📚 Year 12 OCR Further Mathematics: Quick Reference Formula & Theorem Handbook | Year 12 OCR 进阶数学：公式定理速查手册
This concise handbook collates the essential formulas, theorems and key results from the Year 12 OCR Further Mathematics curriculum. Use it for quick revision, problem‑solving and exam preparation. Every entry is presented in English followed by its Chinese equivalent to support bilingual learners.
这本速查手册汇集了 Year 12 OCR 进阶数学课程中最核心的公式、定理与关键结论，适合快速复习、解题和备考。每个条目均以英文在前、中文在后的配对形式呈现，方便双语学习者对照使用。
1. Complex Numbers – Basic Operations | 复数 – 基本运算
A complex number has the form z = x + iy, where x, y ∈ ℝ he real part is Re(z) = x, and the imaginary part is Im(z) = y. The complex conjugate is z̅ = x − iy.
复数写作 z = x + iy，其中 x, y ∈ ℝ 且 i² = −1。实部 Re(z) = x，虚部 Im(z) = y。复共轭为 z̅ = x − iy。
Addition and subtraction are performed component‑ n is achieved by multiplying numerator and denominator by the conjugate of the denominator.
加减法按实部与虚部分别运算：(a + ib) ± (c + id) = (a ± c) + i(b ± d)。乘法利用 i² = −1：(a + ib)(c + id) = (ac − bd) + i(ad + bc)。除法需将分子分母同乘分母的共轭复数。
Equality of complex numbers: a + ib = c + id ⟺ Two complex numbers are equal exactly when their real and imaginary parts are each equal.
复数相等的条件：a + ib = c + id ⟺ a = c 且 b = d。只有当实部与虚部分别相等时，两个复数才相等。
2. Argand Diagrams and Modulus–Argument Form | 阿尔冈图与模–幅角形式
in a plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.
阿尔冈图将复数 z = x + iy 表示为平面上的点 (x, y)。横轴为实轴，纵轴为虚轴。
The modulus of z is |z| = √x² + y² axis, usually chosen in (−π, π] or [0, 2π). Thus z = x + iy = r(cosθ + i sinθ), where r = |z| and θ = arg(z).
z 的模 |z| = √x² + y²，表示到原点的距离。幅角 arg(z) 是从正实轴量起的角度 θ，通常取值在 (−π, π] 或 [0, 2π) 内。因此 z = x + iy = r(cosθ + i sinθ)，其中 r = |z|，θ = arg(z)。
When multiplying two complex numbers in modulus–argument form: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂) (modulo 2π). For division, |z₁/z₂| = |z₁|/|z₂| and arg(z₁/z₂) = arg(z₁) − arg(z₂).
用模–幅角形式做乘法时：|z₁z₂| = |z₁||z₂| 且 arg(z₁z₂) = arg(z₁) + arg(z₂)（模 2π）。除法满足 |z₁/z₂| = |z₁|/|z₂|，arg(z₁/z₂) = arg(z₁) − arg(z₂)。
The complex conjugate in polar form: if z = r(cosθ + i sinθ), then z̅ = r(cosθ − i sinθ) = r(cos(−θ) + i sin(−θ)). Thus |z̅| = r and arg(z̅) = −θ.
极坐标形式下的共轭复数：若 z = r(cosθ + i sinθ)，则 z̅ = r(cosθ − i sinθ) = r(cos(−θ) + i sin(−θ))，因此 |z̅| = r 且 arg(z̅) = −θ。
3. De Moivre's Theorem and Trigonometric Applications | 棣莫弗定理与三角学应用
De Moivre' ). It provides a powerful tool for deriving trigonometric identities and for raising complex numbers to powers.
棣莫弗定理：对任意整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。它为推导三角恒等式以及复数的乘方提供了强有力的工具。
To find powers of a complex number, write z in modulus–argument form: zⁿ = rⁿ (cos(nθ) + i sin(nθ)). For fractional powers (roots), z1/n = r1/n [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)] for k = 0, 1, 2, …, n−1.
求复数的幂次时，先写成模–幅角形式：zⁿ = rⁿ (cos(nθ) + i sin(nθ))。对于分数次幂（开方），z1/n = r1/n [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]，k = 0, 1, 2, …, n−1。
nsions for sin(nθ) and cos(nθ) can be obtained by expanding the binomial (cosθ + i sinθ)ⁿ and equating real and imaginary parts.
