A-Level CCEA Further Maths: Formula & Theorem Quick Reference | A-Level CCEA 进阶数学：公式定理速查手册

📚 A-Level CCEA Further Maths: Formula & Theorem Quick Reference | A-Level CCEA 进阶数学：公式定理速查手册

This concise reference handbook pulls together the key formulas, identities, and theorems required for the CCEA A-Level Further Mathematics specification. Use it as a companion for last‑minute revision, homework checks, and problem‑solving practice across all pure further topics.

这本简明速查手册汇集了 CCEA A-Level 进阶数学考纲所需的核心公式、恒等式与定理。可作为考前冲刺、作业核对与各纯数专题解题练习的随身伴侣。

1. Complex Numbers | 复数

Cartesian form: z = x + iy, where x, y ∈ ℝ and i² = –1.

笛卡儿形式： z = x + iy，x, y ∈ ℝ，i² = –1。

Modulus: |z| = √(x² + y²).

模： |z| = √(x² + y²)。

Argument: arg z = θ, where tan θ = y/x, with –π < θ ≤ π (principal value). For x = 0, arg z = π/2 if y > 0, –π/2 if y < 0.

辐角： arg z = θ，满足 tan θ = y/x，主值范围 –π < θ ≤ π。当 x = 0 时，y > 0 则 arg z = π/2，y < 0 则 arg z = –π/2。

Conjugate: z̄ = x – iy. Then z·z̄ = |z|² and z + z̄ = 2x.

共轭复数： z̄ = x – iy。满足 z·z̄ = |z|² 且 z + z̄ = 2x。

Polar form: z = r(cos θ + i sin θ) = r e^(iθ), where r = |z|, θ = arg z.

极形式： z = r(cos θ + i sin θ) = r e^(iθ)，其中 r = |z|, θ = arg z。

Multiplication/Division: If z₁ = r₁e^(iθ₁), z₂ = r₂e^(iθ₂), then z₁z₂ = r₁r₂ e^(i(θ₁+θ₂)) and z₁/z₂ = (r₁/r₂) e^(i(θ₁–θ₂)).

乘法与除法： 若 z₁ = r₁e^(iθ₁), z₂ = r₂e^(iθ₂)，则 z₁z₂ = r₁r₂ e^(i(θ₁+θ₂))，z₁/z₂ = (r₁/r₂) e^(i(θ₁–θ₂))。

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2. De Moivre’s Theorem & Roots of Unity | 棣莫弗定理与单位根

De Moivre’s theorem: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) for any integer n.

棣莫弗定理： 对任意整数 n，(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)。

nth roots of unity: Solutions to z^n = 1 are zₖ = e^(2πik/n), k = 0, 1, 2, …, n–1. In polar form: zₖ = cos(2πk/n) + i sin(2πk/n).

n 次单位根： z^n = 1 的解为 zₖ = e^(2πik/n)，k = 0, 1, 2, …, n–1。极形式为 zₖ = cos(2πk/n) + i sin(2πk/n)。

Sum of roots of unity: The sum of all n nth roots of unity is 0. If ω ≠ 1 is an nth root of unity, then 1 + ω + ω² + … + ω^(n–1) = 0.

单位根之和： 所有 n 次单位根之和为 0。若 ω ≠ 1 为 n 次单位根，则 1 + ω + ω² + … + ω^(n–1) = 0。

Powers of sin and cos: sin^n θ and cos^n θ can be expressed as sums of sines/cosines of multiple angles using De Moivre and binomial expansion.

正弦余弦的幂： sin^n θ 与 cos^n θ 可利用棣莫弗定理和二项式展开表示为倍角正弦/余弦的和。

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3. Matrices & Determinants | 矩阵与行列式

Order and transpose: A is m × n; its transpose Aᵀ has (i,j) entry aⱼᵢ. A matrix is symmetric if Aᵀ = A, skew‑symmetric if Aᵀ = –A.

阶与转置： 矩阵 A 为 m × n；其转置 Aᵀ 的第 (i,j) 元为 aⱼᵢ。若 Aᵀ = A，则 A 对称；若 Aᵀ = –A，则 A 反对称。

Matrix multiplication: (AB)ᵢⱼ = ∑ₖ aᵢₖ bₖⱼ. Conformability: A (m × n) and B (n × p) produce C (m × p).

矩阵乘法： (AB)ᵢⱼ = ∑ₖ aᵢₖ bₖⱼ。可乘条件：A 为 m × n，B 为 n × p，结果 C 为 m × p。

Determinant (2×2): det⎛⎝a b; c d⎞⎠ = ad – bc.

行列式 (2×2)： det⎛⎝a b; c d⎞⎠ = ad – bc。

Determinant (3×3): Use Sarrus’s rule or cofactor expansion: det A = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃, where Cᵢⱼ = (–1)^(i+j) Mᵢⱼ.

