📚 A-Level WJEC Mathematics: End-of-Term Revision Outline | A-Level WJEC 数学：期末复习提纲
As the end of term approaches, A-Level WJEC Mathematics students need a structured revision plan to consolidate key concepts across Pure Mathematics, Statistics, and Mechanics. This outline highlights essential topics, core techniques, and revision strategies to ensure thorough preparation for internal assessments and final examinations.
随着期末临近，A-Level WJEC 数学学生需要一个结构化的复习计划，以巩固纯数学、统计学和力学的关键概念。本提纲强调重要主题、核心技巧和复习策略，确保为校内测评和最终考试做好充分准备。
1. Algebra and Functions | 代数与函数
Master polynomial division, the factor theorem, and the remainder theorem to simplify cubic and quartic expressions. Apply these to find factors and sketch graphs.
掌握多项式除法、因式定理和余数定理，用以化简三次和四次表达式，并应用于寻找因式和绘制草图。
Use partial fractions effectively, covering linear, repeated linear, and quadratic denominators. This skill underpins binomial expansions and integration.
熟练运用部分分式，涵盖线性、重复线性和二次分母。这一技能是二项展开和积分的基础。
Expand (1 + x)ⁿ for rational n using the binomial theorem, and state the validity condition |x| < 1. Write expansions up to the x³ term using factorial notation.
利用二项式定理对有理指数 n 展开 (1 + x)ⁿ，并说明有效性条件 |x| < 1。使用阶乘表示法写出至 x³ 项的展开式。
Manipulate functions: determine domain, range, composition and inverse functions. Graph transformations, including translations y = f(x) + a, stretches y = af(x), and reflections y = -f(x), are essential.
处理函数：确定定义域、值域、复合函数和反函数。图像变换包括平移 y = f(x) + a、伸缩 y = af(x) 和反射 y = -f(x) 是必考内容。
Solve equations involving modulus functions |ax + b| = c and sketch modulus graphs. Revise critical points and rewriting as piecewise functions.
求解含绝对值函数的方程 |ax + b| = c 并绘制模图像。复习临界点以及将函数改写为分段形式的方法。
2. Trigonometry | 三角函数
Work confidently with radians: convert between degrees and radians, calculate arc length s = rθ and sector area A = ½r²θ.
熟练使用弧度制：进行角度与弧度的换算，计算弧长 s = rθ 和扇形面积 A = ½r²θ。
Use the fundamental identities sin²θ + cos²θ = 1, tanθ = sinθ/cosθ and apply the compound-angle, double-angle formulas to simplify expressions and solve equations.
利用基本恒等式 sin²θ + cos²θ = 1, tanθ = sinθ/cosθ，并运用和角公式、倍角公式化简表达式和解方程。
Express a sinθ + b cosθ in the forms R sin(θ ± α) or R cos(θ ± α). Identify the maximum/minimum values and solve related equations.
将 a sinθ + b cosθ 表示为 R sin(θ ± α) 或 R cos(θ ± α) 的形式。识别最大值/最小值并求解相关方程。
Solve trigonometric equations in a given interval, including those involving sec, cosec, cot. Handle multiple angles and use CAST diagrams or graphs.
求解给定区间内的三角函数方程，包括含有 sec、cosec、cot 的方程。处理多倍角并使用 CAST 图或图像求解。
Review the small-angle approximations sinθ ≈ θ, cosθ ≈ 1 − θ²/2, tanθ ≈ θ for θ in radians. Apply these in limit problems.
复习小角近似公式 sinθ ≈ θ, cosθ ≈ 1 − θ²/2, tanθ ≈ θ（θ 以弧度计），并应用于极限问题。
3. Exponentials and Logarithms | 指数与对数
Understand the relationship between exponentials and natural logarithms: eˡⁿˣ = x, ln eˣ = x. Differentiate and integrate eˣ and functions of the form eᵏˣ.
