📚 Edexcel A-Level Mathematics: Core Skills for Mixed Practice Set pdfjoiner_(4)-228 | Edexcel A-Level 数学：混合练习组 pdfjoiner_(4)-228 核心技能
This article revises the essential techniques that regularly appear in Edexcel A-Level Mathematics mixed question sets, including the one labelled pdfjoiner_(4)-228. It is designed to help you consolidate pure mathematics, statistics and mechanics before the final examination.
本文复习 Edexcel A-Level 数学混合题组（包括标记为 pdfjoiner_(4)-228 的练习）中经常出现的核心技巧，帮助你在考前巩固纯数学、统计和力学。
1. Algebraic Methods and Binomial Expansion | 代数方法与二项式展开
In Edexcel A-Level Pure Mathematics, you must be able to simplify rational expressions, use the factor theorem, and expand binomial expressions of the form (1 + x)ⁿ for rational n when |x| < 1.
在 Edexcel A-Level 纯数学中，你必须会化简有理式、使用因式定理，并在 |x| < 1 时展开形如 (1 + x)ⁿ 的二项式，其中 n 为有理数。
The binomial expansion is (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + …
二项展开式为 (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + n(n − 1)(n − 2)x³/3! + …
- English: The expansion is valid only when |x| < 1 for non-integer n. Chinese: 当 n 不是整数时，该展开仅在 |x| < 1 时成立。
- English: When n is a positive integer, the expansion terminates after n + 1 terms. Chinese: 当 n 为正整数时，展开式在 n + 1 项后终止。
- English: Use partial fractions before expanding rational expressions such as 1/((1 + x)(2 − x)). Chinese: 展开形如 1/((1 + x)(2 − x)) 的有理式前，先分解为部分分式。
2. Functions, Graphs and Transformations | 函数、图像与变换
Edexcel questions often ask you to identify the domain and range of a function, find an inverse function, or sketch transformations of a given graph.
Edexcel 试题经常要求确定函数的定义域和值域、求反函数，或画出给定图像经过变换后的草图。
For a one-to-one function f(x), the inverse is found by writing y = f(x), swapping x and y, and then solving for y. The graph of f⁻¹(x) is a reflection of y = f(x) in the line y = x.
对于一一对应的函数 f(x)，求反函数的方法是先写 y = f(x)，交换 x 与 y，再解出 y。反函数 f⁻¹(x) 的图像是 y = f(x) 关于直线 y = x 的对称图形。
Common transformations include y = f(x) + a, y = f(x + a), y = a f(x) and y = f(ax), where a is a constant.
常见变换包括 y = f(x) + a、y = f(x + a)、y = a f(x) 和 y = f(ax)，其中 a 为常数。
- English: y = f(x) + a shifts the graph vertically by a units. Chinese: y = f(x) + a 将图像竖直平移 a 个单位。
- English: y = f(x + a) shifts the graph horizontally by −a units. Chinese: y = f(x + a) 将图像水平平移 −a 个单位。
- English: y = a f(x) stretches the graph vertically by scale factor a. Chinese: y = a f(x) 将图像沿竖直方向伸缩 a 倍。
3. Trigonometric Identities and Equations | 三角恒等式与方程
Trigonometry is a core area in Edexcel A-Level Mathematics. You need to know exact values for 0°, 30°, 45°, 60° and 90°, and be able to solve equations in a given interval.
三角学是 Edexcel A-Level 数学的核心内容。你需要记住 0°、30°、45°、60° 和 90° 的精确值，并能在给定区间内解三角方程。
sin²θ + cos²θ = 1
sin²θ + cos²θ = 1
tan θ = sin θ / cos θ
tan θ = sin θ / cos θ
The addition formulae are sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B. These are useful for proving identities and solving equations such as sin(θ + 30°) = cos θ.
和角公式为 sin(A ± B) = sin A cos B ± cos A sin B 以及 cos(A ± B) = cos A cos B ∓ sin A sin B。这些公式常用于证明恒等式和解形如 sin(θ + 30°) = cos θ 的方程。
- English: Always check for extra solutions in the required interval after using inverse trigonometric functions. Chinese: 使用反三角函数后，务必在给定区间内检查是否有多余解。
- English: Write equations in the form sin θ = k, cos θ = k or tan θ = k before using the unit circle. Chinese: 先化成 sin θ = k、cos θ = k 或 tan θ = k 的形式，再用单位圆求解。
4. Exponentials and Logarithms | 指数与对数
Edexcel A-Level pure mathematics requires fluent use of exponential and logarithmic functions, including the natural exponential eˣ and natural logarithm ln x.