行列式 (3×3)： 使用 Sarrus 法则或余子式展开：det A = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃，其中 Cᵢⱼ = (–1)^(i+j) Mᵢⱼ。

Inverse (2×2): If A = ⎛⎝a b; c d⎞⎠ and det A ≠ 0, then A⁻¹ = (1/det A)⎛⎝d –b; –c a⎞⎠.

逆矩阵 (2×2)： 若 A = ⎛⎝a b; c d⎞⎠ 且 det A ≠ 0，则 A⁻¹ = (1/det A)⎛⎝d –b; –c a⎞⎠。

Eigenvalues: Solve det(A – λI) = 0. For 2×2, λ satisfies λ² – (trace)λ + det = 0.

特征值： 求解 det(A – λI) = 0。对 2×2 矩阵，λ 满足 λ² – (迹)λ + det = 0。

Eigenvectors: For each λ, solve (A – λI)v = 0 to obtain non‑zero vectors v.

特征向量： 对每个 λ，解 (A – λI)v = 0 得到非零向量 v。

Diagonalisation: If A has n linearly independent eigenvectors, then P⁻¹AP = D, where D is diagonal with eigenvalues on the diagonal, and columns of P are the eigenvectors.

对角化： 若 A 有 n 个线性无关的特征向量，则 P⁻¹AP = D，其中 D 为对角矩阵，对角元为特征值，P 的列为对应的特征向量。

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4. Vectors & 3D Geometry | 向量与三维几何

Vector (cross) product: a × b = (|a||b| sin θ) n̂, where n̂ is a unit vector perpendicular to both a and b. In component form: a × b = (a₂b₃ – a₃b₂, a₃b₁ – a₁b₃, a₁b₂ – a₂b₁).

向量积（叉积）： a × b = (|a||b| sin θ) n̂，n̂ 为垂直于 a 和 b 的单位向量。分量形式：a × b = (a₂b₃ – a₃b₂, a₃b₁ – a₁b₃, a₁b₂ – a₂b₁)。

Scalar triple product: a · (b × c) = det[a b c]. It gives the signed volume of the parallelepiped defined by a, b, c.

标量三重积： a · (b × c) = det[a b c]，表示由 a, b, c 确定的平行六面体的带符号体积。

Equation of a line: Vector form: r = a + t b, where a is a point on the line and b is a direction vector. Cartesian form: (x–x₀)/l = (y–y₀)/m = (z–z₀)/n.

直线方程： 向量式：r = a + t b，a 为直线上一点，b 为方向向量。笛卡儿式：(x–x₀)/l = (y–y₀)/m = (z–z₀)/n。

Equation of a plane: Scalar product form: r · n = p, where n is a normal vector and p = a · n. Cartesian form: ax + by + cz = d.

平面方程： 点积式：r · n = p，n 为法向量，p = a · n。笛卡儿式：ax + by + cz = d。

Distance from point to plane: For plane r·n = p and point P with position vector p₀, distance = |p₀·n – p| / |n|.

点到平面的距离： 对平面 r·n = p 与位置向量为 p₀ 的点 P，距离 = |p₀·n – p| / |n|。

Intersection of lines/planes: Solve the vector equations simultaneously. Two lines may be skew, intersecting or parallel; two planes intersect in a line (if not parallel).

直线/平面的交： 联立向量方程求解。两条直线可能异面、相交或平行；两平面（不平行时）相交于一直线。

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5. Hyperbolic Functions | 双曲函数

Definitions: sinh x = (e^x – e^(–x))/2, cosh x = (e^x + e^(–x))/2, tanh x = sinh x / cosh x.

定义： sinh x = (e^x – e^(–x))/2，cosh x = (e^x + e^(–x))/2，tanh x = sinh x / cosh x。

Reciprocals: sech x = 1/cosh x, cosech x = 1/sinh x, coth x = 1/tanh x.

倒数函数： sech x = 1/cosh x，cosech x = 1/sinh x，coth x = 1/tanh x。

Fundamental identity: cosh² x – sinh² x = 1.

基本恒等式： cosh² x – sinh² x = 1。

Other identities: 1 – tanh² x = sech² x; coth² x – 1 = cosech² x.

其他恒等式： 1 – tanh² x = sech² x；coth² x – 1 = cosech² x。

Derivatives: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x.

导数： d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech² x。

Inverse hyperbolic functions: arsinh x = ln(x + √(x² + 1)); arcosh x = ln(x + √(x² – 1)), x ≥ 1; artanh x = ½ ln((1+x)/(1–x)), |x| < 1.

反双曲函数： arsinh x = ln(x + √(x² + 1))；arcosh x = ln(x + √(x² – 1))，x ≥ 1；artanh x = ½ ln((1+x)/(1–x))，|x| < 1。

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6. Polar Coordinates | 极坐标

Conversion: x = r cos θ, y = r sin θ, r = √(x² + y²), θ = arctan(y/x) [with quadrant adjustment]. r ≥ 0 normally.

转换： x = r cos θ，y = r sin θ，r = √(x² + y²)，θ = arctan(y/x)（需根据象限调整）。通常 r ≥ 0。

Area enclosed by a polar curve: A = ½ ∫[θ₁,θ₂] r² dθ.

