📚 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
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. The sum of the roots is -b/a and the product of the roots is c/a.
对于二次方程 a x² + b x + c = 0（a ≠ 0），其解由求根公式给出。判别式 Δ = b² - 4ac 决定根的性质：Δ > 0 有两个不相等的实根，Δ = 0 有一个重根，Δ < 0 则有一对共轭复根。两根之和为 -b/a，两根之积为 c/a。
x = (-b ± √b² - 4ac) / (2a) Δ = b² - 4ac
rem states that when f(x) is divided by (x - a), the remainder is f(a); more generally, dividing by (b x - a) leaves the remainder f(a/b).
因式定理指出：当且仅当 f(a) = 0 时，(x - a) 是多项式 f(x) 的因式。余数定理指出：f(x) 除以 (x - a) 的余数为 f(a)；更一般地，除以 (b x - a) 的余数为 f(a/b)。
ces are aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a⁰ = 1 (for a ≠ 0), and a1/n = ⁿ√a. These laws underpin almost every algebraic manipulation in the course.
指数基本法则是：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，a⁰ = 1（a ≠ 0），以及 a1/n = ⁿ√a。这些法则几乎是本课程所有代数变形的基础。
e-of-base rule is logₐ b = log_c b / log_c a, which lets you evaluate any logarithm on a calculator.
对数法则与指数法则相对应：logₐ(xy) = logₐ x + logₐ y，logₐ(x/y) = logₐ x - logₐ y，logₐ xⁿ = n logₐ x，logₐ a = 1，logₐ 1 = 0。换底公式为 logₐ b = log_c b / log_c a，它可以让你用计算器求任意对数的值。
The binomial theorem for a rational exponent n states that (1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ... valid for |x| < 1. For a positive integer n, the expansion of (a + b)ⁿ is Σ ⁿCᵣ aⁿ⁻ʳ bʳ summed from r = 0 to n, where ⁿCᵣ = n!/(r!(n - r)!) and the general term is ⁿCᵣ aⁿ⁻ʳ bʳ.
有理指数 n 的二项式定理为：(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ...，仅当 |x| < 1 时成立。对于正整数 n，(a + b)ⁿ 的展开式为 Σ ⁿCᵣ aⁿ⁻ʳ bʳ（r 从 0 到 n 求和），其中 ⁿCᵣ = n!/(r!(n - r)!)，通项为 ⁿCᵣ aⁿ⁻ʳ bʳ。
Partial fractions decompose a rational expression into simpler pieces. A distinct linear factor (x - a) contributes A/(x - a); a repeated linear factor (x - a)² contributes A/(x - a) + B/(x - a)²; and an irreducible quadratic factor (x² + px + q) contributes (Bx + C)/(x² + px + q).
部分分式把一个有理式分解为若干更简单的分式。一个互异线性因式 (x - a) 对应 A/(x - a)；一个重复线性因式 (x - a)² 对应 A/(x - a) + B/(x - a)²；一个不可约二次因式 (x² + px + q) 则对应 (Bx + C)/(x² + px + q)。
2. Functions and Graphs | 函数与图像
ecked by the horizontal line test. On a graph, the inverse is the reflection of y = f(x) in the line y = x, and f(f⁻¹(x)) = f⁻¹(f(x)) = x.
函数 f 把定义域中的每个元素映射到值域中唯一的元素。逆函数 f⁻¹ 仅当 f 是一一映射时才存在，可用水平线检验来判断。图像上，逆函数是 y = f(x) 关于直线 y = x 的反射，且 f(f⁻¹(x)) = f⁻¹(f(x)) = x。
A composite function fg means 'apply g first, then f' s the set of x in the domain of g for which g(x) lies in the domain of f. In general fg ≠ gf, so the order of composition matters.
复合函数 fg 表示 '先作用 g，再作用 f'，因此 fg(x) = f(g(x))。fg 的定义域是 g 的定义域中那些使 g(x) 落在 f 的定义域内的 x。一般地 fg ≠ gf，所以复合的顺序很重要。
The standard graph transformations are: vertical translation f(x) + a, horizontal translation f(x + a), vertical stretch a f(x) (stretch when a > 1, compress when 0 < a < 1), horizontal stretch f(ax), reflection in the x-axis -f(x), reflection in the y-axis f(-x), and the modulus transformation |f(x)|, which reflects any part of the curve below the x-axis upwards.
