A-Level Mathematics Core Formula Summary & Memory Techniques | A-Level 数学核心公式汇总与记忆技巧

📚 A-Level Mathematics Core Formula Summary & Memory Techniques | A-Level 数学核心公式汇总与记忆技巧

This guide consolidates the most essential formulas across Pure Mathematics, Trigonometry, Calculus, Vectors, Numerical Methods, and Probability & Statistics. Each formula is paired with a memory technique or exam-focused tip to help you retain it under pressure.

本指南汇总了纯数学、三角函数、微积分、向量、数值方法以及概率与统计中最核心的公式。每个公式均配有记忆技巧或备考重点提示，帮助你在考试压力下快速、准确地回忆与应用。

1. Algebraic Manipulation & Quadratic Equations | 代数变形与二次方程

The Quadratic Formula | 二次方程求根公式

x = (−b ± √(b² − 4ac)) / 2a

For ax² + bx + c = 0, the discriminant Δ = b² − 4ac determines the nature of roots: Δ > 0 gives two distinct real roots; Δ = 0 gives one repeated root; Δ < 0 gives no real roots.

对于 ax² + bx + c = 0，判别式 Δ = b² − 4ac 决定根的性質：Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 无实根。

Memory Tip | 记忆技巧: Sing it to the tune of “Pop Goes the Weasel”: “Minus b, plus or minus the square root, of b squared minus 4ac, all over 2a.”

记忆技巧：把求根公式唱成《Pop Goes the Weasel》的旋律：”Minus b, plus or minus the square root, of b squared minus 4ac, all over 2a.”

Laws of Indices | 指数运算法则

aᵐ × aⁿ = aᵐ⁺ⁿ — same base, add powers | 同底数相乘，指数相加
aᵐ ÷ aⁿ = aᵐ⁻ⁿ — same base, subtract powers | 同底数相除，指数相减
(aᵐ)ⁿ = aᵐⁿ — power of a power, multiply | 幂的幂，指数相乘
a⁻ⁿ = 1/aⁿ — negative power means reciprocal | 负指数表示倒数
a^(1/n) = ⁿ√a — fractional power means root | 分数指数表示开方

Memory Tip: When multiplying, connect the bases and “add” the exponents; when dividing, “subtract” them. Think of counting steps on a number line.

记忆技巧：相乘时底数不变、指数”相加”；相除时指数”相减”。想象在数轴上数步数即可。

2. Exponentials & Logarithms | 指数与对数

Definition | 定义

y = aˣ ⇔ logₐ y = x

The logarithm is the inverse operation of exponentiation. The base a must be positive and not equal to 1.

对数是指数运算的反函数。底数 a 必须为正且不等于 1。

Key Logarithm Rules | 对数核心法则

logₐ (xy) = logₐ x + logₐ y — “multiplication becomes addition” | 乘法变加法
logₐ (x/y) = logₐ x − logₐ y — “division becomes subtraction” | 除法变减法
logₐ (xᵏ) = k logₐ x — “power brings down” | 指数提前
logₐ a = 1, logₐ 1 = 0 | 底数的对数为1，1的对数为0

Change of Base | 换底公式

logₐ b = logₓ b / logₓ a

Memory Tip: The change-of-base formula reads like a fraction: the log of the top number over the log of the bottom number, both in the new base. “Top over bottom, same new base.”

记忆技巧：换底公式形如分数——新的底数下，上面对数除以底数对数。”上除下，底相同。”

3. Trigonometry Identities | 三角恒等式

Pythagorean Identity | 毕达哥拉斯恒等式

sin²θ + cos²θ = 1

Dividing by cos²θ gives tan²θ + 1 = sec²θ. Dividing by sin²θ gives 1 + cot²θ = csc²θ. | 两边同除以 cos²θ 得 tan²θ + 1 = sec²θ；同除以 sin²θ 得 1 + cot²θ = csc²θ。

Compound Angle Formulas | 和角公式

sin(A ± B) = sinA cosB ± cosA sinB

cos(A ± B) = cosA cosB ∓ sinA sinB

tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)

Memory Tip: For sin(A+B): “sine cosine, cosine sine — same sign.” For cos(A+B): “cosine cosine, sine sine — opposite sign.” The word “sine” starts with the same letter as “same” (for sin), and “cosine” with “change/opposite” (for cos).

记忆技巧：sin(A±B) 展开为”正弦余弦、余弦正弦，符号相同”；cos(A±B) 展开为”余弦余弦、正弦正弦，符号相反”。利用首字母：sin 与 same（相同）都以 s 开头；cos 与 change/opposite（相反）对应。

Double Angle Formulas | 二倍角公式

sin2θ = 2sinθcosθ

cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

Memory Tip: Sin double angle is simply 2sinθcosθ — picture “2 smiling faces side by side”. Cos double angle has three forms; if you remember the first (cos² − sin²), you can derive the other two from the Pythagorean identity.

