📚 Core Pure Maths Revision: Key Techniques for Edexcel A-Level | Edexcel A-Level 数学核心技巧复习
This revision guide consolidates the essential algebraic, calculus, and trigonometric techniques required for Edexcel A-Level Mathematics (Pure Maths). Each section pairs a concise English explanation with a Chinese translation, allowing you to review key ideas quickly and accurately before your exams.
本复习指南整合了 Edexcel A-Level 数学（纯数部分）所必需的代数、微积分和三角学核心技巧。每一节均采用英文简述与中文翻译配对的形式，帮助你在考前快速、准确地回顾关键概念。
1. Quadratic Functions and the Discriminant | 二次函数与判别式
For a quadratic equation ax² + bx + c = 0, the discriminant is Δ = b² − 4ac. If Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one repeated root; if Δ < 0, there are no real roots. The solution is given by x = (−b ± √Δ) / (2a).
对于二次方程 ax² + bx + c = 0，判别式为 Δ = b² − 4ac。若 Δ > 0，则有两个不等实根；若 Δ = 0，则有一个重根；若 Δ < 0，则没有实根。求根公式为 x = (−b ± √Δ) / (2a)。
When completing the square, rewrite ax² + bx + c as a(x + b/(2a))² + (c − b²/(4a)). This form reveals the turning point of the parabola: (−b/(2a), c − b²/(4a)).
配方时，将 ax² + bx + c 改写为 a(x + b/(2a))² + (c − b²/(4a))。这种形式直接给出抛物线的顶点坐标：(−b/(2a), c − b²/(4a))。
Δ = b² − 4ac, x = (−b ± √Δ) / (2a)
2. Inequalities and Set Notation | 不等式与集合记号
Solving linear inequalities follows the same rules as equations, except that multiplying or dividing by a negative number reverses the inequality sign. For quadratic inequalities, first find the critical values by solving the corresponding equation, then test intervals on a number line.
解线性不等式与解方程规则相同，唯一例外是：当两边同时乘以或除以负数时，不等号方向必须反转。对于二次不等式，先解对应方程得到临界值，再在数轴上测试各区间。
Set notation is commonly tested: {x : x < a} ∪ {x : x > b} represents the union of two disjoint intervals, while {x : a ≤ x ≤ b} denotes a closed interval. A strict inequality uses parentheses in interval notation: (a, b); a non-strict inequality uses brackets: [a, b].
集合记号是常考内容：{x : x < a} ∪ {x : x > b} 表示两个不相交区间的并集，而 {x : a ≤ x ≤ b} 表示闭区间。严格不等式在区间记号中用圆括号 (a, b)，非严格不等式用方括号 [a, b]。
3. Coordinate Geometry: Straight Lines and Circles | 坐标几何：直线与圆
The gradient of a line through points (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁) / (x₂ − x₁). The equation of a line can be written as y − y₁ = m(x − x₁) or in the general form ax + by + c = 0. Two lines are perpendicular if the product of their gradients is −1.
过点 (x₁, y₁) 与 (x₂, y₂) 的直线斜率为 m = (y₂ − y₁) / (x₂ − x₁)。直线方程可写为点斜式 y − y₁ = m(x − x₁) 或一般式 ax + by + c = 0。两条直线垂直当且仅当它们的斜率乘积为 −1。
The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r². The perpendicular bisector of any chord passes through the centre. To find the intersection of a line and a circle, substitute the linear equation into the circle equation and solve the resulting quadratic.
