📚 Edexcel A-Level Maths FC Paper 158: Pure Core Topic Revision | Edexcel A-Level 数学 FC Paper 158 纯数核心考点复习
This revision guide covers the pure mathematics techniques that are tested most frequently in Edexcel A-Level Mathematics papers such as FC Paper 158. Work through each section as a focused checklist before attempting timed past-paper practice.
本复习指南涵盖 Edexcel A-Level 数学试卷（如 FC Paper 158）中最常考查的纯数技巧。每一节都可以作为限时真题训练前的重点自测清单。
1. Algebraic Techniques | 代数技巧
Confident algebraic manipulation speeds up almost every later question, especially when simplifying surds, rationalising denominators, and applying index laws.
熟练的代数运算能加快几乎所有后续题目的解答，尤其是化简根式、有理化分母和运用指数律。
For example, 8²ᐟ³ is interpreted as (∛8)² = 4, while 1/√3 can be rationalised to √3/3.
例如，8²ᐟ³ 可以理解为 (∛8)² = 4，而 1/√3 可以有理化为 √3/3。
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a² – b² = (a + b)(a – b)
Use these laws before expanding or cancelling so that fractions become simpler rather than more complicated.
在展开或约分之前先使用这些法则，可以让分式变得更简单，而不是更复杂。
2. Functions and Graph Transformations | 函数与图像变换
Functions questions often combine domain, range, inverse functions, and graph transformations in one command.
函数题常把定义域、值域、反函数和图像变换组合在同一个小问中。
To find an inverse, write y = f(x), swap x and y, then rearrange for y. The inverse only exists if the original function is one-to-one.
求反函数时，先写 y = f(x)，交换 x 与 y，再整理出 y。只有原函数是一一对应时反函数才存在。
f(x + a): shift left; f(x – a): shift right; f(x) + b: shift up; f(x) – b: shift down
When combining transformations, apply horizontal changes before vertical changes unless a specific order is stated.
组合变换时，除非题目明确指定顺序，一般先处理水平方向的变化，再处理垂直方向的变化。
3. Quadratics and Inequalities | 二次函数与不等式
The discriminant tells you how many real roots a quadratic has, and completing the square gives both the vertex and the solution set.
判别式可以判断二次方程实根的个数，而配方法既能给出顶点，也能给出解集。
Δ = b² – 4ac
If Δ > 0 there are two distinct real roots, if Δ = 0 there is one repeated root, and if Δ < 0 there are no real roots.
若 Δ > 0，则有两个不同实根；若 Δ = 0，则有一个重根；若 Δ < 0，则没有实根。
For quadratic inequalities such as x² – 5x + 6 > 0, factorise to (x – 2)(x – 3) > 0 and use a sketch to identify x < 2 or x > 3.
对于 x² – 5x + 6 > 0 这类二次不等式，先分解为 (x – 2)(x – 3) > 0，再借助图像确定 x < 2 或 x > 3。
4. Coordinate Geometry and Circles | 坐标几何与圆
Straight-line questions usually require the gradient formula and the perpendicular gradient rule m₁m₂ = -1.
直线题通常需要梯度公式和垂直梯度关系 m₁m₂ = -1。
A perpendicular bisector passes through the midpoint and has gradient equal to the negative reciprocal of the original segment.
垂直平分线经过中点，并且其梯度是原线段梯度的负倒数。
(x – a)² + (y – b)² = r²
For circle tangents, use the fact that the radius to the point of contact is perpendicular to the tangent line.
对于圆的切线，要利用切点半径与切线垂直这一性质。
5. Trigonometry and Identities | 三角函数与恒等式
Trigonometry questions require you to move between degrees and radians comfortably and to solve equations within a given interval.
三角题要求你能够在角度制与弧度制之间自如转换，并在给定区间内解三角方程。
sin²θ + cos²θ = 1; tanθ = sinθ / cosθ
Use the unit circle or CAST diagram to find all solutions in the required range, rather than stopping at the first calculator answer.