记忆技巧：sin 二倍角就是 2sinθcosθ——想象两个并列的”笑脸”。cos 二倍角有三种形式；只要记住第一种 cos²θ − sin²θ，再用 sin²θ + cos²θ = 1 即可推出另外两种。

4. Differentiation | 微分

Power Rule | 幂法则

d/dx (xⁿ) = n xⁿ⁻¹

Product Rule | 乘积法则

d/dx (uv) = u’v + uv’

Quotient Rule | 商法则

d/dx (u/v) = (u’v − uv’) / v²

Chain Rule | 链式法则

dy/dx = dy/du × du/dx

Memory Tip for Product Rule: “Derivative of the first times the second, plus the first times the derivative of the second.” Say it as: “First d-second, plus second d-first.”

乘积法则记忆技巧：”前导后不导，加前不导后导。” 即 “first d-second, plus second d-first”。

Memory Tip for Quotient Rule: “Low d-high minus high d-low, over the square of what’s below.” Arrange the numerator in the same order as the quotient: high/low → (low d-high − high d-low)/low².

商法则记忆技巧：”低导高减高导低，除以分母的平方。” 分子顺序与商一致：高/低 → (低 d-高 − 高 d-低) / 低²。

Standard Derivatives | 常见导数

d/dx (eˣ) = eˣ — the exponential function is its own derivative | 指数函数导数为自身
d/dx (ln x) = 1/x | 自然对数的导数为 1/x
d/dx (sin x) = cos x; d/dx (cos x) = −sin x | 正弦导数为余弦，余弦导数为负正弦

Memory Tip:”sin goes to cos (positive), cos goes to −sin (negative).” The cycle continues: −sin → −cos → sin.

记忆技巧：”sin 变 cos（正），cos 变 −sin（负）。” 循环继续：−sin → −cos → sin。

5. Integration | 积分

Power Rule for Integration | 积分幂法则

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1

Special Case | 特殊情况

∫ 1/x dx = ln|x| + C

Standard Integrals | 常见积分

∫ eˣ dx = eˣ + C | 自然指数的积分等于自身
∫ sin x dx = −cos x + C | 正弦积分为负余弦
∫ cos x dx = sin x + C | 余弦积分为正弦
∫ sec² x dx = tan x + C | sec² 积分为 tan

Memory Tip: Integration is “reverse differentiation” — add 1 to the power and divide by the new power. When in doubt, differentiate your answer to check it matches the integrand.

记忆技巧：积分是”微分的逆运算”——指数加 1，再除以新指数。不确定时，对答案求导即可验证。

Integration by Parts | 分部积分法

∫ u dv = uv − ∫ v du

Memory Tip: Choose u using the acronym LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) — pick u in that priority order to make the remaining integral simpler.

记忆技巧：用首字母缩写 LIATE（对数、反三角、代数、三角、指数）选择 u——按此优先级选取，可使余下积分更简单。

6. Differential Equations | 微分方程

Separable Variables | 变量分离法

dy/dx = f(x)g(y) → ∫ 1/g(y) dy = ∫ f(x) dx

Rearrange to gather all y terms with dy on one side and all x terms with dx on the other, then integrate both sides. | 将所有含 y 的项与 dy 放在一边，含 x 的项与 dx 放在另一边，然后两边分别积分。

Integrating Factor (First-Order Linear) | 积分因子（一阶线性）

dy/dx + P(x)y = Q(x) → IF = e^(∫P(x)dx)

Multiply both sides by the integrating factor, and the left side becomes the derivative of (IF × y).

两边同乘积分因子，左侧恰好变为 (IF × y) 的导数。

Memory Tip: The integrating factor formula looks like “e to the integral of P”. Once found, the equation collapses into a simple product rule differentiation — a satisfying check.

记忆技巧：积分因子就是”e 的 P 的积分次方”。求出之后，原方程立刻化简为简单的乘积求导，非常奇妙。

7. Vectors | 向量

Dot Product | 点积

a · b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃

Angle Between Vectors | 向量夹角

cosθ = (a · b) / (|a||b|)

If a · b = 0, the vectors are perpendicular. | 若 a · b = 0，则两向量垂直。

Cross Product (3D) | 叉积（三维）

a × b = |a||b|sinθ n̂

The cross product produces a vector perpendicular to both a and b. Its magnitude equals the area of the parallelogram formed by the two vectors.

