📚 A-Level Maths: End-of-Term Revision Checklist | A-Level 数学：期末复习提纲
As the end of term approaches, A-Level Mathematics students face the challenge of consolidating a vast syllabus. This revision checklist breaks down the core topics—from pure mathematics to mechanics and statistics—into manageable sections. Use it to identify areas of strength and weakness, and to structure your final review sessions before the exams.
随着期末临近，A-Level 数学学生面临着梳理庞杂考纲的挑战。这份复习提纲将核心主题——从纯数学到力学与统计——拆解为易于掌控的模块。用它来识别你的强项与薄弱环节，并帮你在考试前规划最终的复习节奏。
1. Algebra and Polynomials | 代数与多项式
Master polynomial factorisation, the remainder theorem, and algebraic division. Be able to factorise cubic and quartic expressions by identifying integer roots using the factor theorem.
熟练掌握多项式因式分解、余数定理和代数长除法。能够利用因式定理找出整数根，对三次和四次多项式进行因式分解。
Revisit the binomial expansion for rational and negative exponents, paying attention to the validity range |x| < 1. Practise expanding (1 + x)ⁿ and more complex binomials like (a + bx)ⁿ.
重温有理指数和负指数的二项式展开，注意有效范围 |x| < 1。练习展开 (1 + x)ⁿ 以及 (a + bx)ⁿ 这类更复杂的二项式。
Ensure fluency with partial fractions, including repeated linear factors and irreducible quadratic denominators. This skill is essential for integration and series expansions later in the syllabus.
确保熟练进行部分分式分解，包括重复线性因子和不可约二次分母。这项技巧对于后续的积分与级数展开至关重要。
2. Functions and Graphs | 函数与图像
Understand domain, range, and inverse functions. Be prepared to sketch graphs of rational functions, including vertical and horizontal asymptotes, after applying algebraic manipulation.
理解定义域、值域和反函数。准备好绘制有理函数的图像，包括通过代数变形后确定垂直渐近线和水平渐近线。
Know how to combine functions and how to find composite functions fg(x). Recognise that the order matters: fg(x) is not generally equal to gf(x).
知道如何组合函数以及如何求复合函数 fg(x)。要认识到顺序的重要性：通常 fg(x) ≠ gf(x)。
Transformations of graphs—translations, stretches, and reflections—must be second nature. Remember that y = f(ax) is a horizontal stretch by factor 1/a, while y = af(x) is a vertical stretch by factor a.
图像变换——平移、伸缩和对称——必须成为本能。记住 y = f(ax) 是水平缩放 1/a 倍，而 y = af(x) 是垂直缩放 a 倍。
3. Trigonometry | 三角函数
Revise exact trigonometric values for 0°, 30°, 45°, 60°, and 90° in surd form. Know the definitions and graphs of sec θ, csc θ and cot θ.
复习 0°, 30°, 45°, 60°, 90° 的精确三角函数值（根式形式）。掌握 sec θ, csc θ 和 cot θ 的定义与图像。
Be confident with trigonometric identities such as sin²θ + cos²θ = 1, and derived forms 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ. Use them to solve equations and prove identities.
熟练运用三角恒等式，如 sin²θ + cos²θ = 1，以及派生式 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ。用它们解方程和证明恒等式。
Handle double-angle formulae: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ, and tan 2θ = 2tan θ / (1 – tan²θ). These are frequently required in integration and solving trigonometric equations.
掌握倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ，以及 tan 2θ = 2tan θ / (1 – tan²θ)。这些在积分和解三角方程中经常用到。
4. Exponentials and Logarithms | 指数与对数
Know the relationship between the exponential function eˣ and the natural logarithm ln x. Be able to solve equations where the unknown appears as a power, using logarithms to bring the variable down.
理解指数函数 eˣ 与自然对数 ln x 的关系。能够求解未知数出现在指数位置的方程，利用对数将变量下降到系数位置。
Differentiate and integrate exponential functions effortlessly: d/dx (eᵏˣ) = keᵏˣ, and ∫ eᵏˣ dx = (1/k) eᵏˣ + C. Recognise that the derivative of ln x is 1/x, and ∫ 1/x dx = ln|x| + C.
轻松对指数函数求导和积分：d/dx (eᵏˣ) = keᵏˣ, ∫ eᵏˣ dx = (1/k) eᵏˣ + C。记住 ln x 的导数是 1/x，而 ∫ 1/x dx = ln|x| + C。
Apply exponential growth and decay models to real-world problems. Formulate differential equations of the form dx/dt = kx or dx/dt = k(A – x) and solve them by separating variables.
