📚 GCSE Eduqas Further Maths: Quick Reference Formulas & Theorems | GCSE Eduqas 进阶数学：公式定理速查手册
This quick-reference handbook brings together the essential formulas, identities and theorems needed for the GCSE Eduqas Further Mathematics course. Use it for last-minute revision, as a checklist, or to refresh your memory during practice.
本速查手册汇集了 GCSE Eduqas 进阶数学课程中所有核心公式、恒等式与定理，适合考前速查、自测梳理，也可在做题时随时翻阅巩固。
1. Algebraic Foundations | 代数基础
Multiplication law of indices: aᵐ × aⁿ = aᵐ⁺ⁿ
指数乘法法则：aᵐ × aⁿ = aᵐ⁺ⁿ
Division law of indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
指数除法法则：aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power of a power: (aᵐ)ⁿ = aᵐⁿ
幂的乘方：(aᵐ)ⁿ = aᵐⁿ
Zero index: a⁰ = 1 (a ≠ 0)
零指数：a⁰ = 1（a ≠ 0）
Negative index: a⁻ⁿ = 1 / aⁿ
负指数：a⁻ⁿ = 1 / aⁿ
Surds multiplication: √a × √b = √ab
根式乘法：√a × √b = √ab
Surds division: √a / √b = √a/b
根式除法：√a / √b = √a/b
Rationalising a denominator with a single surd: 1/√a = √a / a
分母有理化（单项根式）：1/√a = √a / a
Rationalising a binomial denominator: use (a + √b)(a − √b) = a² − b
二项根式分母有理化：利用 (a + √b)(a − √b) = a² − b
2. Quadratics & Polynomials | 二次与多项式
Quadratic formula: if ax² + bx + c = 0, then x = [−b ± √b² − 4ac] / 2a
二次公式：若 ax² + bx + c = 0，则 x = [−b ± √b² − 4ac] / 2a
Discriminant: Δ = b² − 4ac
判别式：Δ = b² − 4ac
Δ > 0: two distinct real roots; Δ = 0: one repeated real root; Δ < 0: no real roots
Δ > 0：两个不等实根；Δ = 0：一个重根；Δ < 0：无实根
Completed square form: a(x − h)² + k, vertex (h, k)
配方法形式：a(x − h)² + k，顶点为 (h, k)
Factor theorem: (x − p) is a factor of f(x) ⇔ f(p) = 0
因式定理：(x − p) 是 f(x) 的因式 ⇔ f(p) = 0
Polynomial division by a linear factor yields quotient q(x) and remainder r
多项式除以一次因式，可得商式 q(x) 与余数 r
Sum and product of roots for x² + bx + c = 0: sum = −b, product = c
二次方程根的和与积：对于 x² + bx + c = 0，和 = −b，积 = c
3. Coordinate Geometry: Straight Lines | 坐标几何：直线
Distance between two points: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
两点距离公式：d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
中点公式：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Gradient between two points: m = (y₂ − y₁) / (x₂ − x₁)
两点间斜率：m = (y₂ − y₁) / (x₂ − x₁)
Equation of a straight line: y − y₁ = m(x − x₁)
直线点斜式：y − y₁ = m(x − x₁)
Alternative form: y = mx + c, where c is the y-intercept
斜截式：y = mx + c，其中 c 为 y 轴截距
Parallel lines have equal gradients: m₁ = m₂
平行线斜率相等：m₁ = m₂
Perpendicular lines satisfy: m₁ × m₂ = −1
垂直线斜率满足：m₁ × m₂ = −1
4. Coordinate Geometry: Circles | 坐标几何：圆
Standard circle equation: (x − a)² + (y − b)² = r², centre (a, b), radius r
圆的标准方程：(x − a)² + (y − b)² = r²，圆心 (a, b)，半径为 r
General form: x² + y² + 2gx + 2fy + c = 0, centre (−g, −f), radius √g² + f² − c
一般式：x² + y² + 2gx + 2fy + c = 0，圆心 (−g, −f)，半径 √g² + f² − c
Tangent to a circle: the radius to the point of contact is perpendicular to the tangent
圆的切线：切点处的半径与切线垂直
Gradient of radius to point of tangency T obeys mrad × mtang = −1
切点半径的斜率与切线斜率满足 m_半径 × m_切线 = −1
Distance from a point to the centre equals radius for a point on the circle
圆上一点到圆心的距离等于半径
5. Trigonometry Basics | 三角函数基础
SOH CAH TOA: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
SOH CAH TOA：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边
Exact values:
sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2
cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½
tan 30° = 1/√3, tan 45° = 1, tan 60° = √3
特殊角精确值：
sin 30° = ½，sin 45° = √2/2，sin 60° = √3/2
cos 30° = √3/2，cos 45° = √2/2，cos 60° = ½
tan 30° = 1/√3，tan 45° = 1，tan 60° = √3
Sine rule: a / sin A = b / sin B = c / sin C
正弦定理：a / sin A = b / sin B = c / sin C
Cosine rule: a² = b² + c² − 2bc cos A
余弦定理：a² = b² + c² − 2bc cos A
Area of a triangle: ½ ab sin C
三角形面积公式：½ ab sin C
6. Trigonometric Identities & Equations | 三角恒等式与方程
Fundamental identity: sin²θ + cos²θ = 1
基本恒等式：sin²θ + cos²θ = 1
Tangent identity: tan θ = sin θ / cos θ
正切恒等式：tan θ = sin θ / cos θ
Solving equations: rearrange using identities, then find angles in the given interval
解三角方程：利用恒等式整理，然后在指定区间内求角
General solution for sin θ = k: θ = 180° − θ_ref + 360°n or θ = θ_ref + 360°n
sin θ = k 的通解：θ = θ_ref + 360°n 或 θ = 180° − θ_ref + 360°n
For cos θ = k: θ = ± θ_ref + 360°n; for tan θ = k: θ = θ_ref + 180°n
cos θ = k 的通解：θ = ± θ_ref + 360°n；tan θ = k 的通解：θ = θ_ref + 180°n
7. Binomial Expansion & Sequences | 二项式展开与数列
Binomial theorem: (a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ, for r = 0 to n
二项式定理：(a + b)ⁿ = Σ (ⁿCᵣ) aⁿ⁻ʳ bʳ，r 从 0 到 n
ⁿCᵣ = n! / [r!(n − r)!]