Edexcel A-Level 纯数学要求熟练运用指数函数与对数函数，包括自然指数 eˣ 和自然对数 ln x。
| English rule | 中文法则 |
| ln(ab) = ln a + ln b | ln(ab) = ln a + ln b |
| ln(a/b) = ln a − ln b | ln(a/b) = ln a − ln b |
| ln aⁿ = n ln a | ln aⁿ = n ln a |
A common exam skill is to take logarithms of both sides to solve equations such as 3ˣ = 5ˣ⁻¹ or to model exponential growth and decay.
常见考试技巧是对等号两边取对数，以解形如 3ˣ = 5ˣ⁻¹ 的方程，或建立指数增长与衰减模型。
The derivative of eˣ is eˣ, and the derivative of ln x is 1/x. The integral of 1/x with respect to x is ln|x| + C.
eˣ 的导数是 eˣ，ln x 的导数是 1/x。1/x 对 x 的积分是 ln|x| + C。
5. Differentiation Techniques | 微分技巧
Differentiation is tested through tangents, normals, stationary points, optimisation and rates of change. You need to apply the product rule, quotient rule and chain rule confidently.
微分通过切线、法线、驻点、优化问题和变化率进行考查。你需要熟练运用乘法法则、除法法则和链式法则。
d/dx (xⁿ) = nxⁿ⁻¹
d/dx (xⁿ) = nxⁿ⁻¹
d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x
d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x
The product rule is d/dx (uv) = u dv/dx + v du/dx. The quotient rule is d/dx (u/v) = (v du/dx − u dv/dx) / v².
乘法法则为 d/dx (uv) = u dv/dx + v du/dx。除法法则为 d/dx (u/v) = (v du/dx − u dv/dx) / v²。
The chain rule is d/dx [f(g(x))] = f′(g(x)) g′(x). Use it for composite functions such as sin(3x), ln(x² + 1) and e²ˣ.
链式法则为 d/dx [f(g(x))] = f′(g(x)) g′(x)。将其用于 sin(3x)、ln(x² + 1) 和 e²ˣ 等复合函数。
- English: At a stationary point, dy/dx = 0. Use the second derivative to classify maxima and minima. Chinese: 在驻点处 dy/dx = 0。用二阶导数判断极大值与极小值。
- English: For a tangent, use y − y₁ = m(x − x₁), where m is the gradient. Chinese: 求切线时用 y − y₁ = m(x − x₁)，其中 m 为斜率。
6. Integration and Area Under Curves | 积分与曲线下面积
Integration is the reverse of differentiation. Edexcel questions test indefinite integrals, definite integrals, area between curves and the trapezium rule.
积分是微分的逆运算。Edexcel 试题考查不定积分、定积分、曲线之间的面积以及梯形法则。
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C，n ≠ −1
∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C
The area between a curve y = f(x), the x-axis and the lines x = a and x = b is given by ∫ₐᵇ f(x) dx, provided the function does not cross the x-axis in [a, b].
曲线 y = f(x)、x 轴以及直线 x = a 与 x = b 所围成的面积由 ∫ₐᵇ f(x) dx 给出，前提是函数在 [a, b] 内不穿过 x 轴。
If the curve lies below the x-axis, take the absolute value of the integral. For area between two curves, integrate the difference f(x) − g(x).
如果曲线位于 x 轴下方，则对积分取绝对值。求两条曲线之间的面积时，对差 f(x) − g(x) 进行积分。
- English: Always add the constant of integration for indefinite integrals. Chinese: 不定积分务必加上积分常数 C。
- English: Use integration by substitution or integration by parts when the integrand is a product or composite function. Chinese: 当被积函数为乘积或复合函数时，使用换元积分法或分部积分法。
7. Vectors and 3D Coordinate Geometry | 向量与三维坐标几何
Vector questions in Edexcel A-Level Mathematics include finding magnitudes, unit vectors, scalar products and angles between vectors in two or three dimensions.
Edexcel A-Level 数学中的向量题包括求模长、单位向量、数量积以及二维或三维向量之间的夹角。
|a| = √(x² + y² + z²)
|a| = √(x² + y² + z²)
The scalar product is a·b = |a||b| cos θ, where θ is the angle between a and b. In component form, a·b = x₁x₂ + y₁y₂ + z₁
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