📚 Edexcel A-Level Maths Core Pure Revision Guide | Edexcel A-Level 数学核心纯数复习指南
This revision guide covers the high-yield pure mathematics topics that appear consistently in Edexcel A-Level Maths Papers 1 and 2. It is designed to help you consolidate key methods, avoid common errors, and produce clear, exam-ready solutions.
本复习指南涵盖 Edexcel A-Level 数学卷一和卷二反复出现的高频纯数考点，旨在帮助你巩固核心方法、规避常见错误，并写出清晰、符合考试要求的解答。
1. Algebra and Functions | 代数与函数
Many Edexcel problems begin with algebraic manipulation: simplifying rational expressions, factorising cubics, or completing the square. You must be able to rewrite a quadratic in the form a(x + p)² + q and identify the vertex (−p, q).
许多 Edexcel 题目以代数变形开篇：化简有理式、因式分解三次多项式或配平方。你必须能把二次式改写为 a(x + p)² + q，并识别顶点 (−p, q)。
The discriminant Δ = b² − 4ac decides the nature of roots. If Δ > 0, there are two real roots; if Δ = 0, there is one repeated root; if Δ < 0, there are no real roots. This is often linked to the number of intersections between a line and a curve.
判别式 Δ = b² − 4ac 决定根的性质。若 Δ > 0，则有两个实根；若 Δ = 0，则有一个重根；若 Δ < 0，则没有实根。这一点常与直线和曲线的交点个数结合考查。
For hidden quadratics, substitute y = x² or y = eˣ to turn an expression into a standard quadratic, then solve for the original variable.
对于隐藏二次型，可令 y = x² 或 y = eˣ 将其转化为标准二次方程，再解出原变量。
2. Coordinate Geometry and Parametric Equations | 坐标几何与参数方程
You should be confident converting between Cartesian and parametric forms, such as x = at² and y = 2at for a parabola. The gradient is found using dy/dx = (dy/dt) / (dx/dt).
你应熟练掌握笛卡尔形式与参数形式之间的转换，例如抛物线 x = at²、y = 2at。求斜率时使用 dy/dx = (dy/dt) / (dx/dt)。
For circles, use the centre-radius form (x − a)² + (y − b)² = r² and apply the tangent condition that the radius is perpendicular to the tangent at the point of contact.
对于圆，要使用圆心-半径式 (x − a)² + (y − b)² = r²，并运用切线与半径在切点处垂直这一条件。
When finding the equation of a tangent or normal, first determine the gradient, then substitute the known point into y − y₁ = m(x − x₁).
求切线或法线方程时，先确定斜率，再将已知点代入 y − y₁ = m(x − x₁)。
3. Trigonometry | 三角学
Know exact values for sin, cos and tan at 0°, 30°, 45°, 60°, 90° and their radian equivalents. Edexcel frequently sets non-calculator questions on these values.
熟记 0°、30°、45°、60°、90° 及其弧度制下 sin、cos、tan 的精确值。Edexcel 经常在非计算器题中考查这些内容。
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |
Core identities include sin² θ + cos² θ = 1, tan θ = sin θ / cos θ, and the double-angle formulae sin 2θ = 2 sin θ cos θ and cos 2θ = cos² θ − sin² θ.
核心恒等式包括 sin² θ + cos² θ = 1、tan θ = sin θ / cos θ，以及倍角公式 sin 2θ = 2 sin θ cos θ 和 cos 2θ = cos² θ − sin² θ。
When solving trigonometric equations, always check the given interval and find all solutions within 0 ≤ θ < 2π or 0 ≤ x < 360°.
解三角方程时，务必检查给定区间，并在 0 ≤ θ < 2π 或 0 ≤ x < 360° 内找出所有解。
4. Sequences and Series | 数列与级数
Arithmetic sequences use uₙ = a + (n − 1)d and Sₙ = n/2 [2a + (n − 1)d]. Geometric sequences use uₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.
等差数列使用 uₙ = a + (n − 1)d 和 Sₙ = n/2 [2a + (n − 1)d]。等比数列使用 uₙ = arⁿ⁻¹ 和 Sₙ = a(1 − rⁿ)/(1 − r)，其中 r ≠ 1。
For an infinite geometric series, the sum to infinity is S∞ = a / (1 − r), and this formula is valid only when |r| < 1.
对于无穷等比级数，无穷和为 S∞ = a / (1 − r)，该公式仅在 |r| < 1 时有效。
Sigma notation questions often require you to split the expression, use standard sum formulae for Σr and Σr², then combine the results.
求和符号题经常要求你先拆分表达式，使用 Σr 和 Σr² 的标准求和公式，再合并结果。
5. Binomial Expansion | 二项式展开
For rational n, the binomial expansion is (1 + x)ⁿ = 1 + nx + n(n − 1)/2! x² + n(n − 1)(n − 2)/3! x³ + …, valid for |x| < 1.
对于有理数 n，二项式展开为 (1 + x)ⁿ = 1 + nx + n(n − 1)/2! x² + n(n − 1)(n − 2)/3! x³ + …，在 |x| < 1 时有效。
Remember to factor out constants first, for example (4 + 3x)⁻¹ = 4⁻¹ (1 + 3x/4)⁻¹, before expanding.
展开前记得先提取常数，例如 (4 + 3x)⁻¹ = 4⁻¹ (1 + 3x/4)⁻¹。
To estimate roots or reciprocals, substitute a small value of x into the expansion and state the range of validity clearly.
估算根号或倒数时，将较小的 x 值代入展开式，并明确写出其有效范围。
6. Differentiation | 微分
Chain rule: if y = f(g(x)), then dy/dx = f ‘(g(x)) × g ‘(x). Product rule: if y = uv, then dy/dx = u dv/dx + v du/dx. Quotient rule: if y = u/v, then dy/dx = (v du/dx − u dv/dx) / v².
链式法则：若 y = f(g(x))，则 dy/dx = f ‘(g(x)) × g ‘(x)。乘法法则：若 y = uv，则 dy/dx = u dv/dx + v du/dx。除法法则：若 y = u/v，则 dy/dx = (v du/dx − u dv/dx) / v²。
For parametric differentiation, use dy/dx = (dy/dt) / (dx/dt). For implicit differentiation, differentiate each term with respect to x and collect dy/dx terms.
参数方程求导用 dy/dx = (dy/dt) / (dx/dt)。隐函数求导则对每一项关于 x 求导，并合并含 dy/dx 的项。
Connected rates of change use the chain rule in the form dV/dt = dV/dr × dr/dt, and Edexcel often requires careful unit conversion.
相关变化率问题使用链式法则 dV/dt = dV/dr × dr/dt，Edexcel 常要求仔细进行单位换算。
7. Integration | 积分
Reverse chain rule helps with integrals of the form ∫ f ‘(g(x)) g ‘(x) dx = f(g(x)) + C. Look for a function and its derivative as a factor.
逆链式法则用于 ∫ f ‘(g(x)) g ‘(x) dx = f(g(x)) + C 型积分。要识别一个函数及其导数作为因子的结构。
Definite integrals give the signed area: ∫ₐᵇ f(x) dx = F(b) − F(a). If the curve crosses the x-axis, split the integral to avoid negative areas cancelling.
定积分表示带符号的面积：∫ₐ
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