叉积产生同时垂直于 a 和 b 的向量，其大小等于两向量构成的平行四边形面积。

Memory Tip: Dot “cos” gives a scalar (a number); Cross “sin” gives a vector. Remember: “Dot is scalar, cross is vector.”

记忆技巧：点积用 cos，结果为标量（数值）；叉积用 sin，结果为向量。”点积是标量，叉积是向量。”

8. Numerical Methods | 数值方法

Trapezium Rule | 梯形法则

∫ₐᵇ f(x)dx ≈ h/2 [(y₀ + yₙ) + 2(y₁ + y₂ + … + yₙ₋₁)]

where h = (b − a)/n. | 其中 h = (b − a)/n。

Memory Tip: “Half h times (first + last + twice the middle).” The first and last y-values are only counted once, all middle values get doubled.

记忆技巧：”二分之一 h 乘以（首项加末项，加二倍中间项）。” 首末各算一次，中间全部翻倍。

Newton-Raphson Iteration | 牛顿-拉夫森迭代法

xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)

Memory Tip: “Current x minus the fraction f over f-prime.” The formula literally says: draw the tangent at xₙ, follow it to where it crosses the x-axis, and that’s your next approximation.

记忆技巧：”当前 x 减去 f 除以 f′。” 这个公式的本质是：在 xₙ 处画切线，沿切线找到它与 x 轴的交点，即为下一个近似值。

9. Probability & Statistics | 概率与统计

Conditional Probability | 条件概率

P(A|B) = P(A ∩ B) / P(B)

Memory Tip: The intersection goes on top, and the condition (given) goes on bottom. “Given makes the denominator.”

记忆技巧：交集的概率放分子，条件（given 后面的事件）放下。”given 后面的事件作分母。”

Mean & Variance of a Distribution | 分布的均值与方差

E(X) = Σ x P(X = x)

Var(X) = E(X²) − [E(X)]²

Binomial Distribution | 二项分布

X ~ B(n, p): P(X = r) = C(n,r) pʳ (1−p)ⁿ⁻ʳ

E(X) = np, Var(X) = np(1−p)

Normal Distribution | 正态分布

X ~ N(μ, σ²): Z = (X − μ) / σ

Standardize any normal variable by subtracting the mean and dividing by the standard deviation, then use the standard normal table. | 将任何正态变量减均值、除标准差即可标准化，然后用标准正态分布表查值。

Memory Tip for Binomial: “n choose r, times p to the r, times q to the n minus r” where q = 1 − p. The exponents always sum to n.

二项分布记忆技巧：”C(n,r) 乘以 p 的 r 次方，再乘以 q 的 n−r 次方”，其中 q = 1 − p。两个指数之和恒为 n。

10. Memory Techniques & Exam Tips | 记忆技巧与备考策略

1. The U-Curve Method | U形曲线法

Sleep is crucial for memory consolidation. Study the hardest formula blocks early in the session and again just before bed — your brain processes them during deep sleep. | 睡眠对记忆巩固至关重要。在复习阶段开始时以及睡前复习最难的公式块，大脑会在深度睡眠中加工这些信息。

2. Active Recall with a Formula Sheet | 主动回忆法与公式卡

Write out every formula from memory onto a blank sheet. Check your sheet against the original; mark errors in red. Repeat daily until you achieve 100% accuracy three times in a row.

不看课本，凭记忆将全部公式默写在一张空白纸上；再与原文对照，用红笔标出错误；每天重复，直到连续三次全部正确。

3. Chunking & Mnemonics | 分块与助记符

Break long formulas into 3–4 meaningful chunks. For example, the quotient rule: “low d-high / minus high d-low / over low squared” — three rhythmic beats that match the formula structure.

将长公式拆分为 3-4 个有意义的片段。例如商法则：”低导高 / 减高导低 / 比低方”——三个节奏片段，正好对应公式结构。

4. The Explanation Effect | 讲解效应

Teach the formula to a classmate or even to yourself aloud. If you can explain why the discriminant works, you will never forget when to use it. | 将公式讲解给同学听，甚至朗声讲给自己听。如果能解释清楚判别式为何有效，你就永远不会忘记何时使用它。

5. Past Paper Pattern Recognition | 真题模式识别

A-level exams repeatedly test the same formula families. Sort past paper questions by formula type and identify which formulas appear most often — prioritize those in the final two weeks before the exam.

A-Level 考试会反复考查同样的公式族。将真题按公式类型分类，统计出现频率最高的公式——考前两周优先复习它们。

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A-Level Mathematics Core Formula Summary & Memory Techniques | A-Level 数学核心公式汇总与记忆技巧