📚 IGCSE AQA Further Maths: Quick Reference Formula & Theorem Handbook | IGCSE AQA 进阶数学：公式定理速查手册
This concise handbook gathers the essential formulas and theorems required for the IGCSE AQA Further Mathematics specification. Use it for rapid revision, familiarising yourself with notation, and reinforcing key results before the examination.
本速查手册汇集了 IGCSE AQA 进阶数学大纲所要求的核心公式与定理。可用于快速复习、熟悉符号，并在考前巩固重点结论。
1. Algebra: Key Formulas & Theorems | 代数：核心公式与定理
The factor theorem states that for a polynomial f(x), if f(a) = 0, then (x − a) is a factor. Conversely, if (x − a) is a factor, then f(a) = 0.
因式定理：对于多项式 f(x)，若 f(a) = 0，则 (x − a) 为一个因式。反之，若 (x − a) 是因式，则 f(a) = 0。
The remainder theorem: when f(x) is divided by (x − a), the remainder is f(a).
余式定理：f(x) 除以 (x − a) 时，余数等于 f(a)。
The binomial expansion for positive integer n is (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + …; for (a + b)ⁿ it is Σ [nCr aⁿ⁻ʳ bʳ] from r = 0 to n.
正整数指数二项展开：(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + …；(a + b)ⁿ = Σ [nCr aⁿ⁻ʳ bʳ]，r 从 0 到 n。
Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
二次方程求根公式：x = [−b ± √(b² − 4ac)] / 2a
Discriminant Δ = b² − 4ac. If Δ > 0, two distinct real roots; Δ = 0, one repeated real root; Δ < 0, no real roots.
判别式 Δ = b² − 4ac。Δ > 0 有两个不等实根；Δ = 0 有一个重实根；Δ < 0 无实根。
Completing the square: x² + bx = (x + b/2)² − (b/2)².
配方法：x² + bx = (x + b/2)² − (b/2)²。
2. Functions | 函数
A function f maps each input x from the domain to exactly one output f(x) in the range. The notation f : x → f(x) is often used.
函数 f 将定义域中的每个输入 x 映射到值域中唯一的输出 f(x)。常用记号 f : x → f(x)。
Composite function: (f ∘ g)(x) = f(g(x)), applying g first, then f.
复合函数：(f ∘ g)(x) = f(g(x))，先应用 g，再应用 f。
Inverse function f⁻¹ exists only if f is one‑to‑one. It satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. The graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x.
反函数 f⁻¹ 仅当 f 是一一映射时才存在，满足 f(f⁻¹(x)) = x 且 f⁻¹(f(x)) = x。y = f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的反射。
Domain and range: the set of all allowable inputs is the domain; the set of all possible outputs is the range. For example, f(x) = √x has domain x ≥ 0 and range f(x) ≥ 0.
定义域是所有允许输入的集合；值域是所有可能输出的集合。例如 f(x) = √x 的定义域为 x ≥ 0，值域为 f(x) ≥ 0。
3. Coordinate Geometry | 坐标几何
The gradient of the line joining (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁) / (x₂ − x₁).
连接两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m = (y₂ − y₁) / (x₂ − x₁)。
Equation of a straight line: y − y₁ = m(x − x₁) or y = mx + c, where c is the y‑intercept.
直线方程：y − y₁ = m(x − x₁) 或 y = mx + c，其中 c 为 y 轴截距。
Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1 (provided neither is vertical).
平行线斜率相等；垂直线满足 m₁ × m₂ = −1（假设均非竖直）。
Midpoint of two points: ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点中点：((x₁ + x₂)/2, (y₁ + y₂)/2)。
Distance between two points: √[(x₂ − x₁)² + (y₂ − y₁)²].
两点间距离：√[(x₂ − x₁)² + (y₂ − y₁)²]。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).
圆心 (a, b)、半径 r 的圆方程为 (x − a)² + (y − b)² = r²。一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (−g, −f)，半径为 √(g² + f² − c)。
4. Trigonometry | 三角学
Radians: π rad = 180°. To convert degrees to radians, multiply by π/180. Arc length s = rθ, sector area A = ½ r²θ (θ in radians).
弧度制：π rad = 180°。度数转弧度乘以 π/180。弧长 s = rθ，扇形面积 A = ½ r²θ（θ 为弧度）。
Basic identities: tan θ = sin θ / cos θ, and sin²θ + cos²θ ≡ 1.
基本恒等式：tan θ = sin θ / cos θ，以及 sin²θ + cos²θ ≡ 1。
Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A, or cos A = (b² + c² − a²) / 2bc.
