📚 Year 11 Edexcel Further Maths: Formula & Theorem Quick Reference Handbook | Year 11 Edexcel 进阶数学：公式定理速查手册
This handbook provides a concise collection of the essential formulas, theorems, and key results you will need for the Year 11 Edexcel Further Mathematics course. Use it for quick revision, homework support, and exam preparation.
本手册整理了 Year 11 Edexcel 进阶数学课程所需的核心公式、定理和关键结论，供你快速复习、作业辅助与备考使用。
1. Surds and Indices | 根式与指数
The laws of indices: am × an = am+n.
指数运算法则：am × an = am+n。
Raising a power to a power: (am)n = amn.
幂的乘方：(am)n = amn。
Negative and rational indices: a−n = 1 / an; am/n = n√(am).
负指数与有理指数：a−n = 1 / an；am/n = n√(am)。
Surd rules: √(ab) = √a × √b; √(a/b) = √a / √b (a, b ≥ 0). Rationalising a denominator: multiply by the conjugate, e.g. 1/(√a+√b) = (√a−√b)/(a−b).
根式规则：√(ab) = √a × √b；√(a/b) = √a / √b (a, b ≥ 0)。分母有理化：乘以共轭式，例如 1/(√a+√b) = (√a−√b)/(a−b)。
2. Quadratic Equations | 二次方程
The standard form of a quadratic equation is ax² + bx + c = 0, with a ≠ 0.
二次方程的标准形式为 ax² + bx + c = 0，且 a ≠ 0。
The quadratic formula: x = [ −b ± √(b² − 4ac) ] / (2a)
求根公式：x = [ −b ± √(b² − 4ac) ] / (2a)
The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 (two distinct real roots), Δ = 0 (one repeated real root), Δ < 0 (no real roots).
判别式 Δ = b² − 4ac 决定根的性质：Δ > 0（两个不等实根），Δ = 0（一个重实根），Δ < 0（无实根）。
Sum and product of roots (α, β): α + β = −b/a, αβ = c/a.
根与系数的关系（α, β）：α + β = −b/a，αβ = c/a。
3. Polynomials and Factor Theorem | 多项式与因式定理
For polynomial p(x), the Remainder Theorem states: when p(x) is divided by (x − a), the remainder is p(a).
对于多项式 p(x)，余式定理指出：p(x) 除以 (x − a) 的余数为 p(a)。
The Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0.
因式定理：(x − a) 是 p(x) 的因式，当且仅当 p(a) = 0。
Long division and synthetic division can be used to factorise cubics and higher-degree polynomials.
可使用长除法或综合除法对三次及更高次多项式进行因式分解。
When solving polynomial equations, factorise fully and apply the zero-product principle.
解多项式方程时，先彻底分解因式，再应用零乘积原理。
4. Coordinate Geometry | 坐标几何
Distance between two points (x₁,y₁) and (x₂,y₂): d = √[(x₂−x₁)² + (y₂−y₁)²].
两点 (x₁,y₁) 与 (x₂,y₂) 间的距离：d = √[(x₂−x₁)² + (y₂−y₁)²]。
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2).
中点：M = ((x₁+x₂)/2, (y₁+y₂)/2)。
Gradient (slope) of a line: m = (y₂−y₁)/(x₂−x₁).
直线的斜率：m = (y₂−y₁)/(x₂−x₁)。
Equation of a straight line: y − y₁ = m(x − x₁) or y = mx + c. Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
直线方程：y − y₁ = m(x − x₁) 或 y = mx + c。平行线斜率相等；垂直线满足 m₁ × m₂ = −1。
Equation of a circle with centre (h,k) and radius r: (x − h)² + (y − k)² = r²
圆心 (h,k)、半径 r 的圆方程：(x − h)² + (y − k)² = r²
5. Sequences and Series | 数列与级数
Arithmetic sequence: n-th term un = a + (n−1)d. Sum of first n terms: Sn = n/2 [2a + (n−1)d] or Sn = n/2 (a + l), where l is the last term.
等差数列：第 n 项 un = a + (n−1)d。前 n 项和：Sn = n/2 [2a + (n−1)d] 或 Sn = n/2 (a + l)，其中 l 为末项。
Geometric sequence: n-th term un = arn−1. Sum of first n terms: Sn = a(1−rn)/(1−r) for r ≠ 1. Sum to infinity (convergent, |r| < 1): S∞ = a/(1−r).
