Edexcel Mathematics: Last-Minute Revision Notes | Edexcel 数学：考前冲刺笔记

📚 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。

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1. Algebraic Manipulation and Simplification | 代数运算与化简

Always check for common factors when factorising expressions. For quadratics, use the method of splitting the middle term or recognise special forms 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。

When simplifying algebraic fractions, factorise numerators and denominators fully, then cancel common factors. Remember index laws: aᵐ × aⁿ = aᵐ⁺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。

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2. Functions and Graphs | 函数与图像

The domain of a function is the set of input values for which it is defined; the range is the set of possible outputs. For composite function fg(x), apply 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。

Recall key transformations: y = f(x − a) + b shifts the graph a units right and b units up. y = af(x) stretches 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.

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。

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3. Quadratics and Inequalities | 二次函数与不等式

For quadratic ax² + bx + c = 0, discriminant Δ = b² − 4ac. If Δ > 0: two distinct real roots; Δ = 0: one repeated real root; 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。

Completing the square rewrites ax² + bx + 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。

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4. Coordinate Geometry | 坐标几何

The equation of a straight line through (x₁, y₁) with gradient m is y − y₁ = m(x − x₁). The gradient between two points is m = (y₂ − y₁)/(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。

Distance between (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)。

A circle with centre (a, b) and radius r has equation (x − a)² + (y − b)² = r². The general form x² + y² + 2gx + 2fy + c = 0 gives centre (−g, −f) and radius √(g² + f² − c) provided g² + f² − c > 0. A tangent to a circle is perpendicular to the radius at the point of contact. 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。

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5. Trigonometry | 三角学

Know exact values of sin, cos, tan for 30°, 45°, 60° and related angles. Always be careful whether the 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。

Use the CAST diagram to identify the signs of trig ratios in each quadrant. Key identities: tanθ = sinθ/cosθ, sin²θ + cos²θ = 1. To solve equations like 2 sin²θ − sinθ − 1 = 0, substitute x = sinθ and factor

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