将指数增长和衰减模型应用于实际问题。建立形如 dx/dt = kx 或 dx/dt = k(A – x) 的微分方程，并通过分离变量法求解。
5. Differentiation | 微积分：求导
Review the product rule, quotient rule, and chain rule thoroughly. For composite functions, the chain rule dy/dx = dy/du × du/dx is fundamental and must be applied confidently to functions like sin(kx), eᶠ⁽ˣ⁾, and ln(g(x)).
全面复习乘法法则、除法法则和链式法则。对于复合函数，链式法则 dy/dx = dy/du × du/dx 是基础，必须能自信地应用于 sin(kx), eᶠ⁽ˣ⁾ 和 ln(g(x)) 等函数。
Know how to differentiate parametric equations: dy/dx = (dy/dt) / (dx/dt). Be able to find the equation of a tangent or normal to a curve given in parametric form.
知道如何对参数方程求导：dy/dx = (dy/dt) / (dx/dt)。能够求出由参数形式给出的曲线的切线或法线方程。
Understand differentiation from first principles and its link to the limit definition. While typically only examined on simple functions, this conceptual understanding solidifies your grasp of the derivative.
理解由第一性原理求导及其与极限定义的联系。虽然通常只考简单函数，但这一概念性的理解能巩固你对导数的掌握。
6. Integration | 微积分：积分
Make sure you can integrate standard functions including xⁿ (n ≠ –1), eˣ, 1/x, sin x, cos x, sec²x, and associated linear composites. Reverse differentiation is a key skill.
确保你能对标准函数积分，包括 xⁿ (n ≠ –1), eˣ, 1/x, sin x, cos x, sec²x 及相应的线性组合。逆向求导是一项关键技能。
Use substitution to simplify integrals. When a composite function is multiplied by the derivative of the inner function, substitution with u = inner function works smoothly. Also practise trigonometric substitutions like x = a sin θ.
使用换元法简化积分。当一个复合函数乘以内层函数的导数时，用 u = 内层函数 换元会很顺畅。也要练习三角代换，例如 x = a sin θ。
Integration by parts is crucial for products of different function types. Use the formula ∫ u dv = uv – ∫ v du, and choose u using the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trig, Exponential) as a guideline.
分部积分法对于不同类型函数的乘积至关重要。使用公式 ∫ u dv = uv – ∫ v du，并用 LIATE 规则（对数、反三角、代数、三角、指数）作为选择 u 的指引。
Apply integration to find areas under curves, areas between curves, and volumes of revolution. For volumes, use V = π ∫ y² dx or V = π ∫ x² dy, being careful with limits.
应用积分求曲线下的面积、两曲线之间的面积，以及旋转体体积。体积用 V = π ∫ y² dx 或 V = π ∫ x² dy，需注意积分限的对应。
7. Sequences and Series | 序列与级数
Know the difference between a sequence and a series. Obtain the nth term of arithmetic and geometric sequences, and sum the first n terms. The arithmetic sum formula Sₙ = n/2 (2a + (n–1)d) and geometric sum Sₙ = a(1 – rⁿ)/(1 – r) are essential.
分清序列与级数的区别。会求等差数列和等比数列的第 n 项及前 n 项和。等差数列求和公式 Sₙ = n/2 (2a + (n–1)d) 和等比数列求和公式 Sₙ = a(1 – rⁿ)/(1 – r) 是必备的。
Understand the concept of convergence of a geometric series to infinity. Sum to infinity S∞ = a/(1 – r) exists only when |r| < 1. Apply this to recurring decimals.
理解几何级数无穷项收敛的概念。无穷项和 S∞ = a/(1 – r) 只在 |r| < 1 时存在。将其应用于循环小数。
Use sigma notation Σ confidently, and learn to expand and manipulate simple series. Know that Σ r² = n(n+1)(2n+1)/6 and Σ r³ = [n(n+1)/2]² may be given in formula booklets but you must be able to apply them.
自信地使用 sigma 记号 Σ，并学会展开和处理简单级数。了解 Σ r² = n(n+1)(2n+1)/6 以及 Σ r³ = [n(n+1)/2]²，虽然公式手册可能提供，但必须会应用。
8. Vectors | 向量
Be adept at vector operations: addition, subtraction, scalar multiplication, and calculating magnitudes. The magnitude of vector a = √(x² + y² + z²) is the length in 3D space.
熟练掌握向量运算：加法、减法、数乘和模长计算。向量 a 的模长 = √(x² + y² + z²) 表示三维空间中的长度。
Find and use the dot product a · b = |a||b| cos θ to determine the angle between two vectors. Two vectors are perpendicular if a · b = 0.
求并用点积 a · b = |a||b| cos θ 来确定两向量的夹角。若 a · b = 0，则两向量垂直。
Form vector equations of lines in the form r = a + tb, where a is a position vector and b is a direction vector. Solve problems involving intersections of lines and the shortest distance from a point to a line.