圆心为 (a, b)、半径为 r 的圆方程为 (x − a)² + (y − b)² = r²。任何弦的垂直平分线必过圆心。求直线与圆的交点时，将直线方程代入圆方程，解所得二次方程即可。
(x − a)² + (y − b)² = r²
4. Differentiation: Rules and Applications | 微分：法则与应用
The derivative of xⁿ is nxⁿ⁻¹. The product rule states (uv)’ = u’v + uv’, and the quotient rule states (u/v)’ = (u’v − uv’) / v². The chain rule for composite functions is dy/dx = (dy/du) × (du/dx).
xⁿ 的导数为 nxⁿ⁻¹。乘积法则为 (uv)’ = u’v + uv’，商法则为 (u/v)’ = (u’v − uv’) / v²。复合函数的链式法则是 dy/dx = (dy/du) × (du/dx)。
At a stationary point, dy/dx = 0. Use the second derivative to classify: if d²y/dx² > 0, the point is a local minimum; if d²y/dx² < 0, it is a local maximum. In optimisation problems, form a single-variable expression, differentiate, set the derivative to zero, and verify the nature of the turning point.
在驻点处，dy/dx = 0。利用二阶导数判断极值：若 d²y/dx² > 0，则为局部极小值；若 d²y/dx² < 0，则为局部极大值。在最优化问题中，先建立单变量表达式，求导并令导数为零，再验证驻点的性质。
5. Integration: Indefinite and Definite | 积分：不定积分与定积分
The indefinite integral of xⁿ (for n ≠ −1) is (xⁿ⁺¹)/(n+1) + c. Integration is the reverse process of differentiation. To integrate a sum, integrate each term separately and add the constant of integration for indefinite integrals.
xⁿ（n ≠ −1）的不定积分为 (xⁿ⁺¹)/(n+1) + c。积分是微分的逆运算。对和函数积分时，逐项积分并在不定积分末尾加上积分常数 c。
A definite integral evaluates the signed area between the curve and the x-axis over a given interval: ∫ₐᵇ f(x) dx = F(b) − F(a). If the curve lies below the x-axis, the integral is negative; take the absolute value for the area. For areas enclosed by a curve and the x-axis, identify roots and integrate each region separately.
定积分计算曲线与 x 轴之间在给定区间上的有向面积：∫ₐᵇ f(x) dx = F(b) − F(a)。若曲线位于 x 轴下方，则积分值为负，计算面积时应取绝对值。求曲线与 x 轴围成的面积时，先找根，再分段积分。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ −1)
6. Exponential and Logarithmic Functions | 指数函数与对数函数
The derivative of eˣ is eˣ, and the derivative of ln x (x > 0) is 1/x. The laws of logarithms are: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, and ln(aⁿ) = n ln a. The change of base formula is logₐ b = (ln b)/(ln a).
eˣ 的导数为 eˣ，ln x（x > 0）的导数为 1/x。对数运算法则包括：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，以及 ln(aⁿ) = n ln a。换底公式为 logₐ b = (ln b)/(ln a)。
When solving exponential equations such as aˣ = b, take the natural logarithm of both sides: x = ln b / ln a. In exponential growth and decay models described by N = N₀eᵏᵗ, the rate of change satisfies dN/dt = kN. Remember that e⁻ᵏᵗ represents decay when k > 0.
解形如 aˣ = b 的指数方程时，两边取自然对数：x = ln b / ln a。在指数增长与衰减模型 N = N₀eᵏᵗ 中，变化率满足 dN/dt = kN。注意当 k > 0 时，e⁻ᵏᵗ 表示衰减过程。
7. Trigonometric Identities and Equations | 三角恒等式与三角方程
The two fundamental identities are sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ. The compound angle formulas include sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B.
两个基本恒等式为 sin²θ + cos²θ = 1 与 tan θ = sin θ / cos θ。和角公式包括 sin(A ± B) = sin A cos B ± cos A sin B 以及 cos(A ± B) = cos A cos B ∓ sin A sin B。
To solve a trigonometric equation over a given interval, first isolate the trigonometric function, then find the principal solution using inverse functions. Use the symmetry of the sine, cosine, and tangent graphs to generate all solutions within the required range. Always check the domain: 0 ≤ θ ≤ 180° or in radians 0 ≤ θ ≤ 2π.