等比数列：第 n 项 un = arn−1。前 n 项和：Sn = a(1−rn)/(1−r)，r ≠ 1。无穷等比级数和（收敛，|r| < 1）：S∞ = a/(1−r)。
Binomial expansion for positive integer n: (a+b)n = Σr=0n (nCr) an−r br. The coefficient nCr = n! / [r!(n−r)!].
正整数幂的二项式展开：(a+b)n = Σr=0n (nCr) an−r br。系数 nCr = n! / [r!(n−r)!]。
6. Trigonometry | 三角学
Common exact values (angles in degrees): sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1, sin 45° = √2/2, cos 45° = √2/2, sin 60° = √3/2, cos 30° = √3/2.
常见特殊角的精确值（度）：sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1, sin 45° = √2/2, cos 45° = √2/2, sin 60° = √3/2, cos 30° = √3/2。
Radian measure: 180° = π rad. Arc length s = rθ, sector area A = ½ r²θ (θ in radians).
弧度制：180° = π 弧度。弧长 s = rθ，扇形面积 A = ½ r²θ（θ 以弧度计）。
Fundamental identities: sin²θ + cos²θ = 1; tanθ = sinθ / cosθ.
基本恒等式：sin²θ + cos²θ = 1；tanθ = sinθ / cosθ。
Sine rule: a / sin A = b / sin B = c / sin C. Cosine rule: a² = b² + c² − 2bc cos A.
正弦定理：a / sin A = b / sin B = c / sin C。余弦定理：a² = b² + c² − 2bc cos A。
Area of a triangle: ½ ab sin C.
三角形面积：½ ab sin C。
7. Exponentials and Logarithms | 指数与对数
The exponential function is ex where e ≈ 2.718. Its inverse is the natural logarithm ln x.
指数函数为 ex，其中 e ≈ 2.718。其反函数为自然对数 ln x。
Logarithm definition: if ax = y, then x = loga y. The natural log ln y = loge y.
对数定义：若 ax = y，则 x = loga y。自然对数 ln y = loge y。
Laws of logs: loga (xy) = loga x + loga y; loga (x/y) = loga x − loga y; loga (xn) = n loga x.
对数运算法则：loga (xy) = loga x + loga y；loga (x/y) = loga x − loga y；loga (xn) = n loga x。
Change of base: loga b = logc b / logc a, especially loga b = ln b / ln a
换底公式：loga b = logc b / logc a，特别是 loga b = ln b / ln a
8. Differentiation | 微分
The derivative of y = xn is dy/dx = nxn−1, for any real n.
y = xn 的导数为 dy/dx = nxn−1，n 为任意实数。
Constant multiple rule and sum rule: d/dx [k f(x)] = k f'(x); d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
常数倍法则与加法法则：d/dx [k f(x)] = k f'(x)；d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。
Derivatives of standard functions: d/dx (ekx) = k ekx; d/dx (ln x) = 1/x; d/dx (sin x) = cos x; d/dx (cos x) = −sin x.
标准函数的导数：d/dx (ekx) = k ekx；d/dx (ln x) = 1/x；d/dx (sin x) = cos x；d/dx (cos x) = −sin x。
Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x). Product rule: (uv)’ = u’ v + u v’. Quotient rule: (u/v)’ = (u’ v − u v’) / v².
链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。乘法法则：(uv)’ = u’ v + u v’。除法法则：(u/v)’ = (u’ v − u v’) / v²。
9. Integration | 积分
Integration is the reverse process of differentiation. The indefinite integral of xn (n ≠ −1) is ∫ xn dx = xn+1/(n+1) + C.
积分是微分的逆运算。xn（n ≠ −1）的不定积分为 ∫ xn dx = xn+1/(n+1) + C。
Standard integrals: ∫ ekx dx = (1/k) ekx + C; ∫ 1/x dx = ln|x| + C; ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C.
标准积分：∫ ekx dx = (1/k) ekx + C；∫ 1/x dx = ln|x| + C；∫ sin x dx = −cos x + C；∫ cos x dx = sin x + C。
Definite integral: ∫ab f(x) dx = F(b) − F(a), where F is an antiderivative of f. It represents the signed area between the curve and the x-axis.
