A-Level Maths: Mind Map for Rapid Memorisation | A-Level 数学：思维导图速记

📚 A-Level Maths: Mind Map for Rapid Memorisation | A-Level 数学：思维导图速记

This article distils the entire A-Level Mathematics syllabus into a structured mind-map framework. Each section highlights the essential formulae, concepts and typical problem-solving techniques, enabling you to memorise and recall key content rapidly. Use it as a revision checklist or a daily warm‑up to reinforce your understanding before the exam.

本文将整个 A‑Level 数学大纲浓缩为一幅结构化的思维导图框架。每一节都突显了核心公式、概念和典型解题技巧，帮助你快速记忆和调用关键内容。可将其作为复习清单，或在考前每日热身，以巩固理解。

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1. Algebra & Equations | 代数与方程

ate surds and indices using the laws aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, and a¹⁄ⁿ = ⁿ√a. For surds, √ab = √a √b and rationalise denominators by multiplying the conjugate.

运用 aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，a¹⁄ⁿ = ⁿ√a 等指数法则化简根式和幂。对于根式，√ab = √a √b，通过乘以共轭式使分母有理化。

A quadratic equation ax² + bx + c = 0 (a ≠ 0) is solved by the formula:

x = (–b ± √b² – 4ac) / (2a)

The discriminant Δ = b² – ts). Completing the square rewrites the quadratic as a(x + p)² + q, revealing the vertex (–p, q).

判别式 Δ = b² – 4ac 揭示了根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根（两个复根）。配方可将二次式写成 a(x + p)² + q 的形式，显示出顶点 (–p, q)。

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2. Functions & Transformations | 函数与变换

unction fg(x) = f(g(x)) applies g first, then f. Inverse function f⁻¹(x) satisfies f(f⁻¹(x)) = x = f⁻¹(f(x)), and its graph is a reflection of y = f(x) in the line y = x.

函数 f 将每个输入 x 唯一映射到一个输出 f(x)。定义域是允许的输入集合，值域是可能的输出集合。复合函数 fg(x) = f(g(x)) 先应用 g，再应用 f。反函数 f⁻¹(x) 满足 f(f⁻¹(x)) = x = f⁻¹(f(x))，其图像是 y = f(x) 关于直线 y = x 的反射。

transformations: y = f(x) + a shifts vertically by a; y = f(x + a) shifts horizontally by – s vertically by factor a; y = f(ax) stretches horizontally by factor 1/a. Reflections: y = –f(x) reflects in the x‑axis, y = f(–x) reflects in the y‑axis.

图像变换：y = f(x) + a 纵向平移 a；y = f(x + a) 横向平移 –a；y = a f(x) 纵向拉伸因子 a；y = f(ax) 横向拉伸因子 1/a。反射：y = –f(x) 关于 x 轴对称，y = f(–x) 关于 y 轴对称。

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3. Coordinate Geometry | 坐标几何

In the straight‑line graph y = mx + c, m = (y₂ – y₁)/(x₂ – x₁) is the gradient, c is the y‑intercept. The distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ – x₁)² + (y₂ – y₁)²]. The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). For perpendicular lines, m₁ m₂ = –1.

在直线方程 y = mx + c 中，m = (y₂ – y₁)/(x₂ – x₁) 为斜率，c 是 y 轴截距。点 (x₁, y₁) 与 (x₂, y₂) 之间的距离为 √[(x₂ – x₁)² + (y₂ – y₁)²]。中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。两直线垂直时，m₁ m₂ = –1。

The circle equation (x – a)² + (y – = r² has centre (a, b) and radius r. General form x² + y² + 2gx + 2fy + c = 0 has centre (–g, –f) and radius √g² + f² – c the point of contact; use discriminant Δ = 0 to find tangent lines from an external point.

圆方程 (x – a)² + (y – b)² = r² 圆心为 (a, b)，半径为 r。一般式 x² + y² + 2gx + 2fy + c = 0 的圆心为 (–g, –f)，半径为 √g² + f² – c。圆的切线与过切点的半径垂直；可利用判别式 Δ = 0 求圆外一点的切线方程。

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4. Trigonometry | 三角学

. Exact values for 0°, 30°, 45°, 60°, 90° must be memorised. Fundamental identities: tan θ ≡ sin θ / cos θ, sin²θ + cos²θ ≡ 1, 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ cosec²θ. Sine and cosine rules for non‑right‑angled triangles: a / sin A = b / sin B = c / sin C (Sine Rule); a² = b² + c² – 2bc cos A (Cosine Rule). Area = ½ ab sin C.

