📚 Edexcel A-Level Pure Maths Core Skills: Algebra, Trigonometry & Calculus | Edexcel A-Level 纯数学核心技能：代数、三角函数与微积分
This revision guide brings together the most important Pure Mathematics skills tested in the Edexcel A-Level Maths specification. It covers algebraic manipulation, quadratics, graphs, trigonometry, exponentials, logarithms, differentiation, integration and sequences. Each section is written as a bilingual pair so you can review the concept in English and check the meaning in Chinese, helping you to build both mathematical fluency and exam confidence.
本复习指南汇总了 Edexcel A-Level 数学大纲中 Pure Mathematics 最重要、最常见的核心技能，涵盖代数运算、二次函数、函数图像、三角函数、指数与对数、微分、积分以及数列。每个小节均采用中英文配对讲解，帮助你同时巩固数学概念和考试应答能力。
1. Laws of Indices and Surds | 指数律与根式
The laws of indices must be automatic in A-Level algebra. For a non-zero base a, the key rules are aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. Fractional powers link directly to roots: a¹/² = √a and a¹/³ = ∛a, so aᵐ/ⁿ = ⁿ√(aᵐ).
在 A-Level 代数中，指数律必须熟练到自动反应。对非零底数 a，核心法则为：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，以及 a⁻ⁿ = 1/aⁿ。分数指数直接联系根式：a¹/² = √a，a¹/³ = ∛a，因此 aᵐ/ⁿ = ⁿ√(aᵐ)。
Surds are irrational roots such as √2 or √3. To simplify a surd, factorise the number under the root and take out square factors: √50 = √(25 × 2) = 5√2. To rationalise a denominator like 1/√3, multiply numerator and denominator by √3; for 1/(2 + √3), multiply by the conjugate 2 − √3.
根式是指如 √2 或 √3 这样的无理根。化简根式时，将被开方数分解因数并提取平方因子：√50 = √(25 × 2) = 5√2。要将分母有理化，例如 1/√3，可分子分母同乘 √3；对于 1/(2 + √3)，则同乘共轭式 2 − √3。
- Simplify fully: (16x⁸)¹/² = 4x⁴ for x ≥ 0.
- 完全化简：(16x⁸)¹/² = 4x⁴，其中 x ≥ 0。
- Rationalise 5/(√5 − 1) by multiplying by √5 + 1 to obtain (5√5 + 5)/4.
- 有理化 5/(√5 − 1)：分子分母同乘 √5 + 1，得到 (5√5 + 5)/4。
2. Quadratic Functions and the Discriminant | 二次函数与判别式
A quadratic function has the form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a parabola: the turning point is a minimum if a > 0 and a maximum if a < 0. Completing the square is used to rewrite the function as a(x + p)² + q, giving the vertex (−p, q).
二次函数的一般形式为 f(x) = ax² + bx + c，其中 a ≠ 0。其图像是一条抛物线：当 a > 0 时顶点为最小值点，当 a < 0 时为最大值点。配方法可将函数改写为 a(x + p)² + q，从而直接读出顶点 (−p, q)。
The discriminant D = b² − 4ac determines the nature of the roots of ax² + bx + c = 0. If D > 0 there are two distinct real roots; if D = 0 there is one repeated real root; if D < 0 there are no real roots. This is especially useful when a quadratic is hidden inside a problem.
判别式 D = b² − 4ac 决定方程 ax² + bx + c = 0 根的性质。若 D > 0，方程有两个不同实根；若 D = 0，有一个重根；若 D < 0，则无实根。当二次方程隐含在综合题中时，判别式尤为有用。
x = (−b ± √(b² − 4ac)) / 2a
This is the quadratic formula. Always write the equation in standard form before identifying a, b and c, and be careful with negative coefficients.
这是求根公式。使用前必须先将方程写成标准形式，再确定 a、b、c，并特别小心负系数。
3. Simultaneous Equations and Inequalities | 联立方程与不等式
For two linear equations, elimination or substitution works well. In harder questions, one equation may be linear and the other quadratic: substitute the linear expression into the quadratic, simplify, and solve the resulting equation. Always substitute back to find the other variable.
