Year 12 CAIE Mathematics: Formula & Theorem Quick Reference Handbook | Year 12 CAIE 数学：公式定理速查手册

📚 Year 12 CAIE Mathematics: Formula & Theorem Quick Reference Handbook | Year 12 CAIE 数学：公式定理速查手册

This concise reference compiles the essential formulae, theorems, and standard results required for the CAIE AS Level Mathematics course (Pure Mathematics 1 and supporting topics). It is designed for quick revision, covering quadratics, functions, coordinate geometry, circular measure, trigonometry, sequences, differentiation, integration, vectors, and key identities. Each entry pairs the English statement with its Chinese counterpart for bilingual clarity.

本速查手册汇集了 CAIE AS 数学课程（纯数 1 及相关内容）所需的所有核心公式、定理与标准结果，专为快速复习设计。涵盖二次式、函数、坐标几何、弧度制、三角学、数列、微分、积分、向量以及重要恒等式。每个条目采用中英对照形式，方便双语理解与记忆。

1. Quadratics and Inequalities | 二次方程与不等式

• For a quadratic equation ax² + bx + c = 0 (a ≠ 0), the roots are given by the quadratic formula: x = [ −b ± √b² − 4ac ] / (2a).
  对于二次方程 ax² + bx + c = 0 (a ≠ 0)，求根公式为：x = [ −b ± √b² − 4ac ] / (2a)。
• The discriminant Δ = b² es two distinct real roots, Δ = 0 gives one repeated real root, Δ < 0 gives no real roots.
  判别式 Δ = b² − 4ac 决定根的性质：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: ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a)).
  配方法：ax² + bx + c = a(x + b/(2a))² + (c − b²/(4a))。
• Sum and product of roots: if α and β are roots, then α + β = −b/a, αβ = c/a.
  根的和与积：若 α 与 β 为根，则 α + β = −b/a，αβ = c/a。
• Solving quadratic inequalities: sketch the graph of the quadratic function to determine intervals where f(x) > 0 or f(x) < 0. Remember to reverse the inequality sign when multiplying or dividing by a negative number.
  解二次不等式：画出二次函数草图以确定 f(x) > 0 或 f(x) < 0 的区间。注意当乘以或除以负数时，需反转不等号。

2. Functions and Transformations | 函数与变换

• A function f maps each element x in the domain to exactly one element f(x) in the range. The vertical line test checks if a graph represents a function.
  函数 f 将定义域中的每个元素 x 映射到值域中的唯一元素 f(x)。铅垂线检验可判断图形是否为函数。
• Composite function: (f ∘ g)(x) = f(g(x)). The domain of f ∘ g is the set of x in the domain of g for which g(x) is in the domain of f.
  复合函数：(f ∘ g)(x) = f(g(x))。其定义域为 g 的定义域中使得 g(x) 属于 f 的定义域的 x 的集合。
• Inverse function f⁻¹(x) exists only if f is one‑to‑one. Its graph is the reflection of y = f(x) in the line y = x.
  反函数 f⁻¹(x) 仅在一一映射时存在。其图像为 y = f(x) 关于直线 y = x 的对称图形。
• r a (a > 1 stretches, 0 < a < 1 compresses); y = f(ax) is a horizontal stretch by factor 1/a.
  图像变换：y = f(x) + a 为垂直平移 a；y = f(x + a) 为水平平移 −a；y = a f(x) 为垂直伸缩，因子 a（a > 1 拉伸，0 < a < 1 压缩）；y = f(ax) 为水平伸缩，因子 1/a。
• Modulus function |x|: |x| = x for x ≥ 0, and −x for x < 0. Equations like |ax + b| = c lead to two linear equations; inequalities require careful splitting into regions.
  绝对值函数 |x|：当 x ≥ 0 时 |x| = x，当 x < 0 时 |x| = −x。方程 |ax + b| = c 转化为两个线性方程；不等式需分区域讨论。

