📚 Year 11 OCR Further Maths: Formula & Theorem Quick Reference | Year 11 OCR 进阶数学：公式定理速查手册
This quick-reference handbook condenses every essential formula and theorem from the Year 11 OCR Further Mathematics specification. Use it to reinforce your memory, check key results before exams, and build a solid foundation for A Level study.
本速查手册浓缩了 Year 11 OCR 进阶数学课程中的所有核心公式和定理。你可以用它巩固记忆、考前核查关键结论，并为 A Level 学习打下坚实基础。
1. Algebra and Polynomials | 代数与多项式
The quadratic formula solves ax² + bx + c = 0:
二次公式用于解方程 ax² + bx + c = 0：
x = (-b ± √b² – 4ac) / 2a
The discriminant Δ = b² – 4ac determines the nature of the roots: two real distinct, one repeated, or no real roots.
判别式 Δ = b² – 4ac 决定根的性质：两个相异实根、一个重根或无实根。
Completing the square: x² + bx = (x + b/2)² – (b/2)².
配方法：x² + bx = (x + b/2)² – (b/2)²。
Factor theorem: (x – a) is a factor of f(x) ⇔ f(a) = 0. Remainder theorem: when f(x) is divided by (x – a), the remainder is f(a).
因式定理：(x – a) 是 f(x) 的因式当且仅当 f(a) = 0。余式定理：f(x) 除以 (x – a) 的余数为 f(a)。
p between roots and coefficients for ax² + bx + c = 0: if α, β are the roots, then α + β = –b/a and αβ = c/a.
二次方程 ax² + bx + c = 0 的根与系数关系：若根为 α, β，则 α + β = –b/a 且 αβ = c/a。
The binomial expansion for positive integer n:
正整数指数 n 的二项展开：
(a + b)ⁿ = aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + ... + bⁿ
General term: Tr+1 = ⁿCᵣ aⁿ⁻ʳ bʳ, where ⁿCᵣ = n! / [r!(n – r)!].
通项：Tr+1 = ⁿCᵣ aⁿ⁻ʳ bʳ，其中 ⁿCᵣ = n! / [r!(n – r)!]。
2. Coordinate Geometry and Graphs | 坐标几何与图像
Equation of a straight line: y – y₁ = m(x – x₁) or y = mx + c, where m = (y₂ – y₁)/(x₂ – x₁).
直线方程：y – y₁ = m(x – x₁) 或 y = mx + c，其中 m = (y₂ – y₁)/(x₂ – x₁)。
Distance between two points: √[(x₂ – x₁)² + (y₂ – y₁)²]. Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点间距离：√[(x₂ – x₁)² + (y₂ – y₁)²]。中点坐标：((x₁ + x₂)/2, (y₁ + y₂)/2)。
Equation of a circle with centre (a, b) and radius r: (x – a)² + (y – b)² = r².
圆心为 (a,b)、半径为 r 的圆方程：(x – a)² + (y – b)² = r²。
Quadratic graphs y = ax² + bx + c: shape determined by a (∪ for a > 0, ∩ for a < 0); vertex at x = –b/(2a); discriminant tells intersections with x-axis.
二次函数图像 y = ax² + bx + c：图形开口由 a 决定（a > 0 开口向上 ∪，a < 0 开口向下 ∩）；顶点横坐标 x = –b/(2a)；判别式决定与 x 轴交点个数。
Cubic graphs have one inflection point and often display an S‑shape; the leading coefficient a in y = ax³ + ... controls the long‑term behaviour.
三次函数图像有一个拐点，通常呈 S 形；最高次项系数 a 决定图像的远期走向。
3. Sequences and Series | 数列与级数
Arithmetic progression: nth term uₙ = a + (n – 1)d. Sum of first n terms: Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l).
等差数列：第 n 项 uₙ = a + (n – 1)d。前 n 项和：Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l)。
Geometric progression: nth term uₙ = arⁿ⁻¹. Sum of first n terms: Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1.
等比数列：第 n 项 uₙ = arⁿ⁻¹。前 n 项和：Sₙ = a(1 – rⁿ)/(1 – r)，其中 r ≠ 1。
Sum to infinity for |r| < 1: S∞ = a/(1 – r).
当 |r| < 1 时的无穷等比级数和：S∞ = a/(1 – r)。
Sigma notation: Σ (from k=1 to n) aₖ represents the sum of terms a₁ + a₂ + ... + aₙ.
求和符号：Σ (k=1 到 n) aₖ 表示 a₁ + a₂ + ... + aₙ 的和。
4. Matrices | 矩阵
A 2×2 matrix is written as
| a | b |
| c | d |
2×2 矩阵写作
| a | b |
| c | d |
Multiplication by a scalar: k
| a | b |
| c | d |
| ka | kb |
| kc | kd |
标量乘法：k
| a | b |
| c | d |
| ka | kb |
| kc | kd |
Matrix multiplication:
| a | b |
| c | d |
| e | f |
| g | h |
| ae+bg | af+bh |
| ce+dg | cf+dh |
矩阵乘法：
| a | b |
| c | d |
| e | f |
| g | h |
| ae+bg | af+bh |
| ce+dg | cf+dh |
Identity matrix I =
| 1 | 0 |
| 0 | 1 |
单位矩阵 I =
| 1 | 0 |
| 0 | 1 |
Determinant of A: det(A) = ad – bc. If det(A) = 0 the matrix is singular and has no inverse.
A 的行列式：det(A) = ad – bc。若 det(A) = 0，该矩阵为奇异矩阵，无逆矩阵。
Inverse of a non‑singular matrix: A⁻¹ = 1/(ad – bc)
| d | -b |
| -c | a |
非奇异矩阵的逆矩阵：A⁻¹ = 1/(ad – bc)
| d | -b |
| -c | a |
Solving simultaneous equations: write as AX = B, then X = A⁻¹B when det(A) ≠ 0.
解线性方程组：写成 AX = B，当 det(A) ≠ 0 时 X = A⁻¹B。
5. Trigonometry | 三角学
Sine rule: a/sin A = b/sin B = c/sin C, used for any triangle.
正弦定理：a/sin A = b/sin B = c/sin C，适用于任意三角形。
Cosine rule: a² = b² + c² – 2bc cos A. Useful for finding a side or an angle when you have SAS or SSS.
余弦定理：a² = b² + c² – 2bc cos A。已知两边及其夹角或三边时求边或角。
Area of a triangle: Area = ½ ab sin C (two sides and the included angle).
三角形面积公式：面积 = ½ ab sin C（需要两边及其夹角）。
Pythagorean identity: sin²θ + cos²θ ≡ 1. Tangent identity: tan θ ≡ sin θ / cos θ.
毕达哥拉斯恒等式：sin²θ + cos²θ ≡ 1。正切恒等式：tan θ ≡ sin θ / cos θ。
Exact trigonometric values (commonly tested):
常见精确三角函数值（常考）：
| θ (deg) | sin θ | cos θ | tan θ |
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
Supplementary relationships: sin(180° – θ) = sin θ, cos(180° – θ) = –cos θ.
补角关系：sin(180° – θ) = sin θ, cos(180° – θ) = –cos θ。
6. Vectors | 向量
A vector can be expressed as a column vector
| x |
| y |
向量可表示为列向量
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