理解指数函数与自然对数的关系：eˡⁿˣ = x, ln eˣ = x。对 eˣ 及形如 eᵏˣ 的函数进行微分和积分。
Apply log laws: ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln aᵏ = k ln a. Use these to solve equations where the unknown appears as a power.
应用对数律：ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln aᵏ = k ln a。用以求解未知数出现在指数位置的方程。
Model real-life growth and decay using exponential expressions P = P₀ eᵏᵗ or A = A₀ e⁻ᵏᵗ. Interpret the rate constant k and half-life.
使用指数表达式 P = P₀ eᵏᵗ 或 A = A₀ e⁻ᵏᵗ 建立现实生活中的增长与衰减模型。解释速率常数 k 和半衰期。
Tackle equations that require changing the base, e.g., aˣ = b. Take natural logs of both sides and rearrange. Be prepared for quadratic forms in eˣ.
处理需要换底的方程，例如 aˣ = b。对方程两边取自然对数并整理。准备应对 eˣ 的二次型方程。
4. Differentiation | 微分
Differentiate all standard functions: polynomials, eˣ, ln x, sin x, cos x, tan x, and their reciprocals. Know derivatives of sec x, cosec x, cot x.
对所有标准函数进行微分：多项式、eˣ、ln x、sin x、cos x、tan x 及其倒数。掌握 sec x、cosec x、cot x 的导数。
Apply the chain rule, product rule, and quotient rule accurately to compound functions. Practise identifying the appropriate rule quickly.
准确运用链式法则、乘法法则和除法法则对复合函数求导。练习快速识别适用的求导法则。
Use differentiation to find equations of tangents and normals to a curve, and locate stationary points. Determine the nature of stationary points via the second derivative d²y/dx².
利用微分求曲线的切线和法线方程，并确定驻点位置。通过二阶导数 d²y/dx² 判断驻点性质。
Solve optimisation problems: set up a function from geometric or physical constraints, differentiate, and find maximum or minimum values.
解决最优化问题：根据几何或物理约束建立函数，求导并找出最大值或最小值。
Differentiate functions defined parametrically: for x = f(t), y = g(t), use dy/dx = (dy/dt) / (dx/dt). Also find second derivatives parametrically.
对参数定义的函数进行微分：对于 x = f(t), y = g(t)，使用 dy/dx = (dy/dt) / (dx/dt)。也可参数式地求二阶导数。
5. Integration | 积分
Integrate standard functions: ∫ xⁿ dx, ∫ eˣ dx, ∫ 1/x dx, ∫ sin x dx, ∫ cos x dx, ∫ sec² x dx. Always include the constant of integration + C.
对标准函数积分：∫ xⁿ dx, ∫ eˣ dx, ∫ 1/x dx, ∫ sin x dx, ∫ cos x dx, ∫ sec² x dx。始终不要遗漏积分常数 +C。
Perform definite integration to calculate the area under a curve. Be mindful of areas below the x-axis being negative and split the integral where necessary.
执行定积分以计算曲线下的面积。注意 x 轴下方的面积为负，必要时应拆分积分区间。
Use integration by substitution: when the integrand is of the form f(g(x))g'(x), set u = g(x) and change limits for definite integrals. Reverse chain rule is a fast method.
使用换元积分法：当被积函数形如 f(g(x))g'(x) 时，设 u = g(x) 并变换定积分的上下限。逆链式法则是一种快速方法。
Apply integration by parts: ∫ u dv = uv − ∫ v du. This is essential for products of functions such as x eˣ or x sin x.
应用分部积分法：∫ u dv = uv − ∫ v du。这对于函数乘积（如 x eˣ 或 x sin x）必不可少。
Find the area between two curves by integrating the difference of the upper and lower functions. Solve for intersection points first.
通过对上方与下方函数之差进行积分来求两条曲线之间的面积。先解出交点坐标。
Use numerical methods such as the trapezium rule to approximate definite integrals when an antiderivative is difficult to find. Understand the error reduces with more strips.