3. Coordinate Geometry | 坐标几何

• Distance between points (x₁, y₁) and (x₂, y₂): d = √[(x₂ − x₁)² + (y₂ − y₁)²].
  两点 (x₁, y₁) 与 (x₂, y₂) 间的距离：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 of a straight line: m = (y₂ − y₁) / (x₂ − x₁). Parallel lines have equal gradients; perpendicular lines satisfy m₁ m₂ = −1.
  直线斜率：m = (y₂ − y₁) / (x₂ − x₁)。平行线斜率相等；垂直线斜率乘积为 −1。
• Equation of a straight line: y − y₁ = m(x − x₁) (point‑gradient form); y = mx + c (slope‑intercept form); ax + by + c = 0 (general form).
  直线方程：y − y₁ = m(x − x₁)（点斜式）；y = mx + c（斜截式）；ax + by + c = 0（一般式）。
• Equation of a circle with centre (a, b) and radius r: (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.
  圆的方程，圆心 (a, b)，半径 r：(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。
• The angle between two lines: tan θ = |(m₁ − m₂) / (1 + m₁ m₂)|, provided m₁ m₂ ≠ −1.
  两直线夹角：tan θ = |(m₁ − m₂) / (1 + m₁ m₂)|，其中 m₁ m₂ ≠ −1。

4. Circular Measure | 弧度制

• Radians: 180° = π rad. Conversion: degrees × π/180 → radians; radians × 180/π → degrees.
  弧度与角度：180° = π 弧度。换算：角度 × π/180 → 弧度；弧度 × 180/π → 角度。
• Arc length: s = rθ, with θ in radians.
  弧长公式：s = rθ，其中 θ 以弧度为单位。
• Sector area: A = (1/2) r² θ (θ in radians).
  扇形面积：A = (1/2) r² θ（θ 为弧度）。
• Segment area: area of sector − area of triangle = (1/2) r² (θ − sin θ).
  弓形面积：扇形面积减去三角形面积 = (1/2) r² (θ − sin θ)。
• Length of a chord: chord = 2r sin(θ/2).
  弦长：弦长 = 2r sin(θ/2)。

5. Trigonometry | 三角学

• tios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
  基本三角比：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。
• Exact values for standard angles:
  标准角的精确值：

  θ	0°	30° (π/6)	45° (π/4)	60° (π/3)	90° (π/2)
  sin θ	0	1/2	1/√2 (or √2/2)	√3/2	1
  cos θ	1	√3/2	1/√2	1/2	0
  tan θ	0	1/√3	1	√3	undefined

• Trigonometric identities: sin² θ + cos² θ ≡ 1; tan θ ≡ sin θ / cos θ (cos θ ≠ 0).
  三角恒等式：sin² θ + cos² θ ≡ 1；tan θ ≡ sin θ / cos θ (cos θ ≠ 0)。
• π, period of tan is π. Amplitude for sin and cos is 1; tan has asymptotes at θ = π/2 + kπ.
  正弦、余弦、正切图像：sin 和 cos 的周期为 2π，tan 的周期为 π。sin 和 cos 的振幅为 1；tan 在 θ = π/2 + kπ 处有渐近线。
• Solving trigonometric equations: use CAST diagram or graph to find all solutions in a given interval, applying general solutions where necessary.
  解三角方程：使用 CAST 图或图像在给定区间内求所有解，必要时使用通解公式。
• General solutions: for sin θ = k, θ = nπ + (−1)ⁿ α; cos θ = k, θ = 2nπ ± α; tan θ = k, θ = nπ + α, where α is the principal value and n ∈ ℤ.
  通解：sin θ = k 时，θ = nπ + (−1)ⁿ α；cos θ = k 时，θ = 2nπ ± α；tan θ = k 时，θ = nπ + α，其中 α 为主值，n ∈ ℤ。

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6. Sequences and Series | 数列与级数

