📚 Edexcel A-Level Pure Maths Core Revision | 爱德思A-Level纯数学核心复习
This revision guide covers the essential pure mathematics topics assessed in the Edexcel A-Level Mathematics specification. It is designed to help you consolidate key methods, avoid common errors, and approach exam questions with confidence.
本复习指南涵盖爱德思 A-Level 数学考试大纲中的纯数学核心主题，旨在帮助你巩固关键方法、避免常见错误，并自信地应对考试题目。
1. Algebraic Expressions and Indices | 代数表达式与指数
To simplify expressions with indices, remember the core rules: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. Negative and fractional indices are also common at A-Level, where a⁻ⁿ = 1/aⁿ and a^(1/n) = √[n]a.
化简含指数的表达式时，要记住核心法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，以及 (aᵐ)ⁿ = aᵐⁿ。负指数和分数指数在 A-Level 中也很常见，其中 a⁻ⁿ = 1/aⁿ，a^(1/n) = √[n]a。
When factorising cubic or quartic expressions, look for a common factor first, then use factor theorem or grouping. Edexcel questions often ask you to factorise fully, so check each bracket for further common factors.
因式分解三次或四次表达式时，先寻找公因式，然后使用因式定理或分组法。爱德思考题经常要求完全因式分解，因此要检查每个括号是否还能提取公因式。
(x + a)(x + b) = x² + (a + b)x + ab
2. Quadratics and the Discriminant | 二次函数与判别式
The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves ax² + bx + c = 0. The discriminant Δ = b² − 4ac tells you the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots.
二次公式 x = (−b ± √(b² − 4ac)) / 2a 可以解方程 ax² + bx + c = 0。判别式 Δ = b² − 4ac 说明根的性质：Δ > 0 有两个不同实根，Δ = 0 有一个重实根，Δ < 0 没有实根。
Completing the square is another key technique. Write ax² + bx + c in the form a(x + p)² + q, which helps find the vertex of a parabola and solve quadratic equations when the coefficient of x² is not 1.
配方法是另一项关键技巧。将 ax² + bx + c 写成 a(x + p)² + q 的形式，有助于求抛物线的顶点，并在 x² 的系数不为 1 时解二次方程。
x² + 6x + 5 = (x + 3)² − 4
3. Equations and Inequalities | 方程与不等式
Linear inequalities are solved like equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number. For example, from −2x > 6 you get x < −3.
解线性不等式与解方程类似，但乘以或除以负数时要反转不等号。例如，由 −2x > 6 可得 x < −3。
Quadratic inequalities require a sign diagram or a sketch of the graph. After finding the critical values, test intervals to decide where the quadratic is positive or negative. Always answer in set notation or interval form if specified.
二次不等式需要符号表或图像草图。求出临界值后，检验各区间的正负。如果题目要求，请用集合符号或区间形式作答。
Simultaneous equations often involve one linear and one quadratic equation. Substitute the linear expression into the quadratic, solve the resulting quadratic, and then find the corresponding values of the other variable.
联立方程通常包含一个线性方程和一个二次方程。将线性表达式代入二次方程，解所得二次方程，再求另一个变量的对应值。
4. Graphs and Transformations | 图像与变换
Understanding graph transformations is essential for sketching curves quickly. The transformation y = f(x) + a translates the graph vertically by a units, while y = f(x + a) translates it horizontally by −a units.
理解图像变换对于快速画曲线至关重要。变换 y = f(x) + a 将图像竖直平移 a 个单位，而 y = f(x + a) 将其水平平移 −a 个单位。
Reflections are given by y = −f(x) for reflection in the x-axis and y = f(−x) for reflection in the y-axis. Stretches use y = af(x) and y = f(ax), where a > 1 stretches the graph parallel to the axes.
反射包括关于 x 轴的 y = −f(x) 和关于 y 轴的 y = f(−x)。伸缩变换使用 y = af(x) 和 y = f(ax)，当 a > 1 时图像沿坐标轴方向拉伸。
| Transformation | Effect on y = f(x) |
|---|---|
| Translation up by a | y = f(x) + a |
| Translation left by a | y = f(x + a) |
| Stretch vertically by scale factor a | y = af(x) |
| Stretch horizontally by scale factor 1/a | y = f(ax) |
5. Exponential and Logarithmic Functions | 指数与对数函数
The natural exponential function eˣ and natural logarithm ln x are inverses of each other. Key properties include ln(eˣ) = x for all real x and e^(ln x) = x for x > 0.
自然指数函数 eˣ 和自然对数 ln x 互为反函数。关键性质包括对所有实数 x 有 ln(eˣ) = x，且对 x > 0 有 e^(ln x) = x。
Logarithm laws are vital for solving exponential equations: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, and logₐ(xⁿ) = n logₐx. These laws work for any valid base a.
对数法则对于解指数方程至关重要：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，以及 logₐ(xⁿ) = n logₐx。这些法则适用于任何有效底数 a。
e^(ln 5) = 5 and ln(e²) = 2
Exponential growth and decay models have the form y = A e^(kt). If k > 0 the model represents growth, and if k < 0 it represents decay. You may need to find k from given data using logarithms.
指数增长和衰减模型的形式为 y = A e^(kt)。若 k > 0 表示增长，若 k < 0 表示衰减。你可能需要用对数从给定数据中求出 k。
6. Trigonometric Identities and Equations | 三角恒等式与方程
The fundamental identity sin²θ + cos²θ ≡ 1 is used constantly in A-Level trigonometry. Dividing through by cos²θ or sin²θ gives the derived identities 1 + tan²θ ≡ sec²θ and 1 + cot²θ ≡ cosec²θ.