正弦定理：a/sin A = b/sin B = c/sin C。余弦定理：a² = b² + c² − 2bc cos A，或 cos A = (b² + c² − a²) / 2bc。
Area of a triangle: ½ ab sin C.
三角形面积：½ ab sin C。
Exact trigonometric values (angles in degrees): 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.
特殊角三角函数值（角度制）：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。
5. Sequences and Series | 数列与级数
Arithmetic sequence: nth term uₙ = a + (n − 1)d, where a is the first term and d is the common difference.
等差数列：第 n 项 uₙ = a + (n − 1)d，其中 a 为首项，d 为公差。
Sum of the first n terms of an arithmetic series: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
等差数列前 n 项和：Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)，l 为末项。
Geometric sequence: nth term uₙ = arⁿ⁻¹, where a is the first term and r is the common ratio.
等比数列：第 n 项 uₙ = arⁿ⁻¹，a 为首项，r 为公比。
Sum of the first n terms of a geometric series: Sₙ = a(1 − rⁿ) / (1 − r) for r ≠ 1.
等比数列前 n 项和：Sₙ = a(1 − rⁿ) / (1 − r)，r ≠ 1。
Sum to infinity of a convergent geometric series (|r| < 1): S∞ = a / (1 − r).
无穷递缩等比数列的和（|r| < 1）：S∞ = a / (1 − r)。
Sigma notation: Σ from k=1 to n of uₖ represents the sum u₁ + u₂ + … + uₙ.
求和符号：Σₖ₌₁ⁿ uₖ 表示 u₁ + u₂ + … + uₙ。
6. Exponentials and Logarithms | 指数与对数
Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(m/n) = ⁿ√(aᵐ).
指数律：aᵐ × aⁿ = aᵐ⁺ⁿ；aᵐ / aⁿ = aᵐ⁻ⁿ；(aᵐ)ⁿ = aᵐⁿ；a⁰ = 1；a⁻ⁿ = 1/aⁿ；a^(m/n) = ⁿ√(aᵐ)。
Logarithms: if aˣ = b, then x = logₐ b. Key properties: logₐ (xy) = logₐ x + logₐ y; logₐ (x/y) = logₐ x − logₐ y; logₐ (xⁿ) = n logₐ x; logₐ 1 = 0; logₐ a = 1.
对数：若 aˣ = b，则 x = logₐ b。主要性质：logₐ (xy) = logₐ x + logₐ y；logₐ (x/y) = logₐ x − logₐ y；logₐ (xⁿ) = n logₐ x；logₐ 1 = 0；logₐ a = 1。
Change of base: logₐ b = logₓ b / logₓ a.
换底公式：logₐ b = logₓ b / logₓ a。
The natural logarithm ln x = logₑ x, where e ≈ 2.718. ln e = 1, ln 1 = 0. The function eˣ is its own derivative.
自然对数 ln x = logₑ x，e ≈ 2.718。ln e = 1，ln 1 = 0。函数 eˣ 的导数为其本身。
7. Differentiation | 微分
The derivative f'(x) gives the gradient of the curve y = f(x). For f(x) = xⁿ, f'(x) = nxⁿ⁻¹ (n rational).
导数 f'(x) 给出曲线 y = f(x) 的斜率。对于 f(x) = xⁿ，f'(x) = nxⁿ⁻¹（n 为有理数）。
Constant multiple rule: d/dx [c f(x)] = c f'(x). Sum rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
常数倍法则：d/dx [c f(x)] = c f'(x)。加减法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。
Chain rule: d/dx [f(g(x))] = f'(g(x)) g'(x). Product rule: d/dx [u v] = u’ v + u v’. Quotient rule: d/dx [u/v] = (u’ v − u v’) / v².
链式法则：d/dx [f(g(x))] = f'(g(x)) g'(x)。乘法法则：d/dx [u v] = u’ v + u v’。除法法则：d/dx [u/v] = (u’ v − u v’) / v²。
Second derivative f”(x) is the derivative of f'(x). It describes the rate of change of gradient and helps determine concavity.
二阶导数 f”(x) 是 f'(x) 的导数，描述斜率变化率并用于判断凹凸性。
Stationary points occur where f'(x) = 0. Use the first derivative test (sign change) or second derivative test: if f”(x) > 0, local minimum; if f”(x) < 0, local maximum; if f''(x) = 0, test inconclusive.
驻点出现在 f'(x) = 0 处。用一阶导数符号变化检验，或二阶导数检验：f”(x) > 0 为局部极小；f”(x) < 0 为局部极大；f''(x) = 0 则无法判定。
8. Integration | 积分
Indefinite integration reverses differentiation. ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c, for n ≠ −1.