• Arithmetic progression (AP): nth term u_n = a + (n − 1)d. Sum of first n terms S_n = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term.
  等差数列（AP）：第 n 项 u_n = a + (n − 1)d。前 n 项和 S_n = n/2 [2a + (n − 1)d] = n/2 (a + l)，其中 l 为末项。
• Common difference d = u_n+1 − u_n.
  公差 d = u_n+1 − u_n。
• Geometric progression (GP): nth term u_n = arⁿ⁻¹. Sum of first n terms S_n = a(1 − rⁿ) / (1 − r) for r ≠ 1; if r = 1, S_n = na.
  等比数列（GP）：第 n 项 u_n = arⁿ⁻¹。前 n 项和：当 r ≠ 1 时，S_n = a(1 − rⁿ) / (1 − r)；若 r = 1，则 S_n = na。
• Common ratio r = u_n+1 / u_n.
  公比 r = u_n+1 / u_n。
• Sum to infinity of a convergent GP (|r| < 1): S_∞ = a / (1 − r).
  收敛等比级数（|r| < 1）的无穷和：S_∞ = a / (1 − r)。
• Sigma notation: ∑_k=1ⁿ u_k. Useful sums: ∑ k = n(n+1)/2, ∑ k² = n(n+1)(2n+1)/6, ∑ k³ = [n(n+1)/2]².
  西格玛符号：∑_k=1ⁿ u_k。常用求和公式：∑ k = n(n+1)/2，∑ k² = n(n+1)(2n+1)/6，∑ k³ = [n(n+1)/2]²。

7. Differentiation | 微分

• Power rule: if y = xⁿ, dy/dx = n xⁿ⁻¹ (for rational n).
  幂函数求导：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹（n 为有理数）。
• Constant multiple and sum rules: d/dx [c f(x)] = c f'(x); d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
  常数倍与和差法则：d/dx [c f(x)] = c f'(x)；d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。
• Gradient of a curve: the derivative f'(x) gives the gradient of the tangent at any point. The normal gradient is −1/f'(x).
  曲线的梯度：导数 f'(x) 给出任一点处切线的斜率。法线的斜率为 −1/f'(x)。
• Equation of tangent: y − y₁ = f'(x₁)(x − x₁). Equation of normal: y − y₁ = −1/f'(x₁) (x − x₁) (if f'(x₁) ≠ 0).
  切线方程：y − y₁ = f'(x₁)(x − x₁)。法线方程：y − y₁ = −1/f'(x₁) (x − x₁)（若 f'(x₁) ≠ 0）。
• Second derivative: d²y/dx² = f''(x). It determines concavity and helps classify stationary points.
  二阶导数：d²y/dx² = f''(x)。它决定凹凸性并用于判断驻点类型。
• Stationary points: solve f'(x) = 0. Nature test: if f''(x) > 0 then minimum; if f''(x) < 0 then maximum; if f''(x) = 0 use first derivative sign change test.
  驻点：解 f'(x) = 0。性质判定：若 f''(x) > 0 则为极小值点；若 f''(x) < 0 则为极大值点；若 f''(x) = 0 则用一阶导数符号变化检验。
• Increasing/decreasing functions: f'(x) ≥ 0 ⇒ increasing; f'(x) ≤ 0 ⇒ decreasing.
  函数的增减性：f'(x) ≥ 0 表示函数递增；f'(x) ≤ 0 表示函数递减。