基本恒等式 sin²θ + cos²θ ≡ 1 在 A-Level 三角学中经常使用。将等式两边同除以 cos²θ 或 sin²θ，可以得到导出恒等式 1 + tan²θ ≡ sec²θ 和 1 + cot²θ ≡ cosec²θ。
When solving trigonometric equations, find all solutions in the given interval. Use the period of the function and symmetry properties such as sin(180° − θ) = sin θ and cos(−θ) = cos θ.
解三角方程时，要找出给定区间内的所有解。利用函数的周期性和对称性，如 sin(180° − θ) = sin θ 和 cos(−θ) = cos θ。
sin 30° = ½, cos 60° = ½, tan 45° = 1
Exact values for 0°, 30°, 45°, 60° and 90° must be memorised. They allow you to evaluate trigonometric expressions without a calculator and often appear in proof or equation questions.
必须记住 0°、30°、45°、60° 和 90° 的精确值。这些值让你不用计算器就能计算三角表达式，并经常出现在证明或方程题中。
7. Differentiation Techniques | 微分技巧
The power rule states that if y = xⁿ, then dy/dx = nxⁿ⁻¹. This rule extends to terms with negative and fractional powers, so it is useful for differentiating expressions like 1/x² or √x.
幂法则指出，若 y = xⁿ，则 dy/dx = nxⁿ⁻¹。该法则适用于负指数和分数指数，因此对 1/x² 或 √x 这类表达式求导非常有用。
The chain rule is needed for composite functions: if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). It is often written as dy/dx = dy/du × du/dx by setting u = g(x).
复合函数求导需要使用链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。通常设 u = g(x)，写成 dy/dx = dy/du × du/dx。
The product rule and quotient rule are used for products and quotients of functions. Product rule: if y = uv, then dy/dx = u’v + uv’. Quotient rule: if y = u/v, then dy/dx = (u’v − uv’) / v².
乘积法则和商法则用于函数的乘积和商。乘积法则：若 y = uv，则 dy/dx = u’v + uv’。商法则：若 y = u/v，则 dy/dx = (u’v − uv’) / v²。
d/dx [sin(2x)] = 2cos(2x)
8. Integration and Area Under Curves | 积分与曲线下面积
Integration reverses differentiation. The basic power rule for integration is ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, where n ≠ −1. Do not forget the constant of integration for indefinite integrals.
积分是微分的逆运算。基本的幂函数积分法则是 ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C，其中 n ≠ −1。不定积分不要忘记加积分常数。
Definite integrals give the exact area between a curve and the x-axis. If the curve lies below the x-axis, the integral is negative, so use absolute values or split the interval to find total area.
定积分给出曲线与 x 轴之间的精确面积。如果曲线在 x 轴下方，积分为负，因此求总面积时要取绝对值或分割区间。
∫₀¹ 2x dx = [x²]₀¹ = 1 − 0 = 1
Standard integrals include ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C, ∫ cos x dx = sin x + C, and ∫ sin x dx = −cos x + C. These are frequently combined with substitution or reverse chain rule.
标准积分包括 ∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C，∫ cos x dx = sin x + C，以及 ∫ sin x dx = −cos x + C。它们经常与换元法或反向链式法则结合使用。
9. Vectors in Pure Mathematics | 纯数学中的向量
A vector has both magnitude and direction. In two dimensions, vectors can be written as column vectors or in terms of unit vectors i and j. For example, the vector (3, 4) equals 3i + 4j and has magnitude √(3² + 4²) = 5.
向量既有大小又有方向。在二维空间中，向量可以写成列向量或用单位向量 i 和 j 表示。例如，向量 (3, 4) 等于 3i + 4j，其模长为 √(3² + 4²) = 5。
The position vector of a point P relative to an origin O is written as OP = xi + yj. The vector from point A to point B is given by AB = OB − OA, where OA and OB are position vectors.
点 P 相对于原点 O 的位置向量写作 OP = xi + yj。从点 A 到点 B 的向量由 AB = OB − OA 给出，其中 OA 和 OB 是位置向量。
Parallel vectors are scalar multiples of each other. Two vectors are perpendicular if their dot product is zero: u · v = u₁v₁ + u₂v₂ = 0 in two dimensions.
平行向量互为标量倍数。两个向量垂直当且仅当它们的点积为零：在二维中 u · v = u₁v₁ + u₂v₂ = 0。
10. Sequences and Series | 数列与级数
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 是末项。
A geometric sequence has a common ratio r. The nth term is uₙ = arⁿ⁻¹, and the sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. An infinite geometric series converges to S∞ = a/(1 − r) provided |r| < 1.
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。当 |r| < 1 时，无穷等比级数收敛到 S∞ = a/(1 − r)。
S∞ = a / (1 − r), for |r| < 1
Sigma notation Σ is used to write series compactly. For example, Σ from k=1 to n of (2k + 1) means the sum of all terms 2k + 1 for k = 1, 2, …, n. Break the sum into simpler parts when evaluating.
西格玛符号 Σ 用于简洁地表示级数。例如，Σ 从 k=1 到 n 的 (2k + 1) 表示将 k = 1, 2, …, n 时所有 2k + 1 项相加。计算时可将其拆分为更简单的部分。
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