标准图像变换包括：竖直平移 f(x) + a、水平平移 f(x + a)、竖直伸缩 a f(x)（a > 1 拉伸，0 < a < 1 压缩）、水平伸缩 f(ax)、关于 x 轴反射 -f(x)、关于 y 轴反射 f(-x)，以及取模变换 |f(x)|，它把曲线在 x 轴以下的部分向上翻折。
Modulus equations and inequalities follow two key rules: |x| = a (with a ≥ 0) means x = a or x = -a, while |x| < a means -a < x < a and |x| > a means x < -a or x > a. For |x - b| < a simply replace x by x - b throughout.
含绝对值的方程与不等式遵循两条关键规则：|x| = a（a ≥ 0）意味着 x = a 或 x = -a；|x| < a 意味着 -a < x < a；|x| > a 意味着 x < -a 或 x > a。对于 |x - b| < a，只需把式中的 x 换成 x - b 即可。
The exponential function eˣ and the natural logarithm ln x are inverse functions, so eln x = x for x > 0 and ln(eˣ) = x for all real x. The graph y = eˣ d passes through (0, 1); the graph y = ln x has asymptote x = 0 and passes through (1, 0).
指数函数 eˣ 与自然对数 ln x 互为反函数，因此当 x > 0 时 eln x = x，而对所有实数 x 有 ln(eˣ) = x。图像 y = eˣ 的渐近线为 y = 0，并经过点 (0, 1)；图像 y = ln x 的渐近线为 x = 0，并经过点 (1, 0)。
3. Coordinate Geometry | 坐标几何
₁ + x₂)/2, (y₁ + y₂)/2). The gradient of the line through them is m = (y₂ - y₁)/(x₂ - x₁).
两点 (x₁, y₁) 与 (x₂, y₂) 之间的距离为 √((x₂ - x₁)² + (y₂ - y₁)²)，连接它们的线段中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。过这两点的直线斜率为 m = (y₂ - y₁)/(x₂ - x₁)。
e can be written as y - y₁ = m(x - x₁), as y = mx + c, or in general form ax + by + c = 0.
两条直线平行时 m₁ = m₂，垂直时 m₁ × m₂ = -1。直线方程可以写成 y - y₁ = m(x - x₁)、y = mx + c，或一般式 ax + by + c = 0。
s given by the formula below. This result is extremely useful for finding the height of a triangle when the base is already known.
点 (x₁, y₁) 到直线 ax + by + c = 0 的垂直距离由下式给出。当三角形底边已知时，该结论对求高极为有用。
d = |a x₁ + b y₁ + c| / √a² + b²
e general circle has the form x² + y² + 2gx + 2fy + c = 0, with centre (-g, -f) and radius √g² + f² - c, which exists only when g² + f² - c > 0.
圆心为 (a, b)、半径为 r 的圆的方程为 (x - a)² + (y - b)² = r²。展开后，圆的一般式为 x² + y² + 2gx + 2fy + c = 0，其圆心为 (-g, -f)，半径为 √g² + f² - c，仅当 g² + f² - c > 0 时该圆存在。
For a circle, the perpendicular from the centre to a chord bisects that chord, and the tangent at any point is perpendicular to the radius drawn to that point. These two facts solve most circle problems involving chords, tangents, and lengths.
对于圆，从圆心向弦所作的垂线平分该弦，而任一点处的切线垂直于该点所对应的半径。这两条性质可以解决绝大多数涉及弦、切线与长度的圆的问题。
4. Sequences and Series | 数列与级数
ich can also be written as Sₙ = n/2 (a + l) where l is the last term. The sum is always a quadratic in n.
等差数列的首项为 a、公差为 d，第 n 项为 uₙ = a + (n - 1)d。前 n 项和为 Sₙ = n/2 (2a + (n - 1)d)，也可写成 Sₙ = n/2 (a + l)，其中 l 为末项。该和式总是关于 n 的二次式。
rm uₙ = a rⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1, and when |r| < 1 the series converges to a sum to infinity S∞ = a/(1 - r). When |r| ≥ 1 the series diverges.
