📚 A-Level CCEA Maths: Mind Map Quick Revision | A-Level CCEA 数学：思维导图速记
This article presents a mind-map style quick revision guide covering the core topics of the CCEA A-Level Mathematics syllabus. By visually organising key concepts, formulas and problem-solving techniques, students can reinforce their understanding and recall efficiently.
本文提供一份思维导图风格的速记指南，涵盖 CCEA A-Level 数学大纲的核心主题。通过视觉化梳理关键概念、公式和解题技巧，帮助学生强化理解、高效记忆。
1. Algebra Basics | 代数基础
Recall index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a1/n = ⁿ√a. Rationalising surds often involves multiplying the numerator and denominator by the conjugate.
牢记指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，并且 a1/n = ⁿ√a。分母有理化通常需要分子分母同乘共轭因式。
x−a) is a factor. The Remainder Theorem says the remainder when dividing by (x−a) is f(a).
因式定理：对于多项式 f(x)，若 f(a)=0，则 (x−a) 是一个因式。余数定理：除以 (x−a) 的余数等于 f(a)。
When simplifying rational expressions, cancel common factors only after factorising fully. Always check for restrictions on the variable to avoid division by zero.
化简有理式时，必须先完成因式分解再约去公因式。务必注明变量限制条件，避免分母为零。
2. Functions and Graphs | 函数与图像
ne-to-one; use the horizontal line test. The graph of f⁻¹ is a reflection in the line y=x.
函数将每一个输入映射到唯一输出。定义域是 x 的可能取值集合，值域是产生的 y 值集合。反函数 f⁻¹ 存在的前提是原函数一一对应；可用水平线检验。f⁻¹ 的图像关于直线 y=x 对称。
etches horizontally. Even functions satisfy f(−x)=f(x); odd functions satisfy f(−x)=−f(x).
复合函数如 fg(x) 表示先作用 g 再作用 f。图像变换：y = f(x)+a 纵向平移；y = f(x+a) 横向平移；y = af(x) 纵向伸缩；y = f(ax) 横向伸缩。偶函数满足 f(−x)=f(x)；奇函数满足 f(−x)=−f(x)。
3. Quadratics and Inequalities | 二次方程与不等式
ted root; Δ<0 → no real roots. The sum and product of roots are −b/a and c/a respectively.
对于 ax²+bx+c=0，判别式 Δ = b²−4ac 决定根的性质：Δ>0 → 两个不等实根；Δ=0 → 一个重根；Δ<0 → 无实根。根的和为 −b/a，积为 c/a。
Completing the square rewrites the quadratic as a(x−h)²+k, revealing the vertex (h,k). For quadratic inequalities, sketch the parabola to identify intervals where f(x)>0 or f(x)<0. Solve simultaneous linear and quadratic equations by substitution.
配方法将二次式写成 a(x−h)²+k 的形式，顶点为 (h,k)。解二次不等式时，画出抛物线示意图，找出 f(x)>0 或 f(x)<0 的区间。解线性与二次联立方程使用代入法。
4. Exponentials and Logarithms | 指数与对数
The natural base exponential y=eˣ and natural logarithm y=ln x are inverse functions: ln(eˣ)=x for all x, and eln x=x for x>0. Key log laws: ln(ab)=ln a + ln b, ln(a/b)=ln a − ln b, ln(aⁿ)=n ln a.
自然指数 y=eˣ 与自然对数 y=ln x 互为反函数：对所有 x 有 ln(eˣ)=x，对 x>0 有 eln x=x。关键对数法则：ln(ab)=ln a + ln b，ln(a/b)=ln a − ln b，ln(aⁿ)=n ln a。
To solve exponential equations, take natural logs: e²ˣ=5 ⇒ 2x=ln 5. For equations like aˣ arithmic equations usually need to be simplified to a single log first, then exponentiate.