当原函数难以找到时，使用梯形法则等数值方法近似计算定积分。理解增加分段数可减小误差。
6. Sequences and Series | 数列与级数
Work with arithmetic sequences: find the nth term using a + (n−1)d and sum to n terms using Sₙ = n/2(2a + (n−1)d).
处理等差数列：使用 a + (n−1)d 求第 n 项，使用 Sₙ = n/2(2a + (n−1)d) 求前 n 项和。
Handle geometric sequences: nth term arⁿ⁻¹, sum of first n terms Sₙ = a(1 − rⁿ)/(1 − r). For |r| < 1, sum to infinity S∞ = a/(1 − r).
处理等比数列：第 n 项 arⁿ⁻¹，前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r)。当 |r| < 1 时，无穷项和 S∞ = a/(1 − r)。
Use sigma notation Σ efficiently. Recognise telescoping sums or simple series that can be split into summable parts.
有效使用求和符号 Σ。识别裂项求和或可拆分为可求和部分的简单级数。
Prove statements by mathematical induction: show true for n = 1, assume for n = k, then prove for n = k+1. Common in series and divisibility proofs.
通过数学归纳法证明命题：验证 n = 1 成立，假设 n = k 成立，然后证明 n = k+1 成立。常见于级数和整除性证明。
7. Vectors | 向量
Represent vectors in 2D and 3D using i, j, k notation or column vectors. Calculate magnitude |a| = √(a₁² + a₂² + a₃²) and find unit vectors.
使用 i, j, k 符号或列向量表示二维和三维向量。计算模长 |a| = √(a₁² + a₂² + a₃²) 并求单位向量。
Add and subtract vectors, multiply by scalars. Solve geometric problems involving parallel and collinear vectors.
对向量进行加减和数乘运算。解决涉及平行和共线向量的几何问题。
Compute the scalar (dot) product a·b = |a||b| cos θ. Use this to find the angle between two vectors and test for perpendicularity (a·b = 0).
计算标量积（点积）a·b = |a||b| cos θ。利用点积求两向量夹角并检验垂直性 (a·b = 0)。
Write the vector equation of a line r = a + λb, where a is a position vector and b is the direction. Determine whether two lines intersect.
写出直线的向量方程 r = a + λb，其中 a 是位置向量，b 是方向向量。判断两条直线是否相交。
8. Statistical Sampling and Probability | 统计抽样与概率
Understand sampling methods: random, stratified, systematic, and quota sampling. Be able to critique their advantages and disadvantages in context.
理解抽样方法：随机、分层、系统和配额抽样。能够结合情境评述其优缺点。
Calculate probabilities using tree diagrams, conditional probability P(A|B) = P(A∩B)/P(B) and product rule for independent events P(A∩B) = P(A)P(B).
使用树状图计算概率，利用条件概率 P(A|B) = P(A∩B)/P(B) 和独立事件的乘法规则 P(A∩B) = P(A)P(B)。
Apply the binomial distribution B(n, p): P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ. Use cumulative tables or calculators to find P(X ≤ r) or P(X ≥ r).
应用二项分布 B(n, p)：P(X = r) = ⁿCᵣ pʳ (1−p)ⁿ⁻ʳ。使用累积分布表或计算器求 P(X ≤ r) 或 P(X ≥ r)。
Understand the Poisson distribution Po(λ) and its use as an approximation to the binomial when n is large and p is small.
理解泊松分布 Po(λ) 及其在 n 大 p 小时作为二项分布近似值的应用。
9. Probability Distributions and Hypothesis Testing | 概率分布与假设检验
Work with the normal distribution N(μ, σ²): standardise to Z = (X − μ)/σ. Use the standard normal table to find probabilities and percentiles.
处理正态分布 N(μ, σ²)：标准化为 Z = (X − μ)/σ。使用标准正态表求概率和百分位数。
Appreciate the Central Limit Theorem: for a large sample, the sample mean approximates a normal distribution. This justifies many statistical procedures.