8. Integration | 积分

• Integration as the reverse of differentiation: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, for n ≠ −1.
  积分是微分的逆运算：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c，其中 n ≠ −1。
• Constants and sums: ∫ k f(x) dx = k ∫ f(x) dx; ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx.
  常数与和规则：∫ k f(x) dx = k ∫ f(x) dx；∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx。
• Definite integral: ∫_a^b f(x) dx = F(b) − F(a), where F is an antiderivative of f.
  定积分：∫_a^b f(x) dx = F(b) − F(a)，其中 F 是 f 的一个原函数。
• Area under a curve: area = ∫_a^b y dx for regions above the x‑axis. Take absolute value of the integral or split the interval when the curve crosses the x‑axis.
  曲线下的面积：对于 x 轴上方的区域，面积 = ∫_a^b y dx。若曲线穿过 x 轴，需取积分的绝对值或分段计算。
• Area between two curves: area = ∫_a^b (y_top − y_bottom) dx.
  两曲线间的面积：面积 = ∫_a^b (y_上 − y_下) dx。
• Trapezium rule (approximate integration): ∫_a^b f(x) dx ≈ (h/2)[y₀ + y_n + 2(y₁ + y₂ + ... + y_n−1)], where h = (b − a)/n.
  梯形法则（近似积分）：∫_a^b f(x) dx ≈ (h/2)[y₀ + y_n + 2(y₁ + y₂ + ... + y_n−1)]，其中 h = (b − a)/n。
• Improper integrals with infinite limits are not required in P1, but you may be asked to interpret definite integrals in contexts.
  无穷限的反常积分不在 P1 范围内，但可能与定积分的实际意义结合考查。

9. Vectors in 2D | 二维向量

• A vector is represented as (x, y) or xi + yj, where i and j are unit vectors along the x‑ and y‑axes.
  向量表示为 (x, y) 或 xi + yj，其中 i 与 j 分别为沿 x 轴和 y 轴方向的单位向量。
• Magnitude of vector a = xi + yj: |a| = √x² + y².
  向量 a = xi + yj 的模：|a| = √x² + y²。
• Addition and subtraction: a + b = (x₁+x₂)i + (y₁+y₂)j; a − b = (x₁−x₂)i + (y₁−y₂)j.
  加减法：a + b = (x₁+x₂)i + (y₁+y₂)j；a − b = (x₁−x₂)i + (y₁−y₂)j。
• Scalar multiplication: k a = k(x i + y j) = (kx)i + (ky)j.
  数乘：k a = k(x i + y j) = (kx)i + (ky)j。
• Position vector of a point A: OA = a. Vector from A to B: AB = b − a.
  点 A 的位置向量：OA = a。从 A 到 B 的向量：AB = b − a。
• Unit vector in the direction of a: â = a / |a|.
  方向与 a 相同的单位向量：â = a / |a|。
• Equal vectors have the same magnitude and direction. Parallel vectors are scalar multiples of each other.
  相等向量具有相同的模和方向。平行向量互为数乘。
• Collinearity: points A, B, C are collinear if AB = k BC for some scalar k.
  共线性：若存在标量 k 使得 AB = k BC，则 A、B、C 三点共线。
• Dot product: a · b = x₁x₂ + y₁y₂ = |a||b| is the angle between a and b. The dot product is used to test perpendicularity: a · b = 0.
  点乘：a · b = x₁x₂ + y₁y₂ = |a||b| cos θ，θ 为两向量夹角。点乘用于判断垂直：a · b = 0。

10. Key Identities and Theorems | 重要恒等式与定理

• Binomial expansion for rational index n (where n is a positive integer): (1 + x)ⁿ = ∑_k=0ⁿ (ₙCₖ) x^k = 1 + nx + [n(n−1)/2!] x² + ... + xⁿ, valid for any x. The term independent of x can be found.
  正整数指数 n 的二项式展开：(1 + x)ⁿ = ∑_k=0ⁿ (ₙCₖ) x^k = 1 + nx + [n(n−1)/2!] x² + ... + xⁿ，对任意 x 成立。可求与 x 无关的项。
• Pythagorean identity: sin² θ + cos² θ ≡ 1.
  毕达哥拉斯恒等式：sin² θ + cos² θ ≡ 1。
• Sine rule: a / sin A = b / sin B = c / sin C = 2R (R = circumradius, not always required).
  正弦定理：a / sin A = b / sin B = c / sin C = 2R（R 为外接圆半径，有时不要求）。
• Cosine rule: a² = b² + c² − 2bc cos A.
  余弦定理：a² = b² + c² − 2bc cos A。
• Area of a triangle: Area = (1/2)ab sin C.
  三角形面积公式：面积 = (1/2)ab sin

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