📚 Paper 158: Comprehensive Pure Mathematics Revision | 纯数学综合复习指南
Welcome to this in-depth revision guide tailored for Edexcel A-Level Mathematics Paper 158. This paper tests your ability to apply pure mathematical concepts across algebra, trigonometry, calculus, vectors, sequences, and exponentials/logarithms. We break down each major topic, highlight typical question styles, and provide systematic strategies to ensure you are fully prepared for every challenge the paper may present. Use the paired explanations to strengthen your bilingual understanding of key ideas.
欢迎阅读这篇专为 Edexcel A-Level 数学卷 158 编写的深度复习指南。本试卷考查你在代数、三角学、微积分、向量、数列以及指数对数等领域应用纯数学概念的能力。我们将逐一解析各大主题，点明典型题型，并提供系统化的解题策略，确保你能够从容应对试卷中的任何挑战。通过中英配对讲解，你可以进一步巩固对核心概念的双语理解。
1. Algebraic Manipulation and Polynomials | 代数运算与多项式
Strong algebraic foundations are essential. Be prepared to factorise cubic and quartic expressions, apply the factor theorem, and simplify rational expressions. Polynomial division (long division or synthetic division) often appears in the context of finding remainders and factors.
扎实的代数基础至关重要。你需要熟练地进行三次和四次多项式的因式分解，运用因式定理，并化简有理表达式。多项式除法（长除法或综合除法）常出现在求余式和因式的题目中。
found, divide to obtain a quadratic, then factorise further or use the quadratic formula.
务必先检查是否存在公因式。对三次多项式 f(x) 进行因式分解时，可以尝试常数项的约数（±1、±2、±3）代入验证 f(a)=0。找到一个因式 (x − a) 后，通过除法得到一个二次式，再进一步分解或使用求根公式。
For rational expressions, simplify by factorising numerator and denominator first. State any restrictions on the variable to avoid division by zero. Partial fractions are frequently tested when integrating, so keep them in mind.
对于有理式，先对分子分母分别进行因式分解再化简，并写出使分母不为零的变量限制条件。部分分式在积分中经常考查，需要多加留意。
2. Quadratic Functions and Their Graphs | 二次函数及其图像
tex (h, k) gives the maximum or minimum point, and the line x = h is the axis of symmetry.
二次函数是许多建模题的基础。你需要熟练地在标准式 y = ax² + bx + c 与完全平方式 y = a(x − h)² + k 之间进行转换。顶点 (h, k) 表示最大值点或最小值点，直线 x = h 是对称轴。
e is a repeated real root (line is tangent to curve); if Δ < 0 there are no real roots. Interpret this geometrically when solving intersections of a line and a quadratic curve.
判别式 Δ = b² − 4ac 决定了根的性质：Δ > 0 表示有两个不相等的实根；Δ = 0 表示一个重根（直线与曲线相切）；Δ < 0 表示没有实根。在求解直线与二次曲线的交点时，要用几何意义加以解释。
Be comfortable solving inequalities such as ax² + bx + c > 0 by considering the graph's position relative to the x-axis. Always sketch a quick graph or use a sign diagram to avoid common mistakes.
要能熟练求解形如 ax² + bx + c > 0 的不等式，结合图像相对于 x 轴的位置来判断。应快速画出草图或使用符号表，以避免常见错误。
3. Equations and Inequalities | 方程与不等式
Linear simultaneous equations may be extended to one linear and one quadratic. Solve by substitution: express one variable in terms of the other from the linear equation and substitute into the quadratic. Expect to obtain a quadratic in one variable, which you then solve to find two pairs of solutions.
联立方程组可能从一次联立扩展为“一个一次方程 + 一个二次方程”的形式。通过代入法求解：从一次方程中用一个变量表示另一个变量，再代入二次方程。通常会得到一个关于单一变量的二次方程，进而解出两组解。
Inequalities involving quadratic expressions or rational functions require careful sign analysis. For f(x)/g(x) > 0, identify critical values where the expression equals zero or is undefined, and test intervals. Never multiply across an inequality by a denominator whose sign is unknown.
含有二次式或有理不等式的题目需要仔细分析符号。对于 f(x)/g(x) > 0，先找出使表达式等于零或无定义的临界值，再对各区间进行测试。切记不要对符号未知的分母直接去分母。
4. Coordinate Geometry and Circles | 坐标几何与圆
o return to centre-radius form. Questions often ask for the centre and radius, or ask you to prove that a line is tangent.