利用棣莫弗定理，可得 sin 2θ = 2 sinθ cosθ，cos 2θ = cos²θ − sin²θ，并可通过展开 (cosθ + i sinθ)ⁿ 并比较实部和虚部得到 sin(nθ) 和 cos(nθ) 的展开式。
4. Matrices – Operations and Determinants | 矩阵 – 运算与行列式
f numbers with m rows and n columns. Addition (or subtraction) is defined only for matrices of the same dimensions, by adding corresponding entries element‑wise.
m × n 矩阵是一个 m 行 n 列的矩形数表。只有同型矩阵才能进行加减运算，规则为对应元素分别相加减。
Multiplication of an m × n matrix A by an n × p matrix B yields an m × p matrix C, where Cij = Σ (aik bkj) for k = 1 to n. Matrix multiplication is not commutative in general: AB ≠ BA.
一个 m × n 矩阵 A 与一个 n × p 矩阵 B 相乘，得到 m × p 矩阵 C，其元素 Cij = Σ (aik bkj) (k 从 1 到 n)。矩阵乘法一般不满足交换律：AB ≠ BA。
nant of a square matrix can be used to test for invertibility: A matrix is invertible (non‑singular) if and only if det(A) ≠ 0.
2 × 2 矩阵 A = [[a, b], [c, d]] 的行列式为 det(A) = ad − bc。方阵的行列式可用于判断可逆性：det(A) ≠ 0 时矩阵可逆（非奇异）。
umn using minors, e.g. along the first row: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg).
对于 3 × 3 矩阵，可通过按某一行或某一列的余子式展开计算行列式，例如按第一行展开：det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)。
5. Matrix Inverses and Linear Transformations | 矩阵的逆与线性变换
0. For 3 × 3 matrices, the inverse can be found by the cofactor method or row operations.
2 × 2 矩阵 A = [[a, b], [c, d]] 的逆为 A⁻¹ = (1/(ad−bc)) [[d, −b], [−c, a]]，前提是 ad−bc ≠ 0。3 × 3 矩阵的逆可通过伴随矩阵法或行变换求得。
× 2 matrix. Common transformations include: reflection (e.g. [[1,0],[0,−1]] reflects in x‑ osθ, −sinθ], [sinθ, cosθ]], enlargement scale factor k: [[k,0],[0,k]], shear parallel to x‑axis: [[1,λ],[0,1]].
平面上的线性变换可用 2 × 2 矩阵表示。常见变换有：反射（如 [[1,0],[0,−1]] 关于 x 轴反射），旋转 θ 角：[[cosθ, −sinθ], [sinθ, cosθ]]，放大倍数 k：[[k,0],[0,k]]，平行于 x 轴的剪切：[[1,λ],[0,1]]。
The composition of two transformations corresponds to multiplying their matrices. If transformation B is applied followed by A, the combined matrix is AB. The order matters; the matrix of the transformation applied first is on the right.
两个变换的复合对应于矩阵相乘。若先实施变换 B 再实施变换 A，则合成矩阵为 AB。顺序非常重要，先实施的变换其矩阵在右侧。
6. Mathematical Induction | 数学归纳法
≥ a). The proof has three main steps: Base case, inductive hypothesis, and inductive step.
数学归纳法是用于证明命题 P(n) 对所有正整数 n（或所有 n ≥ a）都成立的方法。证明包括三个主要步骤：奠基步骤、归纳假设和归纳递推。
p: using the assumption P(k), prove that P(k+1) is true. Then by induction, P(n) holds for all n ≥ a.
奠基步骤：验证 P(1)（或最小整数 a）成立。归纳假设：假设对某个任意整数 k ≥ a，P(k) 成立。归纳递推：利用假设 P(k)，证明 P(k+1) 成立。由此通过归纳，P(n) 对所有 n ≥ a 成立。
ility results (e.g., n³ − n is divisible by 6), and inequalities (e.g., 2ⁿ > n for n ≥ 1).
典型应用包括证明求和公式（如 Σ r = n(n+1)/2）、整除性问题（如 n³ − n 可被 6 整除）以及不等式（如当 n ≥ 1 时 2ⁿ > n）。
7. Summation of Series – Standard Results | 级数求和 – 标准结果
Standard summation formulas are essential for evaluating finite series. They are often proved by induction and used in further analysis.
标准求和公式是计算有限级数的核心工具，常通过归纳法证明，并在后续分析中使用。
Σ r from r=1 to n = ½ n(n+1)
求和公式 1：Σ r (r=1 to n) = ½ n(n+1)
Σ r² from r=1 to n = ⅙ n(n+1)(2n+1)
Σ r² (r=1 to n) = ⅙ n(n+1)(2n+1)
Σ r³ from r=1 to n = ¼ n² (n+1)² = (½ n(n+1))²
Σ r³ (r=1 to n) = ¼ n² (n+1)² = (½ n(n+1))²
+ cΣ r. A typical exam question requires splitting a more complicated sum into recognisable standard forms.