ⁿCᵣ = n! / [r!(n − r)!]
Arithmetic sequence n‑th term: uₙ = a + (n − 1)d
等差数列第 n 项：uₙ = a + (n − 1)d
Sum of first n terms: Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l)
前 n 项和：Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)
Geometric sequence n‑th term: uₙ = arⁿ⁻¹
等比数列第 n 项：uₙ = arⁿ⁻¹
Sum of first n terms (r ≠ 1): Sₙ = a(1 − rⁿ) / (1 − r)
等比前 n 项和（r ≠ 1）：Sₙ = a(1 − rⁿ) / (1 − r)
8. Differentiation | 微分
Derivative from first principles: f'(x) = limₕ→₀ [f(x + h) − f(x)] / h
导数定义（第一原理）：f'(x) = limₕ→₀ [f(x + h) − f(x)] / h
Power rule: d/dx (xⁿ) = n xⁿ⁻¹
幂法则：d/dx (xⁿ) = n xⁿ⁻¹
Derivative of a sum: d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
和差求导：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
Gradient of a curve at x = a is f'(a); equation of tangent: y − f(a) = f'(a)(x − a)
曲线在 x = a 处斜率为 f'(a)；切线方程：y − f(a) = f'(a)(x − a)
Normal gradient: −1 / f'(a), equation of normal: y − f(a) = −1/f'(a) (x − a)
法线斜率：−1 / f'(a)，法线方程：y − f(a) = −1/f'(a) (x − a)
Stationary points occur where f'(x) = 0; classify using second derivative or sign test
驻点满足 f'(x) = 0；用二阶导数或符号变化判断极值类型
Second derivative: f''(x) = d²y/dx²; if f''(a) > 0 then minimum, if f''(a) < 0 then maximum
二阶导数 f''(x) = d²y/dx²；f''(a) > 0 为极小值，f''(a) < 0 为极大值
9. Integration | 积分
Integration as reverse of differentiation: ∫ f'(x) dx = f(x) + C
积分是微分的逆运算：∫ f'(x) dx = f(x) + C
Power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1
幂法则积分：∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C，n ≠ −1
Integral of a sum: ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
和函数积分：∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x)
定积分：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F'(x) = f(x)
Area under curve y = f(x) between x = a and x = b (above x‑axis) equals definite integral
曲线 y = f(x) 与 x 轴之间（f(x) ≥ 0）从 a 到 b 的面积等于定积分
Area between two curves: ∫ₐᵇ [top curve − bottom curve] dx
两条曲线间的面积：∫ₐᵇ [上方曲线 − 下方曲线] dx
10. Matrices & Transformations | 矩阵与变换
Matrix addition: add corresponding entries; matrices must have the same order
矩阵加法：对应元素相加，矩阵必须同阶
Scalar multiplication: multiply every entry by the scalar
数乘矩阵：将标量与每个元素相乘
tion AB defined when columns of A = rows of B; entry (i,j) is row i of A dot column j of B
矩阵乘法 AB 要求 A 的列数 = B 的行数；(i, j) 元素是 A 的第 i 行与 B 的第 j 列的点积
Identity matrix I: 2×2 identity =
| 1 | 0 |
| 0 | 1 |
单位矩阵 I：2×2 单位阵 =
| 1 | 0 |
| 0 | 1 |
Determinant of 2×2 matrix
| a | b |
| c | d |
2×2 矩阵
| a | b |
| c | d |
Inverse of 2×2 matrix: M⁻¹ = 1/(ad − bc)
| d | −b |
| −c |
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