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

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

This concise reference collates the essential formulas, identities, and theorems required for the Eduqas A-Level Further Mathematics specification. It is designed for rapid revision, covering Further Pure, Further Mechanics, and Further Statistics topics. Use it alongside past papers to reinforce your knowledge.

这份精简速查手册汇集了Eduqas A-Level进阶数学考试所需的必备公式、恒等式和定理。它专为快速复习设计，涵盖进阶纯数学、进阶力学和进阶统计模块。请结合历年真题使用，以巩固理解。

1. Complex Numbers | 复数

If z = a + bi, the complex conjugate is z* = a - bi.

若 z = a + bi，则其共轭为 z* = a - bi。

The modulus is |z| = √a² + b² and the argument is arg(z) = θ where tanθ = b/a.

模长定义为 |z| = √a² + b²，辐角 arg(z) = θ 满足 tanθ = b/a。

De Moivre's theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ).

棣莫弗定理：(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。

The nth roots of unity are given by e^2πik/n for k = 0, 1, …, n-1.

n 次单位根由 e^2πik/n (k = 0, 1, …, n-1) 给出。

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2. Matrices and Linear Transformations | 矩阵与线性变换

For a 2×2 matrix M = [[a, b], [c, d]], the determinant is det(M) = ad - bc.

对于 2×2 矩阵 M = [[a, b], [c, d]]，行列式为 det(M) = ad - bc。

The inverse is M⁻¹ = (1/det(M)) [[d, -b], [-c, a]], valid when det(M) ≠ 0.

逆矩阵为 M⁻¹ = (1/det(M)) [[d, -b], [-c, a]]，仅在 det(M) ≠ 0 时成立。

Eigenvalues λ satisfy det(M - λI) = 0; eigenvectors v then satisfy (M - λI)v = 0.

特征值 λ 满足 det(M - λI) = 0；之后特征向量 v 满足 (M - λI)v = 0。

A rotation by θ anticlockwise is represented by [[cosθ, -sinθ], [sinθ, cosθ]].

逆时针旋转 θ 的矩阵表示为 [[cosθ, -sinθ], [sinθ, cosθ]]。

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3. Further Algebra and Functions | 进阶代数与函数

Sum of roots for cubic ax³ + bx² + cx + d = 0: α + β + γ = -b/a.

三次方程 ax³ + bx² + cx + d = 0 的根之和：α + β + γ = -b/a。

Sum of pairwise products: αβ + βγ + γα = c/a, product: αβγ = -d/a.

两两乘积之和：αβ + βγ + γα = c/a，根之积：αβγ = -d/a。

Standard summation results: Σᵣ₌₁ⁿ r = ½n(n+1), Σr² = ⅙n(n+1)(2n+1), Σr³ = ¼n²(n+1)².

标准求和公式：Σᵣ₌₁ⁿ r = ½n(n+1)，Σr² = ⅙n(n+1)(2n+1)，Σr³ = ¼n²(n+1)²。

The Maclaurin series expansion: f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ….

麦克劳林展开式：f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + …。

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4. Further Calculus | 进阶微积分

Reduction formula for Iₙ = ∫ sinⁿx dx: Iₙ = -1/n sinⁿ⁻¹x cosx + (n-1)/n Iₙ₋₂.

Iₙ = ∫ sinⁿx dx 的递推公式：Iₙ = -1/n sinⁿ⁻¹x cosx + (n-1)/n Iₙ₋₂。

Arc length of a curve y = f(x) from a to b: s = ∫ₐᵇ √(1 + (dy/dx)²) dx.

曲线 y = f(x) 从 a 到 b 的弧长：s = ∫ₐᵇ √(1 + (dy/dx)²) dx。

Surface area of revolution about the x-axis: S = 2π ∫ₐᵇ y √(1 + (dy/dx)²) dx.

绕 x 轴旋转的曲面面积：S = 2π ∫ₐᵇ y √(1 + (dy/dx)²) dx。

proper integral evaluation requires taking limits, e.g., ∫₁∞ f(x) dx = limₜ→∞ ∫₁ᵗ f(x) dx.

反常积分计算需取极限，例如 ∫₁∞ f(x) dx = limₜ→∞ ∫₁ᵗ f(x) dx。

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5. Vectors | 向量

Dot product: a · b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃.

点积：a · b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃。

Cross product: a × b = (a₂b₃ - a₃b₂)i - (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)k.

叉积：a × b = (a₂b₃ - a₃b₂)i - (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)k。

Vector equation of a line: r = a + t d.

直线向量方程：r = a + t d。

Equation of a plane: r · n = p, where n is the normal vector.

平面方程：r · n = p，其中 n 为法向量。

Distance from point to plane: |(r₀ · n - p)| / |n|.

点到平面距离：|(r₀ · n - p)| / |n|。

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6. Polar Coordinates | 极坐标

Polar form: x = r cosθ, y = r sinθ; r = √x² + y².

极坐标形式：x = r cosθ, y = r sinθ; r = √x² + y²。

Area enclosed by a polar curve r = f(θ): A = ½ ∫ₐᵝ r² dθ.

极坐标曲线 r = f(θ) 所围面积：A = ½ ∫ₐᵝ r² dθ。

Arc length in polar coordinates: s = ∫ₐᵝ √(r² + (dr/dθ)²) dθ.