在单位圆上定义 sin θ，cos θ，tan θ。必须熟记 0°, 30°, 45°, 60°, 90° 的精确值。基本恒等式：tan θ ≡ sin θ / cos θ，sin²θ + cos²θ ≡ 1，1 + tan²θ ≡ sec²θ，1 + cot²θ ≡ cosec²θ。任意三角形的正弦与余弦定理：a / sin A = b / sin B = c / sin C（正弦定理）；a² = b² + c² – 2bc cos A（余弦定理）。面积 = ½ ab sin C。

or use CAST diagram to find all solutions in the required interval. Remember that sin(180° – θ) = sin θ, cos(360° – , tan(180° + θ) = tan θ, and use appropriate periodic transformations for radians (π, 2π).

求解三角方程时，可画简图或使用 CAST 图解法找出指定区间内的所有解。记住 sin(π – θ) = sin θ，cos(2π – θ) = cos θ，tan(π + θ) = tan θ，并根据弧度制周期（π, 2π）进行适当变换。

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5. Sequences & Series | 数列与级数

An arithmetic sequence has a common difference d: uₙ = a + (n – 1)d. Sum of first n terms Sₙ = n/2 [2a + (n – last term. A geometric sequence has a common ratio r: uₙ = arⁿ⁻¹. Sum to n terms Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. For an infinite geometric series with |r| < 1, S∞ = a / (1 – r).

等差数列公差为 d：uₙ = a + (n – 1)d。前 n 项和 Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l)，其中 l 为末项。等比数列公比为 r：uₙ = arⁿ⁻¹。当 r ≠ 1 时，前 n 项和 Sₙ = a(1 – rⁿ)/(1 – r)。若 |r| < 1，无穷等比级数收敛，S∞ = a / (1 – r)。

(from r=1 to n) uᵣ represents the sum of terms. Use standard formulae Σr = ½ n(n+1), Σr² = ⅙ n(n+1)(2n+1), Σr³ = ¼ n²(n+1)² to handle polynomial summations.

求和符号 Σ (r=1 到 n) uᵣ 表示各项之和。利用标准求和公式 Σr = ½ n(n+1)，Σr² = ⅙ n(n+1)(2n+1)，Σr³ = ¼ n²(n+1)² 可处理多项式求和问题。

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6. Exponentials & Logarithms | 指数与对数

The function y = eˣ has derivative eˣ and inverse y = ln x, defined for x > 0. Key laws: ln(ab) = ln a + ln b, ln(a/b) = ln a – x) / (ln a). Exponential growth and decay models are of the form A = A₀ eᵏᵗ; for a > 0, aˣ = eˣˡⁿⁱ ᵃ.

函数 y = eˣ 的导数为其本身，反函数为 y = ln x（定义域 x > 0）。关键法则：ln(ab) = ln a + ln b，ln(a/b) = ln a – ln b，ln(aᵖ) = p ln a。换底公式：logₐ x = (ln x) / (ln a)。指数增长和衰减模型形如 A = A₀ eᵏᵗ；对于任意 a > 0，aˣ = eˣ ˡⁿ ᵃ。

When solving equations, take logs of both sides: e.g. 2ˣ x ln 2 = ln 5 → x = ln 5 / ln 2. Be careful with extraneous solutions when dropping logs.

求解方程时，两端取对数：如 2ˣ = 5 ⇒ x ln 2 = ln 5 ⇒ x = ln 5 / ln 2。去对数时要注意排除增根。

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7. Differentiation | 微分

The derivative f'(x) = limₕ→₀ [f(x+h) – f(x)] / h gives the gradient of y = f(x). Basic rules: d/dx (xⁿ) = n xⁿ⁻¹, 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) = sec² x. Product rule: d/dx (uv) = u'v + uv'. Quotient rule: d/dx (u/v) = (vu' – uv') / v². Chain rule: dy/dx = dy/du × du/dx.

导数 f'(x) = limₕ→₀ [f(x+h) – f(x)] / h 给出曲线 y = f(x) 的斜率。基本公式：d/dx (xⁿ) = n xⁿ⁻¹，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) = sec² x。乘法法则：(uv)' = u'v + uv'。除法法则：(u/v)' = (vu' – uv') / v²。链式法则：dy/dx = dy/du × du/dx。

Second derivative f''(x) tells you about concavity. At a stationary point (f'(x) = 0), if f''(x) > 0 it is a local minimum; if f''(x) < 0 it is a local maximum; if f''(x) = 0, use the first derivative test to check nature.