不定积分是微分的逆运算。∫ xⁿ dx = xⁿ⁺¹/(n+1) + c，n ≠ −1。
Basic rules: ∫ k f(x) dx = k ∫ f(x) dx; ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.
基本法则：∫ k f(x) dx = k ∫ f(x) dx；∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx。
Definite integral from a to b: ∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a), where F'(x) = f(x).
定积分从 a 到 b：∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)，其中 F'(x) = f(x)。
Area between a curve y = f(x) and the x‑axis from x = a to x = b is given by ∫ₐᵇ |f(x)| dx, partitioning where the curve crosses the axis.
曲线 y = f(x) 与 x 轴在 x=a 到 x=b 之间的面积由 ∫ₐᵇ |f(x)| dx 给出，需对曲线穿越 x 轴的部分分段处理。
Area between two curves y = f(x) and y = g(x) from a to b is ∫ₐᵇ (top − bottom) dx = ∫ₐᵇ [f(x) − g(x)] dx, where f(x) ≥ g(x) on [a, b].
两曲线 y = f(x) 与 y = g(x) 在 a 到 b 间的面积：∫ₐᵇ (上 − 下) dx = ∫ₐᵇ [f(x) − g(x)] dx，其中在 [a, b] 上 f(x) ≥ g(x)。
9. Matrices | 矩阵
A matrix is a rectangular array of numbers. Order is rows × columns. Addition and subtraction are element‑wise for matrices of the same order.
矩阵为矩形数字阵列。阶数为行 × 列。同阶矩阵可逐元素相加或相减。
Scalar multiplication: each entry is multiplied by the scalar.
数乘：每个元素乘以该标量。
Matrix multiplication: If A is m×n and B is n×p, then AB is m×p with (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ. Multiplication is not commutative in general.
矩阵乘法：若 A 为 m×n，B 为 n×p，则 AB 为 m×p 矩阵，(AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ。通常不满足交换律。
Identity matrix I of order n has 1s on the main diagonal and 0s elsewhere, satisfying AI = IA = A.
n 阶单位矩阵 I 的主对角线元素为 1，其余为 0，满足 AI = IA = A。
Determinant of a 2×2 matrix A = [[a, b], [c, d]] is det(A) = ad − bc. If det(A) ≠ 0, the matrix is non‑singular and has an inverse.
2×2 矩阵 A = [[a, b], [c, d]] 的行列式 det(A) = ad − bc。若 det(A) ≠ 0，矩阵非奇异，存在逆矩阵。
Inverse of a 2×2 matrix: A⁻¹ = 1/(ad−bc) [[d, −b], [−c, a]]. It satisfies AA⁻¹ = A⁻¹A = I.
2×2 矩阵的逆：A⁻¹ = 1/(ad−bc) [[d, −b], [−c, a]]，满足 AA⁻¹ = A⁻¹A = I。
Solving linear equations AX = B using inverse: X = A⁻¹B, provided A is square and non‑singular.
利用逆矩阵解线性方程组 AX = B：X = A⁻¹B，要求 A 为方阵且可逆。
10. Vectors | 向量
A vector has magnitude and direction. It can be represented as a column vector [x, y] or using unit vectors i, j: v = xi + yj.
向量具有大小和方向。可用列向量 [x, y] 或用单位向量 i, j 表示为 v = xi + yj。
Magnitude of vector v = xi + yj is |v| = √(x² + y²).
向量 v = xi + yj 的模为 |v| = √(x² + y²)。
Addition and subtraction: a + b = [a₁+b₁, a₂+b₂] and similarly for subtraction. Scalar multiplication: k a = [k a₁, k a₂].
加减法：a + b = [a₁+b₁, a₂+b₂]，减法类似。数乘：k a = [k a₁, k a₂]。
Position vector of point A relative to origin O is OA. The vector AB = OB − OA.
点 A 相对于原点 O 的位置向量为 OA。向量 AB = OB − OA。
Dot product (scalar product): for vectors a = [a₁, a₂] and b = [b₁, b₂], a · b = a₁b₁ + a₂b₂ = |a||b| cos θ, where θ is the angle between them.
点积（数量积）：对于 a = [a₁, a₂] 和 b = [b₁, b₂]，a · b = a₁b₁ + a₂b₂ = |a||b| cos θ，θ 为两向量夹角。
Two non‑zero vectors are perpendicular if and only if their dot product is zero.
两个非零向量垂直当且仅当它们点积为零。
Unit vector in the direction of a is â = a / |a|.
沿 a 方向的单位向量为 â = a / |a|。
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