极坐标曲线围成的面积： A = ½ ∫[θ₁,θ₂] r² dθ。

Arc length (polar): s = ∫[θ₁,θ₂] √(r² + (dr/dθ)²) dθ.

弧长（极坐标）： s = ∫[θ₁,θ₂] √(r² + (dr/dθ)²) dθ。

Common curves: Cardioid r = a(1 + cos θ); Limacon r = a + b cos θ; Rose curves r = a cos(nθ) or r = a sin(nθ).

常见曲线： 心形线 r = a(1 + cos θ)；蚌线 r = a + b cos θ；玫瑰线 r = a cos(nθ) 或 r = a sin(nθ)。

Tangents in polar form: If ψ is the angle between the radius vector and the tangent, tan ψ = r / (dr/dθ).

极坐标切线： 若 ψ 为径向与切线的夹角，则 tan ψ = r / (dr/dθ)。

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7. Differential Equations | 微分方程

Separable first order: dy/dx = f(x)g(y) → ∫ (1/g(y)) dy = ∫ f(x) dx.

可分离一阶方程： dy/dx = f(x)g(y) → ∫ (1/g(y)) dy = ∫ f(x) dx。

Linear integrating factor: dy/dx + P(x)y = Q(x). Integrating factor: μ(x) = e^(∫ P(x) dx). Then d/dx (μ y) = μ Q.

线性积分因子法： dy/dx + P(x)y = Q(x)。积分因子 μ(x) = e^(∫ P(x) dx)。则 d/dx (μ y) = μ Q。

Second order homogeneous (constant coefficients): a d²y/dx² + b dy/dx + c y = 0. Auxiliary equation: a m² + b m + c = 0. Roots real distinct: y = A e^(m₁x) + B e^(m₂x); repeated: y = (A + Bx)e^(mx); complex conjugate m = α ± iβ: y = e^(αx)(A cos βx + B sin βx).

二阶常系数齐次方程： a d²y/dx² + b dy/dx + c y = 0。辅助方程 a m² + b m + c = 0。相异实根：y = A e^(m₁x) + B e^(m₂x)；重根：y = (A + Bx)e^(mx)；共轭复根 m = α ± iβ：y = e^(αx)(A cos βx + B sin βx)。

Particular integrals: For f(x) = polynomial, exponential, sin/cos, use trial functions: e.g., for f(x) = p e^(kx) try yₚ = C e^(kx); for f(x) = p sin ωx + q cos ωx try yₚ = C sin ωx + D cos ωx. If overlap with complementary function, multiply by x.

特解： 对 f(x) 为多项式、指数、正弦/余弦，使用试函数。例如 f(x) = p e^(kx) 试 yₚ = C e^(kx)；f(x) = p sin ωx + q cos ωx 试 yₚ = C sin ωx + D cos ωx。若与补函数重叠，乘以 x。

Substitution methods: Given a substitution (e.g., y = v x, x = e^t), reduce order or change variable to solve.

代换法： 给出代换（如 y = v x, x = e^t）可降阶或变换变量求解。

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8. Series & Expansions | 级数与展开

Maclaurin series: f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … valid within the radius of convergence.

麦克劳林级数： f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … 在收敛半径内成立。

Standard Maclaurin series: e^x = ∑ xⁿ/n!; sin x = x – x³/3! + x⁵/5! – …; cos x = 1 – x²/2! + x⁴/4! – …; ln(1+x) = x – x²/2 + x³/3 – … (–1 < x ≤ 1); (1+x)^n = 1 + nx + n(n–1)x²/2! + … (|x| < 1).

标准麦克劳林展开： e^x = ∑ xⁿ/n!；sin x = x – x³/3! + x⁵/5! – …；cos x = 1 – x²/2! + x⁴/4! – …；ln(1+x) = x – x²/2 + x³/3 – … (–1 < x ≤ 1)；(1+x)^n = 1 + nx + n(n–1)x²/2! + … (|x| < 1)。

Summation of finite series: ∑₍ᵣ₌₁₎ⁿ r = n(n+1)/2, ∑₍ᵣ₌₁₎ⁿ r² = n(n+1)(2n+1)/6, ∑₍ᵣ₌₁₎ⁿ r³ = [n(n+1)/2]².

有限级数求和： ∑₍ᵣ₌₁₎ⁿ r = n(n+1)/2，∑₍ᵣ₌₁₎ⁿ r² = n(n+1)(2n+1)/6，∑₍ᵣ₌₁₎ⁿ r³ = [n(n+1)/2]²。

Method of differences: Express the general term as f(r) – f(r+1) or similar, so that the series collapses to a few boundary terms.

差分法： 将通项写成 f(r) – f(r+1) 等形式，使级数折叠为少量边界项。

Binomial expansion for rational index: (a + bx)^n = a^n(1 + (b/a)x)^n, expand using general binomial theorem, valid for |(

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