📚 Formula & Theorem Quick Reference Handbook | 公式定理速查手册
This quick reference guide covers the essential formulas, theorems and rules needed for the Year 10 OCR Further Mathematics course. Use it to check definitions, practise recall and ensure you can apply every piece of mathematics confidently and accurately. Each section pairs English and Chinese explanations so you can master the terminology in both languages.
本速查手册涵盖 Year 10 OCR 进阶数学所需的核心公式、定理和法则，帮助你快速查阅定义、练习默写，并确保能准确灵活地应用每一条数学知识。每个单元都配有中英文对照解释，便于同时掌握双语术语。
1. Algebraic Essentials | 代数基础
Expanding brackets is fundamental: multiply every term inside the bracket by the factor outside. For a double bracket (x + a)(x + b), the expansion gives x² + (a + b)x + ab.
展开括号是基本技能：把括号外的因子与括号内每一项相乘。对于双重括号 (x + a)(x + b)，展开结果为 x² + (a + b)x + ab。
For three linear factors (x + a)(x + b)(x + c), expand two first and then multiply by the third. Always collect like terms at the end.
三个线性因式 (x + a)(x + b)(x + c) 的展开，先展开前两个，再与第三个相乘，最后合并同类项。
| Special products | 特殊乘积 |
| (a + b)² = a² + 2ab + b² | (a + b)² = a² + 2ab + b² |
| (a - b)² = a² - 2ab + b² | (a - b)² = a² - 2ab + b² |
| (a + b)(a - b) = a² - b² | (a + b)(a - b) = a² - b² (平方差) |
st. A quadratic trinomial x² + bx + c factors into (x + p)(x + q) if p + q = b and pq = c.
因式分解是展开的逆运算：首先提取最大公因式。二次三项式 x² + bx + c 若能写成 (x + p)(x + q)，则必有 p + q = b 且 pq = c。
2. Quadratic Equations & Discriminant | 二次方程与判别式
written as ax² + bx + c = 0 with a ≠ 0, the solutions are given by the quadratic formula:
对于形如 ax² + bx + c = 0（a ≠ 0）的任意二次方程，其解由求根公式给出：
x = (-b ± √b² - 4ac) / (2a)
The expression under the square root, b² - 4ac, is called the discriminant (Δ). It determines the nature of the roots.
根号下的表达式 b² - 4ac 称为判别式（Δ），它决定了根的性质。
If Δ > 0, the equation has two distinct real roots.
若 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, there is one repeated real root (a perfect square).
若 Δ = 0，有一个重根（完全平方式）。
If Δ < 0, there are no real roots (complex roots).
若 Δ < 0，方程没有实数根（仅有复数根）。
Completing the square rewrites the quadratic as a(x + p)² + q. For x² + bx, add (b/2)² to complete the square.
配方法将二次式写成 a(x + p)² + q 的形式。对于 x² + bx，加上 (b/2)² 即可配成完全平方。
3. Factor Theorem & Remainder Theorem | 因式定理与余式定理
When a polynomial f(x) is divided by a linear divisor (x - a), the remainder is f(a). This is the Remainder Theorem.
当多项式 f(x) 除以一次式 (x - a) 时，所得的余数为 f(a)。这就是余式定理。
or of f(x) if and only if f(a) = 0. This is a direct consequence of the Remainder Theorem.
因式定理指出，(x - a) 是 f(x) 的因式当且仅当 f(a) = 0。这是余式定理的直接推论。
To factorise a cubic, find one root a by testing factors of the constant term, then use polynomial division or synthetic division to obtain a quadratic factor.
分解三次多项式时，可先通过试根找到常数项的某个因数 a 使 f(a)=0，再用多项式除法或综合除法得到二次因式。
4. Binomial Expansion | 二项式展开
The binomial expansion for (1 + x)ⁿ where n is a positive integer uses Pascal's triangle and the binomial coefficient ⁿCᵣ.