使用单位圆或 CAST 图可以找到所需范围内的所有解，而不是只停留在计算器给出的第一个答案。
For example, solving sinθ = 0.5 gives θ = π/6 and θ = 5π/6 in the interval 0 ≤ θ < 2π.
例如，在 0 ≤ θ < 2π 内解 sinθ = 0.5，得到 θ = π/6 与 θ = 5π/6。
6. Exponentials and Logarithms | 指数与对数
Exponential and logarithmic questions often require you to apply the laws of logarithms in reverse to combine or separate terms.
指数与对数题常要求逆向使用对数律来合并或拆分项。
logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx – logₐy; logₐxⁿ = n logₐx
To solve e²ˣ = 5, take the natural logarithm of both sides to get 2x = ln 5, then divide by 2.
解 e²ˣ = 5 时，两边取自然对数得到 2x = ln 5，再除以 2。
When modelling growth or decay, identify the initial value, the growth factor, and any boundary condition carefully.
建立增长或衰减模型时，要仔细识别初始值、增长因子和任何边界条件。
7. Differentiation and Tangents | 微分与切线
Differentiation questions frequently combine power-rule derivatives with tangent equations and stationary-point analysis.
微分题常将幂函数求导、切线方程和驻点分析组合在一起。
d/dx (xⁿ) = nxⁿ⁻¹
For a curve y = f(x), the tangent at x = a has gradient f'(a) and passes through the point (a, f(a)).
对于曲线 y = f(x)，在 x = a 处的切线梯度为 f'(a)，且切线经过点 (a, f(a))。
At a stationary point f'(x) = 0; use the sign of f'(x) on either side or the second derivative to classify it.
驻点处 f'(x) = 0；通过考察两侧 f'(x) 的符号或使用二阶导数来判断其类型。
8. Integration and Area | 积分与面积
Integration is the reverse of differentiation and is used to find areas, antiderivatives, and original functions from gradient information.
积分是微分的逆运算，用于求面积、原函数以及由梯度信息还原函数。
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, n ≠ -1
Remember to include the constant of integration for indefinite integrals, but remove it when evaluating a definite integral.
不定积分要加上积分常数，而在计算定积分时则不需要写出该常数。
Area under a curve between x = a and x = b is given by ∫ₐᵇ f(x) dx, provided the curve is above the x-axis on that interval.
若曲线在区间 [a, b] 上位于 x 轴上方，则曲线与 x 轴之间的面积由 ∫ₐᵇ f(x) dx 给出。
9. Sequences and Series | 数列与级数
Arithmetic series questions usually require you to identify the first term a and common difference d before applying the sum formula.
等差数列题通常需要先确定首项 a 和公差 d，再应用求和公式。
Sₙ = n/2 [2a + (n – 1)d]
Geometric series questions use the common ratio r, with the sum to infinity valid only when |r| < 1.
等比数列题使用公比 r，且仅当 |r| < 1 时无穷级数求和公式才有效。
S∞ = a / (1 – r), |r| < 1
Check whether a sequence is arithmetic or geometric before applying a formula; mixed patterns are often tested.
使用公式前要先判断数列是等差还是等比；混合型规律也常作为考点。
10. Binomial Expansion and Proof | 二项展开与证明
Binomial expansion questions often require expanding powers up to x³ and identifying the coefficient of a particular term.
二项展开题常要求展开到 x³ 项，并找出某一项的系数。
(1 + x)ⁿ = 1 + nx + [n(n – 1)/2!]x² + [n(n – 1)(n – 2)/3!]x³ + …
For rational n, the expansion is valid when |x| < 1, and terms must be in ascending powers of x.
当 n 为有理数时，展开式在 |x| < 1 时有效，且各项需按 x 的升幂排列。
Proof questions may require exhaustion, counterexample, or direct algebraic deduction; always state your conclusion clearly.
证明题可能要求穷举法、反例或直接代数推导；最后要清晰地写出结论。
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