两个线性方程组成的方程组可用消元法或代入法求解。较难题目中，通常一个方程是线性的，另一个是二次的：将线性表达式代入二次方程，化简后求解所得方程。最后一定要回代求出另一个变量。
When solving inequalities such as x² − 5x + 6 > 0, first solve the corresponding equation to find critical values x = 2 and x = 3. Then test intervals or sketch the graph to determine where the expression is positive. Use a number line and remember that multiplying or dividing by a negative number reverses the inequality sign.
解不等式如 x² − 5x + 6 > 0 时，先解对应方程得到临界值 x = 2 和 x = 3。然后检验区间或绘制图像，判断表达式在哪些区间为正。使用数轴时切记：乘以或除以负数时，不等式方向要反转。
- Solve x + y = 5 and x² + y² = 25. The line meets the circle at (0, 5) and (5, 0).
- 解方程组 x + y = 5 与 x² + y² = 25。直线与圆相交于 (0, 5) 和 (5, 0)。
- Solve x² − 5x + 6 > 0. The solution set is x < 2 or x > 3.
- 解不等式 x² − 5x + 6 > 0。解集为 x < 2 或 x > 3。
4. Graphs and Transformations | 函数图像与变换
Recognising graph transformations saves time in Edexcel papers. For y = f(x), the transformation y = f(x) + a is a vertical translation by a; y = f(x + a) is a horizontal translation by −a; y = af(x) is a vertical stretch by scale factor a; y = f(ax) is a horizontal stretch by scale factor 1/a. Reflections are given by y = −f(x) and y = f(−x).
识别图像变换可以大幅节省考试时间。对 y = f(x) 而言，y = f(x) + a 表示竖直平移 a 个单位；y = f(x + a) 表示水平平移 −a 个单位；y = af(x) 表示竖直拉伸，比例因子为 a；y = f(ax) 表示水平拉伸，比例因子为 1/a。反射变换为 y = −f(x) 和 y = f(−x)。
Sketching reciprocal and modulus functions is often assessed. The graph of y = 1/x has asymptotes at x = 0 and y = 0, while y = |x| reflects the negative part of y = x in the x-axis. Combining transformations requires the correct order: apply horizontal changes first, then stretches, then translations.
倒数和绝对值函数图像也常考。y = 1/x 的图像有渐近线 x = 0 和 y = 0，而 y = |x| 将 y = x 的负值部分关于 x 轴反射。变换的组合需按正确顺序：先水平变化，再拉伸，最后平移。
- y = 2f(x − 3) means translate 3 units right, then stretch vertically by factor 2.
- y = 2f(x − 3) 表示先向右平移 3 个单位，再竖直拉伸 2 倍。
5. Polynomial Division and the Factor Theorem | 多项式除法与因式定理
Polynomial long division is used when simplifying rational expressions or factorising cubics and quartics. For a polynomial P(x), the factor theorem states that if P(a) = 0, then (x − a) is a factor of P(x). This is often combined with the remainder theorem: when P(x) is divided by (x − a), the remainder is P(a).
多项式长除法用于化简有理式或分解三次、四次多项式。对多项式 P(x)，因式定理指出：若 P(a) = 0，则 (x − a) 是 P(x) 的一个因式。它常与余式定理结合使用：P(x) 除以 (x − a) 的余式为 P(a)。
To factorise a cubic such as x³ − 4x² + x + 6, test small integer values of x. Here P(2) = 8 − 16 + 2 + 6 = 0, so (x − 2) is a factor. Divide to get a quadratic, then factorise further if possible.
要分解三次多项式如 x³ − 4x² + x + 6，可先代入小整数值检验。这里 P(2) = 8 − 16 + 2 + 6 = 0，所以 (x − 2) 是因式。进行长除法得到二次式，再继续分解。
x³ − 4x² + x + 6 = (x − 2)(x² − 2x − 3) = (x − 2)(x + 1)(x − 3)
6. Radian Measure and Trigonometric Identities | 弧度制与三角恒等式
In A-Level trigonometry, angles are usually measured in radians. One full turn is 2π radians, so 180° = π rad. The arc length of a sector is rθ and the area of a sector is ½r²θ, where θ is in radians. These formulas only work with radian measure, so convert degrees when necessary.