用 r = a + tb 的形式列出直线向量方程，其中 a 是位置向量，b 是方向向量。解决涉及直线交点以及点到直线最短距离的问题。
9. Mechanics: Kinematics and Newton’s Laws | 力学：运动学与牛顿定律
Displacement, velocity, and acceleration must be understood both as vectors and as scalar quantities in one dimension. Use constant acceleration (suvat) equations: v = u + at, s = ut + ½at², s = vt – ½at², v² = u² + 2as, s = ½(u + v)t.
位移、速度和加速度既要作为一维标量理解，也要作为向量理解。使用匀加速运动公式(suvat)：v = u + at, s = ut + ½at², s = vt – ½at², v² = u² + 2as, s = ½(u + v)t。
Apply Newton’s second law F = ma and resolve forces in perpendicular directions. Free body diagrams are vital for isolating objects and writing equations of motion.
应用牛顿第二定律 F = ma，并在垂直方向上分解力。自由体受力图对于隔离物体并列出运动方程至关重要。
Model friction correctly: F ≤ μR, where μ is the coefficient of friction and R is the normal reaction. In limiting equilibrium or motion, F = μR.
正确建立摩擦模型：F ≤ μR，其中 μ 是摩擦系数，R 是法向反作用力。在临界平衡或运动状态时，F = μR。
10. Statistics: Probability and Distributions | 统计：概率与分布
Calculate probabilities using tree diagrams, Venn diagrams, and two-way tables. Understand mutually exclusive and independent events: P(A ∪ B) = P(A) + P(B) – P(A ∩ B), and for independence P(A ∩ B) = P(A)P(B).
用树状图、维恩图和双向表计算概率。理解互斥事件和独立事件：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)，而对于独立事件，P(A ∩ B) = P(A)P(B)。
Know the binomial distribution X ~ B(n, p) and its conditions. Use the formula P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ and find cumulative probabilities from tables or calculator.
掌握二项分布 X ~ B(n, p) 及其适用条件。使用公式 P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ 并通过表格或计算器求累积概率。
For normal distribution X ~ N(μ, σ²), standardise by Z = (X – μ) / σ and use normal distribution tables. Recognise key probabilities: about 68% within 1σ, 95% within 2σ.
对于正态分布 X ~ N(μ, σ²)，用 Z = (X – μ) / σ 标准化，并查正态分布表。记住关键概率：约 68% 落在 1σ 内，95% 落在 2σ 内。
11. Statistics: Hypothesis Testing | 统计：假设检验
Set up null and alternative hypotheses correctly for binomial tests: H₀: p = p₀, H₁: p < p₀ (or >, or ≠). For two-tailed tests, halve the significance level when comparing to one-tailed critical values.
正确设定二项检验的原假设和备择假设：H₀: p = p₀, H₁: p < p₀ (或 >, 或 ≠)。对于双尾检验，比较单尾临界值时需将显著性水平减半。
Find critical regions and calculate p-values. A result is significant if the test statistic falls in the critical region or if the p-value is less than the significance level α.
找出临界区域并计算 p 值。若检验统计量落入临界区域，或 p 值小于显著性水平 α，则结果显著。
For normal distribution tests with known variance, use the Z-test. Write conclusions in context: do not just say “reject H₀”, but interpret what it means for the original problem.
对于方差已知的正态分布检验，使用 Z 检验。要结合实际背景写结论：不要说“拒绝 H₀”，而要解释它对原始问题的意义。
12. Numerical Methods and Proof | 数值方法与证明
Learn to use iteration formulae such as xₙ₊₁ = g(xₙ) to find roots of equations. Show stairway and cobweb diagrams to illustrate convergence, and know when rearrangement may fail to converge.
学会使用迭代公式如 xₙ₊₁ = g(xₙ) 来求方程根。画出阶梯图与蛛网图来示意收敛性，并知道何种变形可能不收敛。
Understand the Newton-Raphson method xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ) and its quadratic convergence. Apply it to equations where conventional algebra cannot find exact solutions.
理解牛顿-拉弗森法 xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ) 及其二次收敛速度。用它来处理常规代数无法找到精确解的方程。
Proof by deduction, exhaustion, and contradiction are common in pure mathematics. Practise proving algebraic identities and statements about irrationality, such as √2 being irrational.
演绎法、穷举法和反证法是纯数学中常见的证明方法。练习证明代数恒等式以及关于无理数的命题，例如证明 √2 是无理数。
Proof by induction is tested on sequences, divisibility, and matrices. Structure your solution carefully: base case, induction hypothesis, and induction step from k to k+1.
数学归纳法常考数列、整除性和矩阵问题。仔细组织解答：起始情况、归纳假设和由 k 到 k+1 的归纳步骤。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导