📚 Summary of Key Points | 核心要点总结
This article provides a concise revision summary of the key points for A-Level Mathematics (Edexcel), covering Pure Mathematics, Statistics, and Mechanics. It is designed to help you recall crucial formulas, methods, and exam techniques in one place.
本文为 A-Level 数学（爱德思考试局）核心要点总结，涵盖纯数、统计与力学。目的是帮助你在一个页面内快速回顾关键公式、方法与考试技巧。
1. Algebraic Manipulation and Quadratic Functions | 代数运算与二次函数
Completing the square is a central technique: ax² + bx + c = a(x + b/2a)² + (c − b²/4a). The discriminant Δ = b² − 4ac determines the number of real roots: if Δ > 0, two distinct roots; if Δ = 0, one repeated root; if Δ < 0, no real roots.
配方法是核心技巧：ax² + bx + c = a(x + b/2a)² + (c − b²/4a)。判别式 Δ = b² − 4ac 决定实根个数：若 Δ > 0，有两个不等实根；若 Δ = 0，有一个重根；若 Δ < 0，无实根。
When solving inequalities, always consider the critical values where the expression equals zero, and test intervals. For example, solving x² − 5x + 6 > 0 gives x < 2 or x > 3.
解不等式时，务必先找出表达式等于零的临界值，再在区间内测试。例如：解 x² − 5x + 6 > 0 得 x < 2 或 x > 3。
Laws of indices and surds are frequently tested: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, and rationalising the denominator such as 1/√a = √a/a.
指数与根式法则常考：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，以及分母有理化如 1/√a = √a/a。
2. Coordinate Geometry and Circles | 坐标几何与圆
The midpoint formula, gradient formula m = (y₂ − y₁)/(x₂ − x₁), and the equation of a straight line y − y₁ = m(x − x₁) are essential. For perpendicular lines, m₁ × m₂ = −1.
中点公式、斜率公式 m = (y₂ − y₁)/(x₂ − x₁) 以及直线方程 y − y₁ = m(x − x₁) 均为基础。相互垂直的直线满足 m₁ × m₂ = −1。
A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². The expanded form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).
圆心为 (a, b)、半径为 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 √(g² + f² − c)。
To find the intersection of a line and a circle, substitute the line into the circle equation. A tangent to a circle is perpendicular to the radius at the point of contact.
求直线与圆的交点时，将直线代入圆方程。圆的切线在切点处垂直于半径。
3. Trigonometric Identities and Equations | 三角恒等式与方程
The Pythagorean identity sin²θ + cos²θ = 1 and the tangent identity tanθ = sinθ/cosθ are fundamental. You must also know the exact values for 0°, 30°, 45°, 60°, 90° and the corresponding radians.
毕达哥拉斯恒等式 sin²θ + cos²θ = 1 与正切恒等式 tanθ = sinθ/cosθ 是基础。你还需熟记 0°、30°、45°、60°、90° 及其对应弧度下的精确值。
When solving trigonometric equations over a given interval, find all solutions by considering the symmetry of the sine, cosine, and tangent graphs. For example, sinθ = 0.5 gives θ = 30° and 150° in [0°, 360°].
在给定区间内解三角方程时，需结合正弦、余弦、正切图像的对称性求所有解。例如：sinθ = 0.5 在 [0°, 360°] 内的解为 θ = 30° 和 150°。
Key formulas include the compound angle identities: sin(A ± B) = sinA cosB ± cosA sinB, and double angle formulas such as cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.
关键公式包括和角公式：sin(A ± B) = sinA cosB ± cosA sinB，以及倍角公式 cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。
4. Differentiation | 微分
For y = xⁿ, the derivative is dy/dx = nxⁿ⁻¹. The product rule: d(uv)/dx = u dv/dx + v du/dx. The quotient rule: d(u/v)/dx = (v du/dx − u dv/dx)/v².
对于 y = xⁿ，导数为 dy/dx = nxⁿ⁻¹。乘积法则：d(uv)/dx = u dv/dx + v du/dx。商法则：d(u/v)/dx = (v du/dx − u dv/dx)/v²。
The chain rule states dy/dx = dy/du × du/dx. For example, if y = (2x + 1)⁵, then dy/dx = 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴.
链式法则为 dy/dx = dy/du × du/dx。例如：若 y = (2x + 1)⁵，则 dy/dx = 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴。
Stationary points occur when dy/dx = 0. Use the second derivative d²y/dx² to classify: positive indicates a minimum, negative indicates a maximum, and zero requires further investigation.
驻点出现在 dy/dx = 0 处。用二阶导数 d²y/dx² 判断类型：大于零为极小值，小于零为极大值，等于零则需要进一步判断。
5. Integration | 积分
The reverse of differentiation: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1. Definite integrals evaluate the area under a curve between two limits.
积分是微分的逆运算：∫xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1）。定积分用于计算曲线与 x 轴之间的面积。
For areas between a curve and the x-axis, integrate with respect to x. If the curve crosses the x-axis, split the interval and take absolute values of negative areas.
计算曲线与 x 轴围成的面积时，对 x 积分。若曲线穿过 x 轴，需分段取绝对值后再相加。
Key standard integrals include ∫1/x dx = ln|x| + C, ∫cos x dx = sin x + C, and ∫eˣ dx = eˣ + C. Remember to include the constant of integration for indefinite integrals.
常见标准积分包括 ∫1/x dx = ln|x| + C、∫cos x dx = sin x + C 和 ∫eˣ dx = eˣ + C。不定积分不要忘记加上积分常数 C。
6. Exponentials and Logarithms | 指数与对数
Logarithms are the inverse of exponentials: if y = aˣ, then x = logₐy. The natural logarithm ln x has base e, and it satisfies ln(eˣ) = x and e^(ln x) = x.
对数是指数的逆运算：若 y = aˣ，则 x = logₐy。自然对数 ln x 以 e 为底，满足 ln(eˣ) = x 和 e^(ln x) = x。
The laws of logarithms are: logₐ(mn) = logₐm + logₐn, logₐ(m/n) = logₐm − logₐn, and logₐ(mᵏ) = k logₐm.
对数运算法则：logₐ(mn) = logₐm + logₐn，logₐ(m/n) = logₐm − logₐn，logₐ(mᵏ) = k logₐm。
To solve exponential equations, take logarithms of both sides. For example, 2ˣ = 10 gives x = log₁₀10/log₁₀2 ≈ 3.32. In modelling, the equation N = N₀eᵏᵗ describes exponential growth or decay.
解指数方程时，两边取对数。例如：2ˣ = 10 得 x = log₁₀10/log₁₀2 ≈ 3.32。在实际建模中，N = N₀eᵏᵗ 描述指数增长或衰减。
7. Sequences and Series | 数列与级数
An arithmetic sequence has the nth term uₙ = a + (n−1)d and sum Sₙ = n/2[2a + (n−1)d]. A geometric sequence has uₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ)/(1 − r) when r ≠ 1.
等差数列的第 n 项为 uₙ = a + (n−1)d，前 n 项和 Sₙ = n/2[2a + (n−1)d]。等比数列的第 n 项为 uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。
For an infinite geometric series with |r| < 1, the sum to infinity is S∞ = a/(1 − r). If |r| ≥ 1, the series diverges.
当 |r| < 1 时，无穷等比级数的和为 S∞ = a/(1 − r)。若 |r| ≥ 1，级数发散。
The binomial expansion states (a + b)ⁿ = aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + … + bⁿ for positive integer n. For fractional or negative n, the series is valid only for |x| < 1.
二项式展开为 (a + b)ⁿ = aⁿ + nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² + … + bⁿ（n 为正整数）。当指数为分数或负数时，级数仅在 |x| < 1 时有效。
8. Statistical Sampling and Data Presentation | 统计抽样与数据表示
Key sampling methods include simple random sampling, systematic sampling, stratified sampling, and quota sampling. Each has advantages and disadvantages in terms of bias, cost, and convenience.
主要抽样方法包括简单随机抽样、系统抽样、分层抽样和配额抽样。每种方法在偏差、成本与便利性方面各有优缺点。
For grouped data, the mean is estimated using midpoints, the median is found by linear interpolation, and the standard deviation measures spread. The variance is the square of the standard deviation.
对于分组数据，均值用组中值估算，中位数用线性插值法求，标准差衡量离散程度。方差是标准差的平方。
Remember the relationship between quartiles and the interquartile range (IQR = Q₃ − Q₁), which is used to detect outliers. A box plot visually displays the minimum, Q₁, median, Q₃, and maximum.
记住四分位数与四分位距的关系（IQR = Q₃ − Q₁），用于检测异常值。箱线图直观展示最小值、Q₁、中位数、Q₃ 和最大值。
9. Probability and Binomial Distribution | 概率与二项分布
For independent events, P(A ∩ B) = P(A) × P(B). For mutually exclusive events, P(A ∪ B) = P(A) + P(B). Conditional probability is given by P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0.
独立事件满足 P(A ∩ B) = P(A) × P(B)。互斥事件满足 P(A ∪ B) = P(A) + P(B)。条件概率公式为 P(A|B) = P(A ∩ B)/P(B)（要求 P(B) > 0）。
A binomial distribution X ~ B(n, p) models the number of successes in n independent trials with constant probability p. The probability of exactly k successes is P(X = k) = nCk pᵏ(1−p)ⁿ⁻ᵏ.
二项分布 X ~ B(n, p) 描述 n 次独立试验中成功次数，每次成功概率恒为 p。恰好 k 次成功的概率为 P(X = k) = nCk pᵏ(1−p)ⁿ⁻ᵏ。
The mean of a binomial distribution is np, and the variance is np(1−p). Normal approximation to the binomial can be used when np and n(1−p) are both greater than 5, with continuity correction applied.
二项分布的均值为 np，方差为 np(1−p)。当 np 与 n(1−p) 均大于 5 时，可用正态近似，并需进行连续性校正。
10. Mechanics: Kinematics and Forces | 力学：运动学与力
The constant acceleration (SUVAT) equations are: v = u + at, s = ut + ½at², and v² = u² + 2as. Always define positive direction and use consistent units.
匀加速直线运动公式（SUVAT）包括：v = u + at，s = ut + ½at² 和 v² = u² + 2as。解题时应先设定正方向并统一单位。
The three laws of motion: a body remains at rest or moves uniformly unless acted on by a resultant force; F = ma; and every action has an equal and opposite reaction.
牛顿三定律：物体在不受合外力时保持静止或匀速直线运动；F = ma；作用力与反作用力大小相等、方向相反。
When resolving forces, use Fx = F cosθ and Fy = F sinθ. For a body on an inclined plane, the weight component parallel to the plane is mg sinθ and perpendicular is mg cosθ.
分解力时使用 Fx = F cosθ、Fy = F sinθ。斜面上的物体，沿斜面方向的重力分量为 mg sinθ，垂直斜面的分量为 mg cosθ。
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