Pre-U OCR Mathematics: Formula & Theorem Quick Reference Handbook | Pre-U OCR 数学：公式定理速查手册

📚 Pre-U OCR Mathematics: Formula & Theorem Quick Reference Handbook | Pre-U OCR 数学：公式定理速查手册

This concise handbook brings together the essential formulas, theorems, and standard results that you need to master for the Pre-U OCR Mathematics course. Use it as a rapid revision tool, a memory aid before exams, and a first-stop reference when solving problems across all areas of the syllabus – from algebra and trigonometry to calculus, vectors, complex numbers, probability, and mechanics. Each section presents key facts in English immediately followed by their Chinese translation, making it especially useful for bilingual learners or those who benefit from dual-language reinforcement.

这本简明速查手册汇集了 Pre-U OCR 数学课程中必须掌握的核心公式、定理与标准结论。你可以将其用作考前快速复习的工具、强化记忆的利器，以及解答问题时最先查阅的参考。手册涵盖了从代数、三角学到微积分、向量、复数、概率与力学的全部领域，每个知识要点均先以英文呈现，紧接着提供中文翻译，对双语学习者或希望通过双语言巩固理解的学生尤其有帮助。

1. Algebra Essentials | 代数基础

For a quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula x = (-b ± √(b² – 4ac)) / (2a). The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives a pair of complex conjugate roots.

对于二次方程 ax² + bx + c = 0，求根公式为 x = (-b ± √(b² – 4ac)) / (2a)。判别式 Δ = b² – 4ac 决定了根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 则有一对共轭复根。

The fundamental laws of indices are aᵐaⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, and a^(1/n) = ⁿ√a. For logarithms, logₐ(xy) = logₐ x + logₐ y, logₐ(x/y) = logₐ x – logₐ y, logₐ xⁿ = n logₐ x, and the change-of-base rule is logₐ b = log_c b / log_c a.

指数基本法则是：aᵐaⁿ = aᵐ⁺ⁿ，aᵐ/aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，a^(1/n) = ⁿ√a。对数法则有：logₐ(xy) = logₐ x + logₐ y，logₐ(x/y) = logₐ x – logₐ y，logₐ xⁿ = n logₐ x，以及换底公式 logₐ b = log_c b / log_c a。

The binomial theorem for a rational exponent n states that (1 + x)ⁿ = 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + … for |x| < 1. For a positive integer n, the expansion of (a+b)ⁿ is given by Σ ⁿCᵣ aⁿ⁻ʳ bʳ from r=0 to n, where ⁿCᵣ = n!/(r!(n-r)!).

有理指数 n 的二项式定理为：当 |x| < 1 时 (1 + x)ⁿ = 1 + nx + n(n-1)/2! x² + n(n-1)(n-2)/3! x³ + ...。对于正整数 n，(a+b)ⁿ 的展开式为 Σ ⁿCᵣ aⁿ⁻ʳ bʳ (r=0 到 n)，其中 ⁿCᵣ = n!/(r!(n-r)!)。

2. Functions and Graphs | 函数与图像

A function f maps elements from its domain to its range. The inverse function f⁻¹ exists only if f is one-to-one (passes the horizontal line test). Graphically, the inverse is the reflection of f in the line y = x.

函数 f 将定义域中的元素映射到值域。逆函数 f⁻¹ 仅当 f 是一一映射（通过水平线检验）时才存在。图像上，逆函数是原函数关于直线 y = x 的反射。

Composite functions are formed by applying one function after another: fg(x) = f(g(x)). Transformations of the graph y = f(x) include vertical translation f(x) + a, horizontal translation f(x + a), vertical stretch a f(x) (a > 1 stretch, 0 < a < 1 compress), horizontal stretch f(ax), reflection in the x-axis -f(x), and reflection in the y

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