利用 Σ 的线性性质可组合使用：Σ (ar³ + br² + cr) = aΣ r³ + bΣ r² + cΣ r。典型考题常需要将较复杂的和式拆分为以上标准形式。
8. Roots of Polynomials and Coefficient Relationships | 多项式的根与系数关系
/a and αβ = c/a. These relationships are derived from the factorised form a(z−α)(z−β) = 0.
对于二次方程 az² + bz + c = 0 (a ≠ 0)，设两根为 α, β，则 α + β = −b/a，αβ = c/a。这些关系源于因式分解形式 a(z−α)(z−β) = 0。
bz² + cz + d = 0 with roots α, β, γ: Σα = α+β+γ = −b/a, Σαβ = αβ+βγ+γα = c/a, αβγ = −d/a.
对于三次方程 az³ + bz² + cz + d = 0，根为 α, β, γ，则 Σα = α+β+γ = −b/a，Σαβ = αβ+βγ+γα = c/a，αβγ = −d/a。
³ + cz² + dz + e = 0 with roots α, β, γ, δ: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a.
对于四次方程 az⁴ + bz³ + cz² + dz + e = 0，根为 α, β, γ, δ，则 Σα = −b/a，Σαβ = c/a，Σαβγ = −d/a，αβγδ = e/a。
without solving the equation directly. This technique often appears alongside induction or recurrence relations.
这些对称和使我们能够直接计算如 α²+β² = (α+β)² − 2αβ 的表达式，而无需解方程。该技巧常与归纳法或递推关系结合考查。
9. Vector Geometry in 3D | 三维向量几何
A vector in ℝ a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, the dot product is a·b = a₁b₁ + a₂b₂ + a₃b₃.
ℝ³ 中的向量表示为 xi + yj + zk 或列向量。主要运算包括加法、数乘和点积。向量 a = a₁i + a₂j + a₃k 和 b = b₁i + b₂j + b₃k 的点积为 a·b = a₁b₁ + a₂b₂ + a₃b₃。
The magnitude of a is |a| = √a₁² + a₂² + a₃². The angle θ between a and b satisfies cosθ = (a·b) / (|a||b|). Two non‑zero vectors are perpendicular if and only if a·b = 0.
a 的模为 |a| = √a₁² + a₂² + a₃²。a 与 b 的夹角 θ 满足 cosθ = (a·b) / (|a||b|)。两非零向量垂直当且仅当 a·b = 0。
hrough point A with position vector a and parallel to direction vector d is r = a + λ d, λ ∈ ℝ. In Cartesian form, (x−x₀)/d₁ = (y−y₀)/d₂ = (z−z₀)/d₃ (= λ).
过定点 A（位置向量 a）且平行于方向向量 d 的直线向量方程为 r = a + λ d，λ ∈ ℝ。其坐标形式为 (x−x₀)/d₁ = (y−y₀)/d₂ = (z−z₀)/d₃ (= λ)。
The foot of the perpendicular from a point to a line, the shortest distance, and the intersection points can be found using dot‑product conditions and parameter values.
点到直线的垂足、最短距离以及交点均可利用点积条件与参数值求解。
10. Binomial Expansion for Rational Powers | 有理数指数的二项式展开
o a rational index n (n not a positive integer) using the form (1+x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + …, which is valid for |x| < 1. This infinite series representation is crucial for approximations.
二项式展开可推广到有理数指数 n（非正整数），形式为 (1+x)ⁿ = 1 + nx + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + …，此展开在 |x| < 1 时成立。该无穷级数在近似计算中至关重要。
For a general (a + bx)ⁿ, first factor out aⁿ to obtain aⁿ (1 + (b/a)x)ⁿ, then apply the expansion. The expansion is valid for |(b/a)x| < 1, provided a ≠ 0.
对于一般形式 (a + bx)ⁿ，先提取 aⁿ 化为 aⁿ (1 + (b/a)x)ⁿ，再应用展开式。展开有效范围为 |(b/a)x| < 1（a ≠ 0）。
Special cases: when n = −1, the expansion is 1 − x + x² − x³ + … (geometric series). For n = ½, the expansion gives (1+x)½ = 1 + ½x − ⅛x² + … . These series underpin many approximations in calculus and physics.