在指定区间内解三角方程时，先隔离三角函数，再用反三角函数求出主解。利用正弦、余弦和正切图像的对称性，生成给定范围内的所有解。务必注意定义域：角度制 0 ≤ θ ≤ 180° 或弧度制 0 ≤ θ ≤ 2π。
8. Numerical Methods: Iteration and the Trapezium Rule | 数值方法：迭代与梯形法则
The trapezium rule approximates a definite integral by dividing the interval [a, b] into n strips of equal width h = (b − a)/n. The approximation is ∫ₐᵇ y dx ≈ (h/2) × [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ].
梯形法则将区间 [a, b] 等分为 n 条宽度为 h = (b − a)/n 的小条，从而近似计算定积分。近似公式为 ∫ₐᵇ y dx ≈ (h/2) × [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]。
For iterative methods, an equation such as f(x) = 0 may be rearranged into the form x = g(x). Starting from an initial value x₀, compute xₙ₊₁ = g(xₙ) repeatedly until the values converge to a root. The change of sign method can be used to locate a root: if f(a) and f(b) have opposite signs and f is continuous, there is at least one root between a and b.
对于迭代法，方程 f(x) = 0 可以重排为 x = g(x) 的形式。从初值 x₀ 出发，反复计算 xₙ₊₁ = g(xₙ)，直到数值收敛到根。变号法可用于定位根：若 f(a) 与 f(b) 异号且 f 连续，则 (a, b) 之间至少存在一个根。
9. Proof and Algebraic Fractions | 证明与代数分式
Two common proof methods are proof by deduction and proof by exhaustion. For a statement involving even and odd numbers, let n = 2k for even and n = 2k + 1 for odd, then manipulate algebraically to verify the claim.
两种常见的证明方法是演绎证明与穷举证明。对于涉及偶数和奇数的命题，令偶数 n = 2k，奇数 n = 2k + 1，然后通过代数变形验证结论。
When adding or subtracting algebraic fractions, first find the lowest common denominator. Simplify complex fractions by factoring numerators and denominators, then cancel common factors. In exam questions, always state any restrictions on the variable, such as excluding values that make a denominator zero.
进行代数分式的加减时，先求最简公分母。化简复杂分式时，先对分子分母因式分解，再约去公因式。在考试题中，务必注明变量的限制条件，例如排除使分母为零的值。
10. Exam Strategy and Common Pitfalls | 应试策略与常见易错点
Read each question carefully and identify which topic area it tests. Show all intermediate steps clearly, because method marks are awarded even if the final answer is incorrect. When using a calculator, check the angle mode: degrees or radians, as required by the question.
仔细阅读每一道题，判断它考查的知识板块。清晰写出所有中间步骤，因为即使最终答案有误，步骤分仍然可以获得。使用计算器时，务必检查角度模式：根据题目要求选择角度制或弧度制。
- Do not forget the constant of integration + c in indefinite integrals.
求不定积分时不要忘记积分常数 + c。
- When differentiating a product or quotient, identify u and v before applying the rule.
对乘积或商求导时，先确定 u 和 v，再套用相应法则。
- When solving a quadratic inequality, sketch the graph or test intervals; do not simply reverse signs.
解二次不等式时，画草图或测试区间；不可盲目改变不等号方向。
- When using the trapezium rule, the number of strips must equal the number of intervals, and your table of y-values must contain n+1 entries.
使用梯形法则时，小条数必须等于区间数，y 值表中应有 n+1 个数据。
11. Quick Formula Checklist | 公式快速检查清单
| Differentiation | 微分 | d(xⁿ)/dx = nxⁿ⁻¹; d(eˣ)/dx = eˣ; d(ln x)/dx = 1/x |
| Integration | 积分 | ∫ xⁿ dx = xⁿ⁺¹/(n+1) + c |
| Circle equation | 圆方程 | (x − a)² + (y − b)² = r² |
| Log laws | 对数法则 | ln(ab) = ln a + ln b; ln(aⁿ) = n ln a |
| Trapezium rule | 梯形法则 | ∫ₐᵇ y dx ≈ (h/2) × [y₀ + 2(y₁ + … + yₙ₋₁) + yₙ] |
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply