📚 Edexcel Mathematics: Last-Minute Revision Notes | Edexcel 数学：考前冲刺笔记
This concise last-minute revision guide condenses key topics for the Edexcel A Level Mathematics (Pure) syllabus. Use it to quickly recall essential formulas, common pitfalls, and must-know techniques before the exam. Keep calm and focus on understanding each concept.
本文为 Edexcel A Level 数学（纯数）考前冲刺浓缩笔记，涵盖核心公式、常见失分点与必考技巧。考前快速回顾，保持冷静，逐条理解。
1. Algebraic Manipulation and Simplification | 代数运算与化简
rms such as difference of two squares: a² − b² = (a − b)(a + b). Expand brackets carefully, especially when dealing with negative signs.
分解因式时，首先检查公因式。二次式可裂项或识别平方差：a² − b² = (a − b)(a + b)。展开括号时注意负号。
⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a⁰ = 1 (a ≠ 0). Rationalise denominators when they contain surds by multiplying by the conjugate.
化简代数分式时，先完全因式分解分子分母，再约去公因式。牢记指数律：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁻ⁿ = 1/aⁿ，a⁰ = 1（a ≠ 0）。分母含根式时，通过乘以其共轭式进行有理化。
Be aware of common mistakes like misapplying index rules to addition (aᵐ + aⁿ is not aᵐ⁺ⁿ) and forgetting to distribute the negative when subtracting algebraic fractions.
注意常见错误：勿将指数律用于加法（aᵐ + aⁿ 不等于 aᵐ⁺ⁿ），化简分式减法时别漏掉负号分配。
2. Functions and Graphs | 函数与图像
ly g first then f. To find the inverse f⁻¹(x), write y = f(x), swap x and y, then solve for y. Graphically, the inverse is the reflection in the line y = x.
定义域是函数有意义的输入值集合；值域是可能的输出值集合。复合函数 fg(x) 先运算 g 后 f。求反函数时，令 y = f(x)，交换 x 与 y，再解出 y。图像上，反函数关于直线 y = x 对称。
horizontally by factor 1/a. Reflections: y = −f(x) reflects in the x-axis; y = f(−x) reflects in the y-axis.
牢记图像变换：y = f(x − a) + b 将图像向右平移 a 单位、向上平移 b 单位。y = af(x) 垂直方向伸缩 a 倍；y = f(ax) 水平方向伸缩 1/a 倍。反射：y = −f(x) 关于 x 轴反射；y = f(−x) 关于 y 轴反射。
For problems involving y = |f(x)|, reflect the parts below the x-axis upwards. For y = f(|x|), delete the graph for x < 0 and reflect the right-hand side into the left half.
涉及 y = |f(x)| 时，将 x 轴下方的部分向上反射；y = f(|x|) 则删去左侧图像，将右侧图像反射到左侧。
3. Quadratics and Inequalities | 二次函数与不等式
t; Δ < 0: no real roots. The discriminant is essential for determining the number of intersection points between a curve and a line.
对于二次方程 ax² + bx + c = 0，判别式 Δ = b² − 4ac。Δ > 0 两个不等实根；Δ = 0 一个重实根；Δ < 0 无实根。判别式是判断曲线与直线交点个数的关键。
x + c as a(x + p)² + q, where p = b/(2a) and q = c − (b²/(4a)), giving the vertex (−p, q).
配方法将 ax² + bx + c 写成 a(x + p)² + q，其中 p = b/(2a)，q = c − b²/(4a)，顶点为 (−p, q)。
When solving quadratic inequalities, sketch the graph to see where the expression is positive or negative. Use strict inequalities for open intervals. For hidden quadratics, substitute a single variable to form a standard quadratic, solve, then revert the substitution.
解二次不等式时，先画出图像确定多项式正负区间，开口区间用严格不等号。遇到隐藏二次时可令 u = x² 或其他代换，解出标准方程后再回代。
4. Coordinate Geometry | 坐标几何
/(x₂ − x₁). Parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.
过点 (x₁, y₁) 且斜率为 m 的直线方程为 y − y₁ = m(x − x₁)。两点间的斜率 m = (y₂ − y₁)/(x₂ − x₁)。平行线斜率相等，垂直线斜率乘积为 −1。
n (x₁, y₁) and (x₂, y₂): √[(x₂ − x₁)² + (y₂ − y₁)²]. Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).
两点距离：√[(x₂ − x₁)² + (y₂ − y₁)²]。中点坐标：((x₁ + x₂)/2, (y₁ + y₂)/2)。
− b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 gives centre (−g, −f) and radius √g² + f² − c act. To find tangent equations, use discriminant = 0 or the perpendicular gradient method.
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 对应的圆心为 (−g, −f)，半径 √g² + f² − c（需满足 g² + f² − c > 0）。圆的切线垂直于切点处的半径。求切线方程可用判别式等于零或垂直斜率法。
5. Trigonometry | 三角学
he angle is in degrees or radians; π rad = 180°. Arc length l = rθ, sector area A = ½ r²θ.
熟记 30°、45°、60° 及其相关角的正弦、余弦、正切精确值。注意区分角度与弧度制，π 弧度 = 180°。弧长 l = rθ，扇形面积 A = ½ r²θ。
cos²θ = 1. To solve equations like 2 sin²θ − sinθ − 1 = 0, substitute x = sinθ and factor
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