极坐标下的弧长：s = ∫ₐᵝ √(r² + (dr/dθ)²) dθ。

Common curves: cardioid r = a(1 + cosθ), rose r = a cos(nθ).

常见曲线：心形线 r = a(1 + cosθ)，玫瑰线 r = a cos(nθ)。

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7. Hyperbolic Functions | 双曲函数

Function	Definition	Identity
sinh x	½(eˣ - e⁻ˣ)	cosh²x - sinh²x = 1
cosh x	½(eˣ + e⁻ˣ)	sinh(2x) = 2 sinh x cosh x
tanh x	sinh x / cosh x	tanh²x + sech²x = 1

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 are expressible in logs, e.g., arsinh x = ln(x + √x² + 1).

反双曲函数可用对数表示，例如 arsinh x = ln(x + √x² + 1)。

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8. Differential Equations | 微分方程

Separable ODE: dy/dx = g(x)h(y) → ∫ 1/h(y) dy = ∫ g(x) dx.

可分离的一阶常微分方程：dy/dx = g(x)h(y) → ∫ 1/h(y) dy = ∫ g(x) dx。

Integrating factor for linear ODE dy/dx + P(x)y = Q(x): μ = e^{∫ P dx}.

线性微分方程 dy/dx + P(x)y = Q(x) 的积分因子：μ = e^{∫ P dx}。

Homogeneous 2nd order with constant coefficients: ay'' + by' + cy = 0.

常系数齐次二阶方程：ay'' + by' + cy = 0。

Auxiliary equation: am² + bm + c = 0; if roots real and distinct, y = Ae^m₁x + Be^m₂x.

辅助方程：am² + bm + c = 0；若根为相异实根，通解为 y = Ae^m₁x + Be^m₂x。

For repeated root m, y = (A + Bx)e^mx. For complex roots α ± iβ, y = e^αx(A cosβx + B sinβx).

重根 m 时，y = (A + Bx)e^mx；复根 α ± iβ 时，y = e^αx(A cosβx + B sinβx)。

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9. Further Mechanics | 进阶力学

Impulse-momentum principle: I = mv - mu, where I = ∫ F dt.

冲量-动量原理：I = mv - mu，其中冲量 I = ∫ F dt。

k-energy principle: Work done by resultant force = change in kinetic energy (½mv² - ½mu²).

功能原理：合力做的功 = 动能变化量 (½mv² - ½mu²)。

Elastic potential energy stored in a spring: EPE = ½kx², where k is stiffness.

弹簧储存的弹性势能：EPE = ½kx²，k 为劲度系数。

initial speed U at angle α: horizontal range = (U² sin2α)/g, max height = (U² sin²α)/(2g).

以初速 U、角度 α 斜抛运动：水平射程 = (U² sin2α)/g，最大高度 = (U² sin²α)/(2g)。

Motion in a circle: radial accel = v²/r = rω²; tangential accel = rα.

圆周运动：径向加速度 = v²/r = rω²；切向加速度 = rα。

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10. Further Statistics | 进阶统计

Poisson distribution: X ~ Po(λ), P(X=r) = e^-λ λ^r / r!, mean = variance = λ.

泊松分布：X ~ Po(λ)，P(X=r) = e^-λ λ^r / r!，均值 = 方差 = λ。

Geometric distribution: X ~ Geo(p), P(X=r) = p(1-p)^r-1 for r=1,2,…; E(X) = 1/p.

几何分布：X ~ Geo(p)，对 r=1,2,… 有 P(X=r) = p(1-p)^r-1；期望 E(X) = 1/p。

Negative binomial: X ~ NB(r, p), P(X=k) = C(k-1, r-1) p^r (1-p)^k-r for k = r, r+1,….

负二项分布：X ~ NB(r, p)，P(X=k) = C(k-1, r-1) p^r (1-p)^k-r，k = r, r+1,…。

ic: X² = Σ (Oᵢ - Eᵢ)² / Eᵢ; degrees of freedom: (rows-1)(columns-1) for contingency table.

卡方检验统计量：X² = Σ (Oᵢ - Eᵢ)² / Eᵢ；列联表自由度：(行数-1)(列数-1)。

Product moment correlation coefficient: r = S_xy / √S_xx S_yy.

积矩相关系数：r = S_xy / √S_xx S_yy。

Regression line y on x: y = a + bx, where b = S_xy / S_xx and a = ȳ - b x̄.

y 对 x 的回归线：y = a + bx，其中 b = S_xy / S_xx，a = ȳ - b x̄。

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11. Numerical Methods | 数值方法

Newton-Raphson iteration: xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ).

牛顿-拉弗森迭代公式：xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)。

inate rule for approximate integration: ∫ₐᵇ f(x) dx ≈ h Σ f(midpoints), where h = (b-a)/n.

中点规则近似积分：∫ₐᵇ f(x) dx ≈ h Σ f(中点)，其中 h = (b-a)/n。

Simpson's rule: ∫ₐᵇ f(x) dx ≈ h/3 [y₀ + yₙ + 4(y₁+y₃+…) + 2(y₂+y₄+…)], n even.

辛普森法则：∫ₐᵇ f(x) dx ≈ h/3 [y₀ + yₙ + 4(y₁+y₃+…) + 2(y₂+y₄+…)]，要求 n 为偶数。

Euler's method for ODE: yₙ₊₁ = yₙ + h f(xₙ, yₙ).

欧拉法解微分方程：yₙ₊₁ = yₙ + h f(xₙ, yₙ)。

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