy/dx + c y = 0. Solving using the auxiliary equation am² + bm + c = 0 and giving general solutions in terms of distinct real, repeated real or complex conjugate roots.

建立增长与衰减模型（例如，指数增长 dy/dt = ky）。二阶常系数齐次线性微分方程：a d²y/dx² + b dy/dx + c y = 0。使用辅助方程 am² + bm + c = 0 求解，并根据相异实根、重实根或共轭复根给出通解。

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9. Vectors | 向量

Vectors in two and three dimensions, unit vectors i, j, k. Magnitude, direction, addition, scalar multiplication. Position vectors and the vector equation of a line: r = a + λ b. Scalar (dot) product: a · b = |a||b| cos θ; condition for perpendicularity and calculation of angle between vectors.

二维和三维向量，单位向量 i，j，k。向量的模、方向、加法与标量乘法。位置向量与直线的向量方程：r = a + λ b。数量积（点积）：a · b = |a||b| cos θ；垂直条件以及两向量夹角的计算。

The vector equation of a plane: r = a + λ u + μ v, or in scalar product form r · n = p. Finding intersections between lines and planes, distances from a point to a line or plane. Vector cross product only required in Further Mathematics; here the emphasis is on dot product applications.

平面的向量方程：r = a + λ u + μ v，或用数量积形式 r · n = p。求直线与平面的交点、点到直线或平面的距离。叉积仅在进阶数学中要求；此处重点在于点积的应用。

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10. Probability and Statistics | 概率与统计

Basic probability concepts: sample spaces, events, independence, conditional probability P(A|B) = P(A ∩ B)/P(B). Bayes’ theorem is applied informally through tree diagrams and two-way tables. Discrete random variables: probability mass functions, expectation E(X), variance Var(X) and their properties.

基本概率概念：样本空间、事件、独立性、条件概率 P(A|B) = P(A ∩ B)/P(B)。通过树形图和双向表非正式地应用贝叶斯定理。离散随机变量：概率质量函数、期望 E(X)、方差 Var(X) 及其性质。

Standard distributions: binomial B(n, p) and Poisson Po(λ). Conditions for use, probability calculations, mean and variance. Normal distribution N(μ, σ²): use of standardised variable Z = (X − μ)/σ, symmetry and table reading. Approximations: Poisson to binomial (n large, p small) and normal approximation to binomial/Poisson (continuity correction).

标准分布：二项分布 B(n, p) 与泊松分布 Po(λ)。使用条件、概率计算、均值与方差。正态分布 N(μ, σ²)：使用标准化变量 Z = (X − μ)/σ，对称性及查表。近似：泊松近似二项（n 大，p 小）以及正态近似二项/泊松（连续性校正）。

Inferential statistics: confidence intervals for a population mean (variance known or unknown using t-distribution) and for a proportion. Hypothesis testing: null and alternative hypotheses, significance level, one-tailed and two-tailed tests, p-value and conclusions in context.

推断统计：总体均值的置信区间（方差已知或使用 t 分布），以及比例的置信区间。假设检验：原假设与备择假设、显著性水平、单侧与双侧检验、p 值以及在具体情境中得出结论。

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11. Mechanics – Motion and Forces | 力学 – 运动与力

Kinematics in one dimension: displacement, velocity, acceleration, and the use of constant acceleration (suvat) equations: v = u + at, s = ut + ½ at², v² = u² + 2as, s = ½ (u+v)t. Graphical representation: displacement-time and velocity-time graphs; gradient and area interpretations.

一维运动学：位移、速度、加速度，以及匀加速（suvat）方程的使用：v = u + at，s = ut + ½ at²，v² = u² + 2as，s = ½ (u+v)t。图形表示：位移–时间图和速度–时间图；斜率与面积的解释。

Dynamics: Newton’s laws of motion. Forces as vectors, resolving forces, equilibrium and resultant forces. Free body diagrams, friction (F ≤ μR) and motion on inclined planes. Connected particles over pulleys and in lifts, applying F = ma to each particle.

动力学：牛顿运动定律。力作为向量，力的分解，平衡与合力。受力图、摩擦力（F ≤ μR）以及斜面上的运动。通过滑轮连接的物体和电梯中的运动，对每个质点应用 F = ma。

Work, energy and power: work done by a constant force, kinetic energy ½ mv², potential energy mgh, conservation of mechanical energy when only conservative forces act. Power P = Fv for motion at constant speed.

功、能与功率：恒力做功、动能 ½ mv²、势能 mgh、仅受保守力作用时的机械能守恒。匀速运动时功率 P = Fv。

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12. Numerical Methods | 数值方法

Iterative methods for finding roots of an equation f(x) = 0. The sign-change principle to locate intervals containing a root. Fixed-point iteration using rearrangements x = g(x); understanding convergence and staircase/cobweb diagrams.

求方程 f(x) = 0 根的迭代法。利用符号变化原理确定包含根的区间。使用重写形式 x = g(x) 的不动点迭代；理解收敛性及阶梯图/蛛网图。

The Newton-Raphson method: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Derivation, application and conditions for convergence (e.g., sensitivity to initial value). Numerical integration: trapezium rule to approximate ∫ₐᵇ f(x) dx; improving accuracy by increasing the number of strips. Estimating errors in these methods is discussed qualitatively.

牛顿–拉夫森法：xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)。其推导、应用以及收敛条件（例如对初始值的敏感性）。数值积分：梯形法则近似 ∫ₐᵇ f(x) dx；通过增加条带数提高精度。对这些方法的误差进行定性讨论。

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