A-Level 三角函数中角度通常用弧度制。一整圈为 2π 弧度，因此 180° = π rad。扇形的弧长公式为 rθ，面积为 ½r²θ，其中 θ 必须为弧度。这些公式只适用于弧度制，因此必要时需先将角度制转换为弧度制。
Key identities include sin²θ + cos²θ = 1, tanθ = sinθ/cosθ, and the double angle formulas sin2θ = 2sinθcosθ, cos2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1. These are essential for solving equations and proving identities.
核心恒等式包括 sin²θ + cos²θ = 1、tanθ = sinθ/cosθ，以及倍角公式 sin2θ = 2sinθcosθ、cos2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1。这些在解三角方程和证明恒等式时必不可少。
7. Solving Trigonometric Equations | 解三角方程
To solve equations such as sin x = 0.5 for 0 ≤ x < 2π, first find the principal value using the inverse function: x = π/6. Then use the symmetry of the sine graph or CAST diagram to find the second solution in the interval, x = 5π/6.
解方程 sin x = 0.5 且 0 ≤ x < 2π 时，先用反函数求出主值：x = π/6。然后利用正弦图像的对称性或 CAST 图找出区间内的第二个解 x = 5π/6。
For equations involving cos2θ or tanθ, always consider the period. The function sinθ and cosθ have period 2π, while tanθ has period π. After obtaining a general solution, restrict it to the given interval and write all answers exactly in terms of π where possible.
对于包含 cos2θ 或 tanθ 的方程，务必考虑周期。sinθ 和 cosθ 的周期为 2π，tanθ 的周期为 π。得到通解后，根据给定区间取舍，并尽量用 π 的精确值写出所有答案。
- Solve 2cos x + 1 = 0 for 0 ≤ x ≤ 2π. The solutions are x = 2π/3 and x = 4π/3.
- 解方程 2cos x + 1 = 0，0 ≤ x ≤ 2π。解为 x = 2π/3 和 x = 4π/3。
8. Exponentials and Logarithms | 指数与对数
The exponential function y = eˣ has derivative eˣ and is the inverse of the natural logarithm function y = ln x. The graph of y = eˣ passes through (0, 1) and grows rapidly; y = ln x passes through (1, 0) and is defined only for x > 0.
指数函数 y = eˣ 的导数仍是 eˣ，它是自然对数函数 y = ln x 的反函数。y = eˣ 的图像经过 (0, 1) 且增长极快；y = ln x 经过 (1, 0)，且仅在 x > 0 时有定义。
Logarithm laws are frequently tested: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, and ln(aᵏ) = k ln a. To solve an equation such as 3e²ˣ = 15, first divide by 3, take natural logarithms on both sides, then solve for x.
对数运算律经常考查：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln(aᵏ) = k ln a。解方程如 3e²ˣ = 15 时，先除以 3，再两边取自然对数，最后解出 x。
3e²ˣ = 15 → e²ˣ = 5 → 2x = ln 5 → x = 0.5 ln 5
9. Differentiation: Rules and Applications | 微分法则与应用
The derivative gives the gradient of a curve at any point. For y = xⁿ, the derivative is dy/dx = nxⁿ⁻¹. Use this power rule for polynomials and rational powers such as y = √x = x¹/², whose derivative is dy/dx = ½x⁻¹/².
导数表示曲线上任意一点的斜率。对 y = xⁿ，其导数为 dy/dx = nxⁿ⁻¹。这一幂法则适用于多项式以及有理数幂，如 y = √x = x¹/²，其导数为 dy/dx = ½x⁻¹/²。
The product rule, quotient rule and chain rule are required for more complex functions. If y = uv then dy/dx = u dv/dx + v du/dx; if y = u/v then dy/dx = (v du/dx − u dv/dx)/v²; if y = f(g(x)) then dy/dx = f′(g(x)) g′(x). The chain rule is often used with eˣ, ln x, sin x and cos x.
更复杂的函数需要乘积法则、商法则和链式法则。若 y = uv，则 dy/dx = u dv/dx + v du/dx；若 y = u/v，则 dy/dx = (v du/dx − u dv/dx)/v²；若 y = f(g(x))，则 dy/dx = f′(g(x)) g′(x)。链式法则常与 eˣ、ln x、sin x 和 cos x 一起使用。
Differentiation is used to find tangents, normals, stationary points and rates of change. At a stationary point dy/dx = 0; use the second derivative or sign change test to classify it as a local maximum or minimum.
