📚 Pre-U OCR Mathematics: Core Concepts Review | Pre-U OCR 数学：核心知识点梳理
In Pre-U OCR Mathematics, a deep understanding of core concepts is essential for success. This article provides a structured review of the key topics, including advanced algebra, calculus, complex numbers, vectors, and statistical methods. Each section is designed to reinforce your knowledge and highlight important formulas and theorems.
在 Pre-U OCR 数学中，深入理解核心概念是成功的关键。本文对关键主题进行了结构化梳理，涵盖高等代数、微积分、复数、向量和统计方法等。每个部分旨在强化你的知识并突出重要的公式和定理。
1. Advanced Algebra and Functions | 进阶代数与函数
Algebraic manipulation at the Pre-U level requires fluency with polynomial division, partial fractions, and manipulation of rational functions. You must be able to factorise cubic and quartic expressions and apply the remainder and factor theorems.
在 Pre-U 级别，代数操作要求熟练掌握多项式除法、部分分式以及有理函数的处理。你必须能够因式分解三次和四次表达式，并应用余式定理和因式定理。
Understanding the domain and range of composite functions, and finding inverse functions, is fundamental. Sketching graphs of modulus, reciprocal, and piecewise-defined functions is frequently examined.
理解复合函数的定义域和值域，以及求反函数是基础。绘制模函数、倒函数和分段定义函数的图像是常见的考查内容。
The binomial expansion for rational or negative exponents is extended to (1+x)ⁿ for |x|<1, requiring use of the general binomial coefficient.
对于有理数或负指数，二项式展开被推广到 (1+x)ⁿ（其中 |x|<1），需要使用一般二项式系数。
(1 + x)ⁿ = 1 + nx + [n(n-1)/2!] x² + … + [n(n-1)…(n-r+1)/r!] xʳ + …
2. Differentiation Techniques | 微分技巧
The derivative is defined from first principles, but Pre-U candidates must confidently apply rules to combinations of functions: product, quotient, and chain rules, along with implicit and parametric differentiation.
导数由第一原理定义，但 Pre-U 考生必须自信地将法则应用于函数组合：乘法法则、除法法则、链式法则，以及隐函数微分和参数微分。
Standard derivatives for trigonometric, exponential, logarithmic, and inverse trigonometric functions must be memorised. You should be able to differentiate functions such as ln(sin x) or e^(cos 2x) rapidly.
标准导数的三角函数、指数函数、对数函数和反三角函数必须熟记。你应能快速对诸如 ln(sin x) 或 e^(cos 2x) 的函数求导。
Higher-order derivatives and their notation are tested, and applications include finding equations of tangents and normals, rates of change, and optimisation of real-world problems.
高阶导数及其表示法也是考点，应用包括求切线和法线方程、变化率以及现实问题的优化。
d/dx (tan⁻¹ x) = 1/(1 + x²)
3. Integration Methods | 积分方法
Integration is treated as the reverse of differentiation, with emphasis on integrating standard forms and using substitution, integration by parts, and partial fractions. Recognising when to use each technique is crucial.
积分被视为微分的逆运算，重点是对标准形式进行积分并使用换元法、分部积分法和部分分式法。识别何时使用每种技巧至关重要。
Definite integrals are used to calculate areas under curves, areas between two curves, and volumes of revolution about the x- or y-axis. Correct handling of limits and absolute value areas is expected.
定积分用于计算曲线下方面积、两曲线之间的面积以及绕 x 轴或 y 轴旋转的体积。需要正确处理积分限和绝对值面积。
You may also encounter improper integrals, requiring limits to infinity or dealing with unbounded integrands, and integrals that lead to inverse trigonometric functions.
你还可能遇到反常积分，需要处理趋于无穷的极限或无界被积函数，以及导出反三角函数的积分。
∫ (from a to b) u dv = [uv] (from a to b) – ∫ (from a to b) v du
4. Differential Equations | 微分方程
Separable first-order differential equations form the main class solved in Pre-U. You separate variables, integrate both sides, and apply initial conditions to find a particular solution.
可分离的一阶微分方程是 Pre-U 中求解的主要类型。你分离变量，对两边积分，并应用初始条件以求得特解。
You need to interpret differential equations in context, such as population growth, cooling, or radioactive decay. Linear first-order equations using an integrating factor are also part of the syllabus.
你需要在具体情境中解释微分方程，例如人口增长、冷却或放射性衰变。使用积分因子的线性一阶方程也是大纲的一部分。
The method of separation is formally written as dy/dx = g(x)h(y), leading to ∫ 1/h(y) dy = ∫ g(x) dx. Careful handling of constants and domain restrictions is essential.
分离法的正式写法为 dy/dx = g(x)h(y)，从而导出 ∫ 1/h(y) dy = ∫ g(x) dx。仔细处理常数和定义域限制至关重要。
dy/dx = ky ⟹ y = Ae^(kx)
5. Complex Numbers | 复数
Complex numbers extend the real numbers with i² = -1. You perform arithmetic in Cartesian form z = x + iy and represent them on an Argand diagram, interpreting modulus |z| and argument arg(z).
复数用 i² = -1 扩展了实数。你用笛卡尔形式 z = x + iy 进行算术运算，并在 Argand 图上表示它们，解释模 |z| 和辐角 arg(z)。
Euler’s formula e^(iθ) = cos θ + i sin θ links exponential and trigonometric forms. This is used to multiply, divide, and find powers and roots of complex numbers via de Moivre’s theorem.
欧拉公式 e^(iθ) = cos θ + i sin θ 将指数形式与三角形式联系起来。它用于通过棣莫弗定理进行复数的乘、除、乘方和开方。
Solving polynomial equations with real coefficients yields complex conjugate roots. You must also identify loci in the complex plane, such as |z – a| = r (circle) or |z – a| = |z – b| (perpendicular bisector).
求解实系数多项式方程会产生共轭复根。你还必须识别复平面上的轨迹，如 |z – a| = r（圆）或 |z – a| = |z – b|（垂直平分线）。
e^(iθ) = cos θ + i sin θ, (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
6. Vectors and 3D Geometry | 向量与三维几何
Vectors are expressed in i,j,k basis or column notation. Operations include addition, scalar multiplication, dot product and cross product. The dot product tests perpendicularity and calculates angles.
向量用 i,j,k 基底或列记号表示。运算包括加法、标量乘法、点乘和叉乘。点乘用于检验垂直和计算角度。
Vector equations of lines in 3D take the form r = a + tb, while planes are defined by r·n = d or using two direction vectors. Finding intersections and distances between points, lines, and planes is a core skill.
三维直线的向量方程形式为 r = a + tb，而平面由 r·n = d 或使用两个方向向量定义。求点、线、面之间的交点和距离是核心技能。
You may need to determine whether lines intersect, are parallel, or are skew, and to calculate the shortest distance from a point to a line or plane using projection methods.
你可能需要判断直线是相交、平行还是异面，并用投影法计算点到直线或点到平面的最短距离。
a·b = |a||b| cos θ, a×b = |a||b| sin θ n̂
7. Sequences and Series | 数列与级数
Arithmetic and geometric sequences are extended with sigma notation, sum to infinity of a geometric series, and applications to compound interest and decay. You work with recurrence relations and their long-term behaviour.
等差数列和等比数列用求和符号扩展，包括几何级数的无穷和以及复利和衰减的应用。你处理递推关系及其长期行为。
Standard series for ∑r, ∑r², ∑r³ are given and can be combined to sum more complex polynomial series. Proof by induction is a key tool for verifying sum formulas.
∑r、∑r²、∑r³ 的标准级数是已知的，并可组合来求更复杂的多项式级数和。归纳证明是验证和公式的关键工具。
Tests for convergence of series may include comparison test and the ratio test for series of positive terms. Maclaurin series expansions for eˣ, sin x, cos x, and ln(1+x) are used to approximate functions.
级数的收敛性检验可能包括比较判别法和正项级数的比值判别法。eˣ、sin x、cos x 和 ln(1+x) 的麦克劳林级数展开可用于函数近似。
∑ (from r=1 to n) r = n(n+1)/2, ∑ (from r=1 to n) r² = n(n+1)(2n+1)/6
8. Trigonometry | 三角学
Beyond basic solving of trig equations, you use all reciprocal and inverse trig functions, compound angle formulas, double-angle and half-angle identities, and the factor formulae to solve problems.
除了基本三角方程求解，你还使用所有倒数和反三角函数、复合角公式、倍角和半角恒等式以及和差化积公式来解决问题。
Periodic properties, symmetries, and transformations of sin, cos, and tan graphs are analysed. You prove identities such as sec² θ – tan² θ = 1 and express a sin θ + b cos θ in the form R sin(θ + α).
分析了 sin、cos 和 tan 图像的周期性质、对称性和变换。你证明诸如 sec² θ – tan² θ = 1 的恒等式，并将 a sin θ + b cos θ 表示为 R sin(θ + α) 的形式。
Trigonometric integration features heavily; you must recognise standard forms like ∫ sec² θ dθ, and use substitution t = tan(θ/2) for rational functions of sin and cos.
三角积分大量出现；你必须识别如 ∫ sec² θ dθ 的标准形式，并对 sin 和 cos 的有理函数使用代换 t = tan(θ/2)。
sin(A ± B) = sin A cos B ± cos A sin B, cos 2θ = 2 cos² θ – 1
9. Probability and Statistics | 概率与统计
The statistics component covers probability using set notation, conditional probability, and Bayes’ theorem. Discrete random variables, their probability mass functions, expectation E(X) and variance Var(X) are explored.
统计部分涵盖使用集合记号的概率、条件概率和贝叶斯定理。探讨了离散随机变量、其概率质量函数、期望 E(X) 和方差 Var(X)。
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Discrete distributions: binomial B(n,p) and Poisson Po(λ) are used as models, including using them as approximations to each other when conditions are met.
离散分布：二项分布 B(n,p) 和泊松分布 Po(λ) 用作模型，包括在条件满足时相互近似使用。
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The normal distribution N(μ, σ²) is introduced; standardising to Z ~ N(0,1) allows probability calculations. You find unknown μ and σ from given probabilities.
引入了正态分布 N(μ, σ²)；标准化到 Z ~ N(0,1) 可进行概率计算。你从给定的概率求未知的 μ 和 σ。
Hypothesis testing for a binomial proportion or Poisson mean, and for a normal mean with known variance, is covered. You interpret significance levels and p-values.
涵盖了对二项比例或泊松均值，以及方差已知的正态均值进行假设检验。你解释显著性水平和 p 值。
P(X = k) = C(n,k) pᵏ (1-p)ⁿ⁻ᵏ, P(X = k) = (λᵏ e⁻ᵅ)/k!
10. Numerical Methods and Proof | 数值方法与证明
When exact algebraic solutions are impossible, iterative methods such as the Newton-Raphson method xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ) are used to approximate roots. You also consider convergence criteria.
当无法进行精确代数求解时，使用迭代法，如牛顿-拉夫森法 xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ)，来逼近根。你还考虑收敛条件。
Proof by induction is formally assessed: you show the base case, assume true for n=k, and prove for n=k+1. This is applied to sums, divisibility, inequalities, and sequences.
归纳证明被正式考查：你展示基本情形，假设对 n=k 为真，并证明 n=k+1 成立。这应用于求和、整除性、不等式和数列。
Proof by contradiction and direct proof methods also appear, especially in number theory and algebra contexts. You must structure logical arguments clearly.
反证法和直接证明法也出现，特别是在数论和代数语境中。你必须清晰地组织逻辑论证。
Other numerical methods might include the trapezium rule for approximating definite integrals, and locating intervals where a root lies by sign change.
其他数值方法可能包括用梯形法则近似定积分，以及通过符号变化定位根所在的区间。
xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ)
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