定积分：∫ab f(x) dx = F(b) − F(a)，其中 F 是 f 的原函数。它表示曲线与 x 轴之间的带号面积。
Area between two curves: ∫ab [f(x) − g(x)] dx, assuming f(x) ≥ g(x) on [a,b].
两曲线间的面积：∫ab [f(x) − g(x)] dx，假设在 [a,b] 上 f(x) ≥ g(x)。
10. Vectors | 向量
A vector has magnitude and direction. In component form, a = x i + y j or as a column vector [x, y] (vertical). Position vector of point A relative to O is OA.
向量具有大小和方向。分量形式可写为 a = x i + y j 或列向量 [x, y]（竖排）。点 A 相对于 O 的位置向量记为 OA。
Addition and scalar multiplication: a + b = (x₁+x₂)i + (y₁+y₂)j; k a = (k x)i + (k y)j.
加法与数乘：a + b = (x₁+x₂)i + (y₁+y₂)j；k a = (k x)i + (k y)j。
Magnitude (length): |a| = √(x² + y²). A unit vector in the direction of a is â = a / |a|.
向量的模：|a| = √(x² + y²)。与 a 同方向的单位向量为 â = a / |a|。
Scalar (dot) product: a • b = x₁x₂ + y₁y₂ = |a||b| cos θ, where θ is the angle between them. For perpendicular vectors, a • b = 0.
标量积（点积）：a • b = x₁x₂ + y₁y₂ = |a||b| cos θ，θ 为两向量夹角。若 a ⟂ b，则 a • b = 0。
11. Matrices | 矩阵
A 2×2 matrix A = [ a b; c d ] stores numerical data and represents transformations. Determinant: det(A) = |A| = ad − bc.
2×2 矩阵 A = [ a b; c d ] 可存储数据并表示几何变换。行列式：det(A) = |A| = ad − bc。
Inverse of a 2×2 matrix: provided ad − bc ≠ 0, A−1 = 1/(ad−bc) [ d −b; −c a ].
2×2 矩阵的逆（前提 ad − bc ≠ 0）：A−1 = 1/(ad−bc) [ d −b; −c a ]。
Matrix multiplication: AB is found by dot products of rows of A with columns of B. Note: AB ≠ BA in general.
矩阵乘法：AB 由 A 的行与 B 的列作点积得到。注意：一般 AB ≠ BA。
Common transformation matrices: reflection in the x-axis [ 1 0; 0 −1 ]; rotation 90° anticlockwise about origin [ 0 −1; 1 0 ]; enlargement scale factor k [ k 0; 0 k ].
常见变换矩阵：关于 x 轴的反射 [ 1 0; 0 −1 ]；绕原点逆时针旋转 90° [ 0 −1; 1 0 ]；放大（缩放因子 k）[ k 0; 0 k ]。
Using matrices to solve simultaneous linear equations: write as AX = B, then X = A−1B when A is invertible.
利用矩阵解线性方程组：写成 AX = B，若 A 可逆则 X = A−1B。
12. Transformations of Graphs | 图像变换
Given y = f(x), the following transformations apply to the graph:
对于 y = f(x)，图像受到如下变换：
y = f(x) + a → vertical translation by a units (upwards if a > 0).
y = f(x) + a → 沿 y 轴平移 a 个单位（a > 0 时向上）。
y = f(x + a) → horizontal translation by −a units (to the left if a > 0).
y = f(x + a) → 沿 x 轴平移 −a 个单位（a > 0 时向左）。
y = a f(x) → vertical stretch with scale factor a (stretches by a if a > 1).
y = a f(x) → 垂直伸缩，比例因子为 a（a > 1 时拉伸）。
y = f(ax) → horizontal stretch with scale factor 1/a (compresses if a > 1).
y = f(ax) → 水平伸缩，比例因子为 1/a（a > 1 时压缩）。
y = −f(x) → reflection in the x-axis; y = f(−x) → reflection in the y-axis.
y = −f(x) → 关于 x 轴对称；y = f(−x) → 关于 y 轴对称。
When applying multiple transformations, follow the order: scale, reflection, translation, unless otherwise specified.
进行复合变换时，若无特别说明，通常顺序为：伸缩、对称、平移。
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