圆心为 (a, b)、半径为 r 的圆的方程是 (x − a)² + (y − b)² = r²。将其展开成一般式 x² + y² + Dx + Ey + F = 0 后，可以通过配方法还原成标准式。考题常常要求找出圆心和半径，或者证明某直线与圆相切。
he discriminant Δ = 0. Use the geometry of tangents and chords to find lengths and angles.
若直线与圆相切，则圆心到该直线的垂直距离等于半径。另一种方法是把直线方程代入圆的方程，然后令判别式 Δ = 0。要善于运用切线和弦的几何性质来求长度和角度。
5. Trigonometry: Ratios, Identities, and Equations | 三角学：比值、恒等式与方程
Master the graphs of sin θ, cos θ, and tan θ, including their symmetries, periods (360° for sin/cos, 180° for tan), and transformations. Exact values for 0°, 30°, 45°, 60°, 90° must be at your fingertips.
熟练掌握 sin θ、cos θ 和 tan θ 的图像，包括对称性、周期（sin 和 cos 为 360°，tan 为 180°）以及图像变换。30°、45°、60° 等特殊角的精确三角函数值要烂熟于心。
Key identities: sin²θ + cos²θ ≡ 1, and tan θ ≡ sin θ / cos θ. Use these to simplify expressions, solve equations, or prove other identities. Always begin by expressing everything in terms of sin and cos if stuck.
重要恒等式：sin²θ + cos²θ ≡ 1 和 tan θ ≡ sin θ / cos θ。用它们化简表达式、解方程或证明其他恒等式。若遇到困难，可先化为 sin 与 cos 的形式。
emember to adjust for multiples of θ, for example if solving sin(2θ) = 0.5, find 2θ first.
解三角方程时，先用反三角函数求出主值，再借助 CAST 图或图像对称性找出给定区间内所有的解。若方程中含有倍角，如 sin(2θ) = 0.5，应先解出 2θ 再转换。
6. Exponentials and Logarithms | 指数函数与对数函数
The function y = aˣ (with a > 0) and its special case y = eˣ its inverse. Key properties: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln(aⁿ) = n ln a.
函数 y = aˣ（a > 0）及其特殊情况 y = eˣ 是增长和衰减模型的基础。自然对数 y = ln x 是其反函数。核心性质：ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln(aⁿ) = n ln a。
use the method of writing both sides with the same base if possible. For equations like e²ˣ − 5eˣ + 6 = 0, substitute u = eˣ to form a quadratic in u, solve, then return to x via logs.
解指数方程时，可以对方程两边同时取自然对数，或者在可能时将两边化为同底数。遇到 e²ˣ − 5eˣ + 6 = 0 这类方程，可设 u = eˣ 得到 u 的二次方程，解出 u 后再通过对数反求出 x。
inearisation using logs is a key skill: if a relationship is of the form y = kxⁿ or y = abˣ, taking logs produces a linear graph from which constants can be estimated.
用对数实现线性化是一项关键技能：若关系式为 y = kxⁿ 或 y = abˣ，取对数后可得到线性图像，从而估算出常数。
7. Differentiation: Rules and Applications | 微分法：法则与应用
Thoroughly know the derivatives of standard functions: xⁿ → n xⁿ⁻¹, eˣ → eˣ quotient rule must be applied accurately and in the correct order for composite functions.
牢记基本函数的导数：xⁿ → n xⁿ⁻¹、eˣ → eˣ、ln x → 1/x、sin x → cos x、cos x → −sin x。对于复合函数，要准确使用链式法则、乘积法则和商法则，并注意顺序。
ion is used to find gradients of tangents and normals. The tangent at x = a has gradient f'(a). Its equation is y − f(a) = f'(a)(x − a). The normal has gradient −1/f'(a) provided f'(a) ≠ 0.
微分用于求切线和法线的斜率。在 x = a 处切线的斜率为 f'(a)，方程为 y − f(a) = f'(a)(x − a)。法线斜率为 −1/f'(a)（假设 f'(a) ≠ 0）。
Stationary points occur when f' sign change table to classify them as local maxima, minima, or points of inflection. These skills are essential for optimisation problems.
驻点出现在 f'(x) = 0 处。利用二阶导数或符号变化表来判断极大点、极小点或拐点。优化问题就是这些技巧的直接应用。
8. Integration: The Reverse of Differentiation | 积分：微分的逆运算
everses differentiation: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c, for n ≠ −1. The integral of 1/x is ln|x| + c. Memorise the integrals of eˣ egrating expressions like (ax + b)ⁿ using a simple substitution or the reverse chain rule.
不定积分是微分的逆运算：∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c（n ≠ −1）。1/x 的积分是 ln|x| + c。熟记 eˣ、sin x、cos x 等函数的积分，并学会使用简单的代换或逆链式法则来积分 (ax + b)ⁿ 这样的表达式。
where F is an antiderivative. Be careful with signs when the curve lies below the x-axis.