微分也用于求切线、法线、驻点以及变化率。驻点处 dy/dx = 0；利用二阶导数或一阶导数符号变化判定其为局部极大值还是极小值。
10. Integration: Definite and Indefinite | 积分：不定积分与定积分
Integration is the reverse of differentiation. The indefinite integral of xⁿ is xⁿ⁺¹/(n + 1) + C for n ≠ −1. The constant C must be included in indefinite integrals, but it cancels out in definite integrals.
积分是微分的逆运算。xⁿ 的不定积分为 xⁿ⁺¹/(n + 1) + C，其中 n ≠ −1。不定积分必须加上常数 C，但定积分中常数会相互抵消。
The definite integral ∫ₐᵇ f(x) dx represents the signed area between the curve y = f(x), the x-axis and the lines x = a, x = b. Areas below the x-axis are negative, so split the region or use absolute values when finding total area.
定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x)、x 轴以及直线 x = a、x = b 之间的有符号面积。x 轴下方的面积为负，因此求总面积时应分段或加绝对值。
Integration by substitution and integration by parts are also in the Edexcel pure syllabus. For substitution, choose u = g(x), rewrite the integral in terms of u, integrate, then substitute back. For integration by parts, use ∫ u dv = uv − ∫ v du, usually with u as the function that simplifies when differentiated.
换元积分法和分部积分法也在 Edexcel 纯数学大纲内。换元时选取 u = g(x)，将积分全部用 u 表示，积分后再代回。分部积分用公式 ∫ u dv = uv − ∫ v du，通常选择求导后会变简单的函数作为 u。
11. Sequences and Series: Arithmetic and Geometric | 数列与级数：等差与等比
An arithmetic sequence has a common difference d. The nth term is uₙ = a + (n − 1)d, and the sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l), where l is the last term.
等差数列有公差 d。第 n 项为 uₙ = a + (n − 1)d，前 n 项和为 Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
A geometric sequence has a common ratio r. The nth term is uₙ = arⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. If |r| < 1, the sum to infinity exists and is S∞ = a/(1 − r).
等比数列有公比 r。第 n 项为 uₙ = arⁿ⁻¹。前 n 项和为 Sₙ = a(1 − rⁿ)/(1 − r)，其中 r ≠ 1。若 |r| < 1，无穷级数存在和，且 S∞ = a/(1 − r)。
| Arithmetic | Geometric |
| uₙ = a + (n − 1)d | uₙ = arⁿ⁻¹ |
| Sₙ = n/2 [2a + (n − 1)d] | Sₙ = a(1 − rⁿ)/(1 − r) |
Edexcel questions often ask you to prove a sequence is arithmetic or geometric, or to solve for the first term and common difference/ratio given two pieces of information. Write simultaneous equations from the given terms, then solve them carefully.
Edexcel 考题常要求证明一个数列是等差或等比数列，或根据两个已知信息求首项和公差/公比。应根据已知项列出方程组，再仔细求解。
12. Exam Technique and Common Pitfalls | 考试技巧与常见失分点
In Edexcel A-Level Maths, method marks are awarded for showing clear working, so never skip steps. Label your working with correct notation, keep exact values such as π and √2 instead of rounding too early, and give final answers to the accuracy requested in the question.
在 Edexcel A-Level 数学中，步骤清晰可获得方法分，因此不要跳步。使用正确符号书写过程，保留 π、√2 等精确值，不要过早四舍五入，并按照题目要求给出最终结果的精度。
Common pitfalls include forgetting the + C in indefinite integration, mixing up radian and degree mode on calculators, losing sign errors when subtracting polynomials, and forgetting to check the domain of log functions. Always reread the question to ensure your answer addresses the exact demand.
常见失分点包括：不定积分忘记加常数 C，计算器角度模式在弧度制与角度制之间混淆，多项式相减时出现符号错误，以及忘记检查对数函数的定义域。答题后务必重新审题，确保回答符合题目要求。
Use past-paper questions from Edexcel to practise applying these skills in context. Timed practice under exam conditions is the best way to improve speed and accuracy in Pure Mathematics.
平时可使用 Edexcel 历年真题进行情境化训练。限时模拟考试是提高 Pure Mathematics 解题速度和准确率的最佳方法。
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