特例：当 n = −1 时，展开为 1 − x + x² − x³ + …（几何级数）。当 n = ½ 时，有 (1+x)½ = 1 + ½x − ⅛x² + … 。这些级数是微积分和物理学中许多近似计算的基础。
The general term (r+1)ᵗʰ in the expansion of (1+x)ⁿ is [n(n−1)…(n−r+1) / r!] xʳ. When n is not a positive integer, the series never terminates; it is an infinite series.
(1+x)ⁿ 展开中第 r+1 项为 [n(n−1)…(n−r+1) / r!] xʳ。当 n 不是正整数时，级数永不终止，即为无穷级数。
11. Further Calculus Toolkit – Differentiation and Integration Techniques | 进阶微积分工具箱 – 微分与积分技巧
Chain rule, product rule and quotient rule are assumed knowledge, but in Further Mathematics they are applied to more complex functions, including parametric and implicit forms.
链式法则、乘法法则和除法法则是基础要求，但在进阶数学中它们被用于更复杂的函数，包括参数形式和隐函数。
Standard derivatives: d/dx (sin kx) = k cos kx, d/dx (cos kx) = −k sin kx, d/dx (ekx) = k ekx, d/dx (ln|f(x)|) = f'(x)/f(x). These must be memorised for rapid application.
标准导数：d/dx (sin kx) = k cos kx，d/dx (cos kx) = −k sin kx，d/dx (ekx) = k ekx，d/dx (ln|f(x)|) = f'(x)/f(x)。这些务必牢记以便快速运用。
Reverse chain rule (inspection) for integration: ∫ k f'(x) [f(x)]ⁿ dx = k/(n+1) [f(x)]ⁿ⁺¹ + c (n ≠ −1), and ∫ f'(x)/f(x) dx = ln|f(x)| + c. Recognising these forms saves time in integration problems.
积分中的逆向链式法（凑微分）：∫ k f'(x) [f(x)]ⁿ dx = k/(n+1) [f(x)]ⁿ⁺¹ + c (n ≠ −1)，以及 ∫ f'(x)/f(x) dx = ln|f(x)| + c。识别这些形式能大幅提升积分求解速度。
ion: the volume generated when a curve y = f(x) from x = a to x = b is rotated about the x‑axis is V = π ∫ₐᵇ y² dx. For rotation about the y‑axis of x = g(y) from y = c to y = d, V = π ∫_cd x² dy.
旋转体体积：曲线 y = f(x) 在 x = a 到 x = b 区间绕 x 轴旋转所得体积为 V = π ∫ₐᵇ y² dx。若 x = g(y) 在 y = c 到 y = d 绕 y 轴旋转，V = π ∫_cd x² dy。
= d(dy/dx)/dt ÷ dx/dt. These are essential for analysing curves described parametrically.
参数微分：若 x = f(t), y = g(t)，则 dy/dx = (dy/dt)/(dx/dt)。二阶导数 d²y/dx² = d(dy/dx)/dt ÷ dx/dt。这些在研究参数曲线时不可或缺。
12. Second‑Order Recurrence Relations and Particular Solutions | 二阶递推关系与特解
0. If the roots λ₁, λ₂ are real and distinct, the general solution is uₙ = A λ₁ⁿ + B λ₂ⁿ.
二阶齐次线性递推关系为 uₙ = puₙ₋₁ + quₙ₋₂。其特征方程为 λ² − pλ − q = 0。若两根 λ₁, λ₂ 为不等实数，则通解为 uₙ = A λ₁ⁿ + B λ₂ⁿ。
x conjugates r(cosθ ± i sinθ), the real general solution is uₙ = rⁿ (C cos nθ + D sin nθ).
若特征方程有重根 λ，则通解为 uₙ = (A + Bn) λⁿ。当两根为共轭复数 r(cosθ ± i sinθ) 时，实通解为 uₙ = rⁿ (C cos nθ + D sin nθ)。
For a non‑ ₙ₋₁ + quₙ₋₂ + f(n), the general solution is the sum of the complementary solution (from the homogeneous part) and a particular solution. Particular solutions are found by trial functions matching f(n): constant for constant f(n), linear for linear, exponential for exponential, etc.
对于非齐次递推关系，如 uₙ = puₙ₋₁ + quₙ₋₂ + f(n)，通解为齐次部分补解与特解之和。特解通过根据 f(n) 选择试探函数求得：常数对应常数特解，线性对应线性，指数对应指数等。
Given initial conditions, the arbitrary constants in the general solution can be determined to obtain a unique closed‑form solution. This technique links strongly to series and induction problems.
给定初始条件后，可确定通解中的任意常数，得到唯一的封闭解。这一技巧与级数和归纳法问题紧密相连。
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