解指数方程时，两边取自然对数：e²ˣ=5 ⇒ 2x=ln 5。对于 aˣ=b 这类方程，用对数把指数降下来。对数方程通常先化简为单一对数，再两边取指数。
5. Trigonometry | 三角学
θ/cosθ. Double-angle formulas: sin2θ=2sinθcosθ; cos2θ = cos²θ−sin²θ = 2cos²θ−1 = 1−2sin²θ.
弧度制下，π rad = 180°。熟记 0, π/6, π/4, π/3, π/2 处正弦、余弦和正切的精确值。基本恒等式 sin²θ + cos²θ = 1，以及 tanθ = sinθ/cosθ。倍角公式：sin2θ=2sinθcosθ；cos2θ = cos²θ−sin²θ = 2cos²θ−1 = 1−2sin²θ。
θ+π/6) require adjusting the interval accordingly. The R-formula a cosθ ± b sinθ = R cos(θ ∓ α) or R sin(θ ± α) helps find maxima/minima and solve equations.
解三角方程时使用 CAST 图或单位圆。考虑给定区间内所有解。涉及 sin2θ 或 sin(θ+π/6) 等倍角或平移的方程，需相应调整区间。辅助角公式 a cosθ ± b sinθ = R cos(θ ∓ α) 或 R sin(θ ± α) 可用于求最值和解方程。
6. Differentiation | 微分
The derivative from first principles: f' ]/h. Power rule: d/dx (xⁿ) = n xⁿ⁻¹. Chain rule: dy/dx = dy/du × du/dx. Product rule: (uv)' = u'v + uv'. Quotient rule: (u/v)' = (u'v − uv') / v².
导数定义：f'(x) = limₕ→₀ [f(x+h)−f(x)]/h。幂函数求导：d/dx (xⁿ) = n xⁿ⁻¹。链式法则：dy/dx = dy/du × du/dx。乘积法则：(uv)' = u'v + uv'。商法则：(u/v)' = (u'v − uv') / v²。
Standard derivatives: d/dx (eˣ) = eˣ e the second derivative d²y/dx² to classify them (positive → minimum, negative → maximum).
标准导数公式：d/dx (eˣ) = eˣ；d/dx (ln x) = 1/x；d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (tan x) = sec²x。参数方程 x=f(t), y=g(t) 求导：dy/dx = (dy/dt) ÷ (dx/dt)。驻点出现在 dy/dx = 0 处；用二阶导数 d²y/dx² 判断极值（正为极小值，负为极大值）。
7. Integration | 积分
∫ₐᵇ f(x) dx; if the curve goes below the axis, take absolute value or split the interval.
积分是微分的逆运算。不定积分 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，其中 n ≠ −1。定积分计算 [F(x)]ₐᵇ = F(b)−F(a)。曲线与 x 轴之间的面积 = ∫ₐᵇ f(x) dx；若曲线在 x 轴下方，需取绝对值或分段处理。
Use integration by substitution when an integrand resembles f'(g(x)) g' ) and replace dx. Integration by parts: ∫ u dv = uv − ∫ v du; typically choose u following 'LIATE' logarithmic, inverse trig, algebraic, trigonometric, exponential). Standard integrals: ∫ eˣ dx = eˣ + C; ∫ 1/x dx = ln|x| + C; ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C.
当被积函数形如 f'(g(x))g'(x) 时，用换元积分法。令 u = g(x) 并替换 dx。分部积分法：∫ u dv = uv − ∫ v du；通常按“LIATE”顺序选择 u（对数、反三角、代数、三角、指数）。标准积分：∫ eˣ dx = eˣ + C；∫ 1/x dx = ln|x| + C；∫ sin x dx = −cos x + C；∫ cos x dx = sin x + C。
8. Vectors | 向量
A vector has magnitude and direction. In 2D, a = a₁i + a₂j; in 3D, add a₃k. The magnitude |a| = √a₁² + a₂² + a₃². A unit vector is â = a/|a|. The scalar (dot) product a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃. If a·b = 0, the vectors are perpendicular.
向量有大小和方向。二维记为 a = a₁i + a₂j；三维加上 a₃k。模长 |a| = √(a₁
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