对于正整数指数 n，(1 + x)ⁿ 的二项式展开用帕斯卡三角形和二项式系数 ⁿCᵣ 表示。
(1 + x)ⁿ = 1 + ⁿC₁ x + ⁿC₂ x² + … + ⁿCᵣ xʳ + … + xⁿ
More generally, (a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ for r from 0 to n, where ⁿCᵣ = n! / [r!(n - r)!].
更一般地，(a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ，r 从 0 到 n，其中 ⁿCᵣ = n! / [r!(n - r)!]。
For OCR Further Maths you must be able to expand expressions like (2 - 3x)⁵ and find specific terms without expanding fully.
OCR 进阶数学要求能展开形如 (2 - 3x)⁵ 的式子，并能在不完全展开的情况下直接求出第 r+1 项。
5. Laws of Exponents & Logarithms | 指数与对数法则
The laws of indices are essential for simplifying expressions and solving equations.
指数法则是化简表达式和解方程的基础。
| aᵐ × aⁿ = aᵐ⁺ⁿ | 同底幂相乘，指数相加 |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 同底幂相除，指数相减 |
| (aᵐ)ⁿ = aᵐⁿ | 幂的乘方，指数相乘 |
| a⁰ = 1 (a ≠ 0) | 非零数的 0 次幂为 1 |
| a⁻ⁿ = 1 / aⁿ | 负指数表示倒数 |
Logarithms are the inverse of exponentials: if y = aˣ, then x = logₐ(y). The natural logarithm is logₑ or ln, with base e ≈ 2.718.
对数是乘方的逆运算：若 y = aˣ，则 x = logₐ(y)。自然对数以 e ≈ 2.718 为底，记作 ln。
| logₐ(xy) = logₐx + logₐy | 积的对数等于对数之和 |
| logₐ(x/y) = logₐx - logₐy | 商的对数等于对数之差 |
| logₐ(xⁿ) = n logₐx | 幂的对数：指数可提到对数前 |
6. Coordinate Geometry: Lines & Circles | 坐标几何：直线与圆
(x₁, y₁) and (x₂, y₂) is m = (y₂ - y₁)/(x₂ - x₁). The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).
连结两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率为 m = (y₂ - y₁)/(x₂ - x₁)。中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。
rallel lines have equal gradients; perpendicular lines have gradients whose product is -1.
直线方程可写作 y = mx + c（斜截式）或 y - y₁ = m(x - x₁)（点斜式）。平行直线斜率相等；垂直直线斜率之积为 -1。
- b)² = r². Completing the square turns a general circle equation into centre-radius form.
圆心为 (a, b)、半径为 r 的圆方程为 (x - a)² + (y - b)² = r²。将一般式通过配方可化为标准形。
The tangent to a circle at a point is perpendicular to the radius drawn to that point. Use this fact to find tangent equations.
圆上一点的切线与该点处的半径垂直。利用这一性质可求切线方程。
7. Trigonometry: Rules & Identities | 三角学：定理与恒等式
e: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. SOH CAH TOA is the quick memory aid.
在直角三角形中：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。可用口诀辅助记忆。
in C. Use it when you know two angles and one side (AAS) or two sides and a non-included angle (SSA).
任意三角形中，正弦定理联系边与对角的正弦：a/sin A = b/sin B = c/sin C。适用于已知两角一边（AAS）或两边一对角（SSA）。
c² - 2bc cos A. It can also be rearranged to find an angle: cos A = (b² + c² - a²)/(2bc).
余弦定理用于已知两边夹角（SAS）或三边（SSS）：a² = b² + c² - 2bc cos A。变形后可求角：cos A = (b² + c² - a²)/(2bc)。
be found using Area = ½ ab sin C, where a and b are two sides and C is the included angle.
三角形面积可用 ½ ab sin C 计算，其中 a、b 为两边，C 为其夹角。
Key identities to memorise: tan θ = sin θ/cos θ, sin² θ + cos² θ ≡ 1.