定积分给出曲线在两个界限之间与 x 轴围成的面积。微积分基本定理指出 ∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F 是 f 的一个原函数。当曲线部分位于 x 轴下方时，需注意面积的符号。
Area between two curves y = f(x) and y = g(x) is found by integrating the difference: ∫ₐᵇ |f(x) − g(x)| dx. Always sketch the region to determine which function is on top.
两条曲线 y = f(x) 与 y = g(x) 之间的面积通过对差值积分求得：∫ₐᵇ |f(x) − g(x)| dx。务必先画出草图，确定哪条曲线在上方。
9. Sequences and Series | 数列与级数
the first n terms: 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 uₙ = a + (n − 1)d。前 n 项和 Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，其中 l 为末项。
nth term is uₙ = arⁿ⁻¹. The sum of the first n terms: Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. If |r| < 1, the infinite sum S∞ = a/(1 − r) exists.
等比数列有公比 r。第 n 项 uₙ = arⁿ⁻¹。前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r)（r ≠ 1）。若 |r| < 1，则无穷级数和 S∞ = a/(1 − r) 存在。
Questions may involve modelling with sequences, such as compound interest or population growth. Also learn to use the Σ notation for sums and recognise when a series is arithmetic or geometric.
考题可能涉及用数列进行建模，如复利或人口增长。还要学会使用 Σ 符号表示求和，并识别级数是等差数列还是等比数列。
10. Vectors in Pure Mathematics | 纯数学中的向量
n. In two dimensions, a vector can be written as ai + bj or as a column vector. Magnitude: |ai + bj| = √a² + b². A unit vector in the same direction is v / |v|.
向量同时具有大小和方向。在二维空间中，向量可表示为 ai + bj 或列向量。模长：|ai + bj| = √a² + b²。同方向的单位向量为 v / |v|。
nsidering the quadrant. Vector addition follows the triangle law. Understand how to break a velocity or force vector into perpendicular components.
求向量与 x 轴正方向夹角时，可使用 tan θ = b/a，并注意象限。向量加法遵循三角形法则。要理解如何把速度或力分解为垂直分量。
e multiples of each other; find the angle between two vectors using the dot product: u·v = |u||v| cos θ = a₁a₂ + b₁b₂.
向量常与几何结合：通过证明向量成倍数关系来证明三点共线；利用点乘求两向量夹角：u·v = |u||v| cos θ = a₁a₂ + b₁b₂。
11. Binomial Expansion | 二项式展开
For positive integer n, (a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ, where r runs from 0 to n. nCr = n!/(r!(n−r)! term, such as the term independent of x. When the expansion is of the form (1 + kx)ⁿ with |x| small, you can use the series valid for |kx| < 1.
对于正整数 n，(a + b)ⁿ = Σ (nCr) aⁿ⁻ʳ bʳ，r 从 0 到 n 求和。nCr = n!/(r!(n−r)!)。要会求特定的项，比如不含 x 的常数项。若展开式为 (1 + kx)ⁿ 的形式，且 |x| 足够小，可使用对 |kx| < 1 有效的级数展开。
non-positive integer or fractional n, the binomial series is (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + ..., valid for |x| < 1. This is extremely useful for approximations and integrations.
当指数 n 不是正整数时，二项式级数为 (1 + x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + ...，在 |x| < 1 时有效。这在近似计算和积分中极为有用。
12. Integration by Substitution and by Parts | 换元积分法与分部积分法
When an integrand is of the form f(g(x))·g' imits for definite integrals. Common substitutions include u = ax + b, u = sin x, or u = eˣ.
当被积函数形如 f(g(x))·g'(x) 时，设 u = g(x) 可简化积分。要选择合适的 u，并记得对定积分须同时改变积分上下限。常见代换有 u = ax + b、u = sin x、u = eˣ 等。
(the part that can be integrated easily). Classically used for ∫ x sin x dx or ∫ ln x dx.
分部积分法基于微分的乘积法则：∫ u dv = uv − ∫ v du。关键在选取 u（求导后变简单的部分）和 dv（容易积分的部分）。典型的例子如 ∫ x sin x dx 或 ∫ ln x dx。
These techniques are frequently the core of longer Paper 1 questions. Practise recognising the type: substitution for composite functions, parts for products of different function types. Always check your answer by differentiating back.
这些技巧往往是卷一卷二中较复杂题目的核心。多加练习去识别题型：复合函数用代换法，不同函数类型的乘积用分部积分。算出结果后应通过求导来验算。
Published by TutorHao | Pure Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图