等比数列的首项为 a、公比为 r，第 n 项为 uₙ = a rⁿ⁻¹。前 n 项和为 Sₙ = a(1 - rⁿ)/(1 - r)（r ≠ 1）；当 |r| < 1 时级数收敛，无穷和为 S∞ = a/(1 - r)；当 |r| ≥ 1 时级数发散。
The binomial expansion for a positive integer n is written using sigma notation and the binomial coefficient, with the general term allowing you to find any single coefficient without expanding everything. This is one of the most frequently examined skills in the algebra paper.
正整数 n 的二项式展开可用求和记号与二项式系数表示，利用通项可在不展开全部项的情况下求出任意一项系数。这是代数试卷中考查频率最高的技能之一。
(a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ, ⁿCᵣ = n! / (r!(n - r)!)
r + 1). The partial sum then telescopes, leaving only the first and last surviving terms. This technique appears regularly in Pre-U proof questions.
差分法适用于通项可拆成差值形式的级数，例如 uᵣ = f(r) - f(r + 1)。此时部分和会逐项相消，只剩首尾若干项。该技巧在 Pre-U 的证明题中经常出现。
5. Trigonometry | 三角学
nd the sector area is A = ½ r² θ. Small-angle approximations are sin θ ≈ θ, tan θ ≈ θ, and cos θ ≈ 1 - θ²/2 for small θ.
弧度是角的自然度量单位：π 弧度 = 180°，故 1 弧度 ≈ 57.3°。对半径为 r、圆心角为 θ 弧度的扇形，弧长为 s = rθ，面积为 A = ½ r² θ。小角近似为：当 θ 很小时，sin θ ≈ θ，tan θ ≈ θ，cos θ ≈ 1 - θ²/2。
n θ / cos θ, they allow almost any trigonometric expression to be simplified into a single function.
毕达哥拉斯恒等式为 sin²θ + cos²θ = 1、1 + tan²θ = sec²θ、1 + cot²θ = cosec²θ。与 tan θ = sin θ / cos θ 结合，几乎可以把任何三角表达式化简为单一函数。
mpound-angle formulas are sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B, and tan(A ± B) = (tan A ± tan B)/(1 ∓ cos A, cos 2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A, and tan 2A = 2 tan A/(1 - tan²A).
和角公式为：sin(A ± B) = sin A cos B ± cos A sin B，cos(A ± B) = cos A cos B ∓ sin A sin B，tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)。由此立即得到倍角公式：sin 2A = 2 sin A cos A，cos 2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A，tan 2A = 2 tan A/(1 - tan²A)。
on of the form a sin θ + b cos θ can be written as R sin(θ ± α) or R cos(θ ± α), where R = √a² + b² makes it easy to find maximum and minimum values: the maximum is R and the minimum is -R.
任何形如 a sin θ + b cos θ 的表达式都可写成 R sin(θ ± α) 或 R cos(θ ± α)，其中 R = √a² + b²，tan α = b/a。这一形式便于求最大最小值：最大值为 R，最小值为 -R。
d the area is ½ ab sin C. The cosine rule is used when you know three sides or two sides and the included angle; the sine rule is used in all other cases.
在任意三角形 ABC 中，正弦定理为 a/sin A = b/sin B = c/sin C，余弦定理为 a² = b² + c² - 2bc cos A，面积为 ½ ab sin C。已知三边或两边及其夹角时用余弦定理，其余情况用正弦定理。
6. Differentiation | 微分
The derivative is defined as the limit of the gradient of a chord: f' s first-principles definition is examinable and is often required for simple polynomials or for proving a standard result.
导数定义为弦的斜率所趋的极限：f'(x) = lim (h → 0) [f(x + h) - f(x)]/h。这一从基本原理出发的定义属于考查范围，常用于简单多项式或证明某个标准结论。
The standard derivatives you must know by heart are shown below, together with the two inverse trigonometric results that appear in Pre-U papers.
必须熟记的标准导数如下，同时列出 Pre-U 试卷中出现的两个反三角函数结果。
d/dx (xⁿ) = n xⁿ⁻¹ d/dx (eˣ) = eˣ d/dx (ln x) = 1/x
d/dx (sin x) = cos x &
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