二阶导数 f''(x) 表示凹凸性。在驻点（f'(x) = 0）处，若 f''(x) > 0 则为局部极小值；若 f''(x) < 0 则为局部极大值；若 f''(x) = 0，需用一阶导数判定法判断。

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8. Integration | 积分

Integration reverses differentiation: ∫ f(x) dx = F(x) + c where F'(x) = f(x). Indefinite integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ –1), ∫ 1/x dx = ln|x| + c, ∫ eˣ dx = eˣ + c, ∫ sin x dx = – sec² x dx = tan x + c. Integration by substitution and by parts are essential: ∫ u dv = uv – ∫ v du.

积分是微分的逆运算：∫ f(x) dx = F(x) + c，其中 F'(x) = f(x)。不定积分公式：∫ xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ –1)，∫ 1/x dx = ln|x| + c，∫ eˣ dx = eˣ + c，∫ sin x dx = –cos x + c，∫ cos x dx = sin x + c，∫ sec² x dx = tan x + c。换元积分法和分部积分法是关键：∫ u dv = uv – ∫ v du。

the upper. Trapezium rule approximates ∫ₐᵇ f(x) dx ≈ ½ h [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ] where h = (b – a)/n.

定积分 ∫ₐᵇ f(x) dx 给出了从 x = a 到 x = b 曲线下的有符号面积。求两曲线间面积时，用上方函数减去下方函数。梯形法近似估算：∫ₐᵇ f(x) dx ≈ ½ h [y₀ + 2(y₁ + y₂ + ... + yₙ₋₁) + yₙ]，其中 h = (b – a)/n。

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9. Vectors | 向量

A vector has magnitude and direction. In two dimensions, v = a i + b j has magnitude |v| = √a² + b². Position vector OA gives the coordinates of A. The vector AB = b – a. The unit vector in the direction of v is v / |v|. In 3D, add the component k.

向量有大小和方向。在二维中，v = a i + b j，其模长为 |v| = √a² + b²。位置向量 OA 对应点 A 的坐标。向量 AB = b – a。v 方向上的单位向量为 v / |v|。在三维空间中，增加分量 k。

Dot product: a • b = |a||b| (+ a₃b₃ in 3D). It is used to find the angle between vectors and to test orthogonality (a • find intersection or shortest distance, use dot product with the perpendicular direction.

点积（数量积）：a • b = |a||b| cos θ = a₁b₁ + a₂b₂（三维加 a₃b₃）。可用于求向量夹角，检验垂直（a • b = 0）。直线的向量方程为 r = a + t b；求交点或最短距离时，可与垂直方向求点积。

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10. Statistics & Mechanics | 统计与力学

In statistics, probability P(A or B) = P(A) + P(B) – P(A and B); mutually exclusive ⇒ P(A∩B) = 0; independent ⇒ P(A∩B) = P(A)P(B). Conditional probability P(A|B) = P(A∩B) / P(B). Discrete random variables: E(X) = Σ x p(x), Var(X) = E(X²) – [E(X)]². Binomial distribution X ~ B(n, p): P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ, E(X) = np, Var(X) = np(1 – p). Normal distribution X ~ N(μ, σ²): standardise z = (x – μ) / σ and use tables.

统计部分：概率 P(A 或 B) = P(A) + P(B) – P(A 且 B)；互斥时 P(A∩B)=0；独立时 P(A∩B) = P(A)P(B)。条件概率 P(A|B) = P(A∩B) / P(B)。离散随机变量：期望 E(X) = Σ x p(x)，方差 Var(X) = E(X²) – [E(X)]²。二项分布 X ~ B(n, p)：P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ，E(X) = np，Var(X) = np(1 – p)。正态分布 X ~ N(μ, σ²)：标准化 z = (x – μ) / σ，查表求解。

onstant acceleration a: v = u + at, s = ut + ½ at², s = ½ (u + v) t, v² = u² + 2as, s = vt – nal potential energy = mgh. Momentum = mv; conservation of momentum applies in collisions.

力学部分利用匀加速运动的 suvat 公式：v = u + at，s = ut + ½ at²，s = ½ (u + v) t，v² = u² + 2as，s = vt – ½ at²。牛顿第二定律 F = ma 给出合力。做功 = Fd cos θ，动能 = ½ mv²，重力势能 = mgh。动量 = mv；碰撞中动量守恒。

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