必须熟记的恒等式：tan θ = sin θ/cos θ，sin² θ + cos² θ ≡ 1。
Exact trigonometric values for 30°, 45°, 60° are required:
必须记住 30°、45°、60° 的精确三角函数值：
| Angle θ | sin θ | cos θ | tan θ |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
8. Radian Measure | 弧度制
ended at the centre of a circle by an arc equal in length to the radius. π radians = 180°.
弧度是长度等于半径的圆弧所对的圆心角。π 弧度 = 180°。
egrees = (180/π) × radians. You must use radians in calculus with trigonometric functions.
换算公式：弧度 = (π/180)×度数；度数 = (180/π)×弧度。涉及三角函数的微积分运算必须使用弧度。
radians). Sector area: A = ½ r² θ. These formulas only hold when θ is measured in radians.
弧长公式：s = rθ（θ 以弧度为单位）。扇形面积：A = ½ r² θ。这些公式仅在 θ 为弧度时成立。
9. Differentiation | 微分
Differentiation finds the gradient function of a curve. The derivative of f(x) is f'(x) or dy/dx. For a power function f(x) = xⁿ, the rule is: if y = xⁿ, then dy/dx = n xⁿ⁻¹.
微分用来求曲线的梯度函数。f(x) 的导函数记作 f'(x) 或 dy/dx。对幂函数 f(x) = xⁿ，求导法则为：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。
/dx (c) = 0. The derivative of a sum is the sum of the derivatives: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
常数项的导数为零：d/dx (c) = 0。和差的导数等于导数的和差：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。
oint where dy/dx = 0 can be a maximum, minimum or a point of inflection; use the second derivative or a sign test to classify it.
求导后可计算曲线上某点的梯度、驻点（dy/dx = 0），并可写出切线和法线方程。驻点可能是极大值点、极小值点或拐点；可用二阶导数或符号测试加以区分。
10. Integration | 积分
al of xⁿ is ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1 and C is the constant of integration.
积分是微分的逆运算。xⁿ 的不定积分为 ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C，其中 n ≠ -1，C 为积分常数。
The integral of a constant a is ax + C. Sums and differences integrate term by term. Always add the constant C for indefinite integrals.
常数 a 的积分为 ax + C。和差逐项积分。求不定积分时不要忘记加上积分常数 C。
derivative of f(x). The area between a curve and the x-axis is positive when above the axis and negative when below; split the integral if the curve crosses the axis.
定积分用于计算曲线在上下限间与 x 轴围成的有向面积：∫ₐᵇ f(x) dx = F(b) - F(a)，其中 F(x) 是 f(x) 的一个原函数。曲线在 x 轴上方时面积为正，下方时为负；若跨越 x 轴，需分段积分。
11. Matrices | 矩阵运算
A matrix is a rectangular array of numbers. You must be able to add, subtract and multiply matrices where the dimensions are compatible. Multiplication is only possible if the number of columns in the first matrix equals the number of rows in the second.
矩阵是一个数字的矩形阵列。你须掌握矩阵的加法、减法和乘法，前提是维数满足运算要求。仅当第一个矩阵的列数等于第二个矩阵的行数时，乘法方可进行。
e determinant is det(M) = ad - bc. If det(M) = 0, the matrix is singular (non-invertible).
对于 2×2 矩阵 M = [[a, b], [c, d]]，其行列式为 det(M) = ad - bc。若 det(M) = 0，矩阵为奇异矩阵（不可逆）。
/(ad - bc) [[d, -b], [-c, a]]. Check that M M⁻¹ = I, the identity matrix [[1, 0], [0, 1]].
非奇异 2×2 矩阵的逆矩阵为 M⁻¹ = 1/(ad - bc) [[d, -b], [-c, a]]。务必验证 M M⁻¹ = I（单位矩阵 [[1, 0], [0, 1]]）。
ous linear equations. Write the system as Mx = b, then x = M⁻¹b, provided M is invertible.
矩阵可用来表示并求解联立线性方程组。将方程组写成 Mx = b，若 M 可逆，则解为 x = M⁻¹b。
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