📚 AQA A Level Maths: Core Pure Topics Revision | AQA A-Level 数学：核心纯数主题复习
This revision guide covers the core pure mathematics topics that are assessed in AQA A-level Mathematics. It is designed to help you consolidate key formulae, methods and exam techniques before your final assessment.
本复习指南涵盖 AQA A-level 数学中考查的核心纯数主题。它旨在帮助你在最终评估前巩固关键公式、方法和考试技巧。
1. Algebraic Manipulation and Proof | 代数运算与证明
You must be confident simplifying expressions with indices, surds and algebraic fractions. These skills underpin nearly every pure maths question, so practise them until they become automatic.
你必须熟练掌握含有指数、根式和代数分式的表达式化简。这些技能是几乎所有纯数题目的基础，因此要反复练习直到能够自动运用。
aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ
Proof questions often ask you to show that a statement is true by deduction or exhaustion. State each step clearly and justify algebraic moves such as factorising or completing the square.
证明题通常要求你通过演绎法或穷举法证明某个命题为真。每一步都要写清楚，并说明因式分解或配方法等代数操作的依据。
When simplifying an algebraic fraction, factorise the numerator and denominator first, then cancel common factors. Be careful when cancelling: you can only cancel factors, not individual terms.
化简代数分式时，先对分子和分母进行因式分解，然后约去公因式。约分时要特别小心：只能约去因式，不能约去单独的项。
2. Functions and Graphs | 函数与图像
Understand the formal definition of a function: each input has exactly one output. For composite functions fg(x), apply the inner function g first, then the outer function f.
理解函数的正式定义：每个输入恰好对应一个输出。对于复合函数 fg(x)，先应用内层函数 g，再应用外层函数 f。
fg(x) = f(g(x)), f⁻¹(f(x)) = x
c) + d, first deal with the horizontal shift from c, then horizontal stretch from b, then vertical stretch from a, and finally vertical shift from d.
图像变换必须按正确顺序进行。对于 y = af(bx + c) + d，先处理由 c 引起的水平平移，再处理由 b 引起的水平伸缩，然后是由 a 引起的垂直伸缩，最后是由 d 引起的垂直平移。
The modulus function y = |f(x)| reflects any negative part of the graph in the x-axis, while y = f(|x|) reflects the right-hand side in the y-axis.
模函数 y = |f(x)| 将图像中位于 x 轴下方的部分反射到 x 轴上方，而 y = f(|x|) 将右侧图像反射到 y 轴左侧。
3. Coordinate Geometry | 坐标几何
he gradient. Parallel lines have equal gradients, and perpendicular lines have gradients whose product is -1.
直线方程可以写作 y - y₁ = m(x - x₁)，其中 m 是斜率。平行直线斜率相等，垂直直线的斜率乘积为 -1。
d = √((x₂ - x₁)² + (y₂ - y₁)²), midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
². To find a tangent or normal at a point, use the radius gradient and the perpendicular gradient relationship.
圆心为 (a, b)、半径为 r 的圆方程是 (x - a)² + (y - b)² = r²。要找到某一点处的切线或法线，可利用半径斜率和垂直斜率关系。
of a third variable, usually t. To find the gradient, calculate dy/dx = (dy/dt) ÷ (dx/dt).
参数方程用第三个变量（通常为 t）表示 x 和 y。要求梯度，可计算 dy/dx = (dy/dt) ÷ (dx/dt)。
4. Trigonometry | 三角学
les 0°, 30°, 45°, 60° and 90°. Work in radians for calculus questions, where 180° = π rad.
牢记 0°、30°、45°、60° 和 90° 角的正弦、余弦和正切精确值。在微积分题目中使用弧度制，其中 180° = π 弧度。
sin²θ + cos²θ ≡ 1, tanθ ≡ sinθ / cosθ
Solve trigonometric equations by first isolating the trig function, then finding the reference angle, and finally using the symmetrical properties of the sine, cosine and tangent graphs to list all solutions in the required interval.
解三角方程时，先分离出三角函数，然后求出参考角，最后利用正弦、余弦和正切图像的对称性列出所求区间内的所有解。
For a sector of a circle, the arc length is rθ and the area is ½r²θ, where θ is measured in radians.
对于扇形，弧长为 rθ，面积为 ½r²θ，其中 θ 以弧度为单位。
5. Exponentials and Logarithms | 指数与对数
The natural exponential function eˣ and natural logarithm ln x are inverse operations. This means ln(eˣ) = x for all real x, and eln x = x for x > 0.
自然指数函数 eˣ 与自然对数 ln x 互为逆运算。这意味着对所有实数 x，ln(eˣ) = x；对 x > 0，eln x = x。
ln eˣ = x, eln x = x, logₐx = ln x / ln a
To solve equations such as 3ˣ = 20, take logs of both sides to get x ln
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