理解中心极限定理：对于大样本，样本均值近似服从正态分布。这为许多统计程序提供了依据。
Conduct hypothesis tests for the mean of a normal distribution with known variance. State null and alternative hypotheses, calculate test statistic Z, compare to critical values and draw a conclusion.
对方差已知的正态分布均值进行假设检验。陈述原假设与备择假设，计算检验统计量 Z，与临界值比较并得出结论。
Interpret p-values: if p < significance level α, reject H₀. Understand one-tailed and two-tailed tests and choose the correct rejection region.
解释 p 值：如果 p < 显著性水平 α，则拒绝 H₀。理解单尾和双尾检验并选择正确的拒绝域。
10. Kinematics | 运动学
Use the SUVAT equations for constant acceleration in a straight line: v = u + at, s = ut + ½at², s = ½(u + v)t, v² = u² + 2as, s = vt − ½at².
使用匀加速直线运动的 SUVAT 方程：v = u + at, s = ut + ½at², s = ½(u + v)t, v² = u² + 2as, s = vt − ½at²。
Identify the positive direction and treat displacement, velocity, and acceleration as vector quantities. Account for gravity g = 9.8 ms⁻² acting downwards.
确定正方向，将位移、速度和加速度视为矢量。考虑重力 g = 9.8 ms⁻² 向下作用。
Interpret velocity–time graphs: gradient gives acceleration, area under graph gives displacement. Solve problems involving non‑constant acceleration using calculus.
解读速度-时间图像：斜率给出加速度，图像下的面积给出位移。使用微积分解决变加速度问题。
For projectile motion, resolve initial velocity into horizontal and vertical components. Horizontal motion has constant velocity; vertical motion uses SUVAT with a = ±g.
对于抛体运动，将初速度分解为水平和竖直分量。水平方向匀速运动；竖直方向使用 SUVAT 且 a = ±g。
11. Dynamics and Newton’s Laws | 动力学与牛顿定律
Apply Newton’s second law F = ma to connected particles and single bodies. Draw clear force diagrams showing weight, normal reaction, tension, and friction.
对连接体和单个物体应用牛顿第二定律 F = ma。绘制清晰的受力图，标出重力、法向反力、张力和摩擦力。
Understand friction: F ≤ μR, where μ is the coefficient of friction and R is the normal reaction. Use limiting equilibrium to find critical angles.
理解摩擦：F ≤ μR，其中 μ 是摩擦系数，R 是法向反力。利用极限平衡求临界角度。
Solve problems involving pulleys: assume a light inextensible string and a smooth pulley to maintain equal tension and linked accelerations.
解决涉及滑轮的问题：假设轻绳不可伸长且滑轮光滑，以保持张力相等和加速度关联。
Use the impulse–momentum principle: impulse = change in momentum = mv − mu. Work with vector impulses when forces act in two dimensions.
运用冲量-动量原理：冲量 = 动量变化 = mv − mu。当力作用于二维时，处理矢量冲量。
12. Statics and Moments | 静力学与力矩
For a particle in equilibrium, the vector sum of forces is zero: ΣF = 0. Resolve forces into perpendicular components to form simultaneous equations.
对于处于平衡的质点，力的矢量和为零：ΣF = 0。将力分解为垂直分量以建立联立方程。
Understand the moment of a force about a point: moment = force × perpendicular distance. The principle of moments states that for a rigid body in equilibrium, total clockwise moments equal total anticlockwise moments.
理解力对一点的力矩：力矩 = 力 × 垂直距离。力矩原理表明，刚体平衡时总顺时针力矩等于总逆时针力矩。
Solve problems involving non‑parallel forces on a ladder, plank, or beam using both force and moment equations. Include friction at contact points.
利用力和力矩方程解决涉及梯子、木板或横梁等非平行力问题。考虑接触点的摩擦力。
Determine centre of mass for simple uniform shapes and composite bodies. Use symmetry and tabulate separate masses and positions for complex objects.
确定简单均质形状和组合体的质心。利用对称性，对复杂物体列表记录各块质量和位置。
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