📚 A-Level WJEC Mathematics: Last-Minute Revision Notes | A-Level WJEC 数学：考前冲刺笔记
This revision guide is designed for WJEC A-Level Mathematics candidates who need a rapid, exam-focused review of the core pure, statistics and mechanics content. It brings together the key definitions, techniques and common pitfalls in one structured resource, so you can quickly refresh your memory and approach each paper with confidence.
这份复习指南专为 WJEC A-Level 数学考生编写，帮助你在考前快速回顾纯数学、统计和力学的核心内容。它用结构化的方式整理了关键定义、解题方法和常见易错点，让你高效唤醒记忆，以最佳状态迎战考试。
1. Algebraic Manipulation and Polynomials | 代数运算与多项式
The factor theorem links roots to linear factors: if f(a)=0 then (x-a) is a factor. The remainder theorem gives the remainder when f(x) is divided by (x-a) as f(a). Always test small integer values when searching for factors.
因式定理将根与一次因式联系起来：若 f(a)=0，则 (x-a) 是因式。余式定理指出 f(x) 除以 (x-a) 的余数为 f(a)。寻找因式时先代小整数试探。
For partial fractions, check the denominator first. With distinct linear factors like (x+1)(x-2), write A/(x+1) + B/(x-2). Use the cover-up method to quickly find A and B by substituting x = -1 and x = 2. For repeated factors, include all descending powers; for an irreducible quadratic numerator, use Ax+B.
处理部分分式时先检查分母。对于不同的一次因式 (x+1)(x-2)，设 A/(x+1)+B/(x-2)。用遮盖法代入 x=-1 和 x=2 可快速求出 A 和 B。有重因式时要包含所有降幂形式；遇二次不可约因式时分子设为 Ax+B。
The discriminant Δ = b² – 4ac tells you about the roots of ax²+bx+c=0: two distinct real roots (Δ > 0), one repeated root (Δ = 0) or no real roots (Δ < 0). It is also invaluable when finding intersections of a line and a curve or checking if a quadratic is always positive.
判别式 Δ = b² – 4ac 揭示 ax²+bx+c=0 的根的情况：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。在判断直线与曲线交点或二次函数恒正时必须使用。
Keep the laws of indices ready in your mind: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^(1/n) = ⁿ√a. Be careful with negative and fractional powers especially when integrating or differentiating.
牢记指数运算律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，以及 a^(1/n) = ⁿ√a。在微积分中处理负指数和分数指数时要格外小心。
Simplify surds by expressing them in terms of the smallest possible number under the root and always rationalise denominators when required. For example, 1/(√2) becomes (√2)/2.
化简根式时将根号内的数化为最小整数，必要时用分母有理化。例如 1/√2 写成 √2 / 2。
Quadratic formula: x = [-b ± √(b² – 4ac)] / (2a)
2. Coordinate Geometry and Graphs | 坐标几何与图像
The straight line equation y – y₁ = m(x – x₁) is your go-to form. The gradient m = (y₂ – y₁)/(x₂ – x₁). Two lines are parallel if m₁ = m₂ and perpendicular if m₁ × m₂ = -1.
直线方程首选 y – y₁ = m(x – x₁)。斜率 m = (y₂ – y₁)/(x₂ – x₁)。两直线平行当且仅当 m₁ = m₂，垂直当且仅当 m₁ × m₂ = -1。
The distance between points (x₁,y₁) and (x₂,y₂) is √[(x₂ – x₁)² + (y₂ – y₁)²]. The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). These appear frequently in circle problems.
两点 (x₁,y₁) 与 (x₂,y₂) 的距离为 √[(x₂ – x₁)² + (y₂ – y₁)²]。中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。这在圆方程题目中反复出现。
A circle with centre (a,b) and radius r has equation (x – a)² + (y – b)² = r². To find a tangent, use the fact that the radius to the point of contact is perpendicular to the tangent. You can then apply the perpendicular gradient rule and the line equation.
圆心 (a,b)、半径 r 的圆方程为 (x – a)² + (y – b)² = r²。求切线时，利用半径在切点处垂直于切线的性质，再结合垂直线斜率关系写出方程。
To convert parametric equations x = f(t), y = g(t) into a Cartesian equation, eliminate the parameter t. Look for identities like sin²t + cos²t = 1 or rearrange one equation to express t in terms of x and substitute into the other.
要将参数方程 x = f(t), y = g(t) 化为直角坐标方程，关键是消去参数 t。利用 sin²t + cos²t = 1 等恒等式，或由其中一个解出 t 代入另一个。
Graph transformations: f(x + a) shifts left by a; f(x) + a shifts up; f(ax) is a horizontal stretch by scale factor 1/a; a f(x) is a vertical stretch by scale factor a. Watch the direction of the shift carefully.
图像变换：f(x + a) 向左平移 a；f(x) + a 向上平移；f(ax) 水平拉伸因子 1/a；a f(x) 竖直拉伸因子 a。注意平移方向的正负号。
Midpoint: M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )
3. Differentiation – Rules and Applications | 微分法则与应用
Start with the power rule: if y = xⁿ then dy/dx = n xⁿ⁻¹. This extends to any constant multiple and sums. For a product y = uv, use the product rule: dy/dx = u’v + uv’.
幂函数求导基础：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。对常数倍和和差直接推广。乘积求导用乘法法则：dy/dx = u’v + uv’。
The quotient rule for y = u/v is dy/dx = (u’v – uv’) / v². Remember to keep the order – the numerator derivative minus term comes first. The chain rule handles composite functions: if y = f(g(x)) then dy/dx = f'(g(x)) g'(x).
商的求导法则：y = u/v，dy/dx = (u’v – uv’) / v²。分子必须保持 u’v 减去 uv’ 的顺序。复合函数用链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。
Key standard derivatives: d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec² x. When the argument is kx, don’t forget the chain rule provides a factor of k.
必须记住的标准导数：d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (tan x) = sec² x。当自变量是 kx 时，链式法则会多乘一个因子 k。
Derivatives tell you about gradients. Set dy/dx = 0 to find stationary points. Use the second derivative: d²y/dx² > 0 indicates a minimum, < 0 a maximum; if it equals 0, check the sign change of dy/dx to classify the point. A point of inflection occurs where curvature changes.
导数描述梯度。令 dy/dx = 0 求驻点。用二阶导数判断：d²y/dx² > 0 为极小值，< 0 为极大值；若等于零，需检验 dy/dx 的符号变化。拐点是曲率改变处。
Connected rates of change use the chain rule: dA/dt = (dA/dr)(dr/dt). Always identify the linking variable. This is common in WJEC applied contexts such as expanding circles or filling tanks.
相关变化率利用链式法则：dA/dt = (dA/dr)(dr/dt)。首先找出联系变量。WJEC 常出现圆面积增长、容器注水等情境。
The tangent at x=a has equation y – f(a) = f'(a)(x – a). The normal gradient is -1/f'(a). A quick sketch can prevent sign errors.
在 x=a 处的切线方程为 y – f(a) = f'(a)(x – a)。法线斜率为 -1/f'(a)。画个简图可避免符号错误。
Chain rule: dy/dx = (dy/du)(du/dx)
4. Integration – Techniques and Areas | 积分技巧与面积
Integration reverses differentiation. The general power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, valid for n ≠ -1. For n = -1, ∫ (1/x) dx = ln|x| + C. Always remember to add the constant of integration for indefinite integrals.
积分是微分的逆运算。幂法则通式：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C，n ≠ -1。当 n = -1 时，∫ (1/x) dx = ln|x| + C。不定积分务必加上积分常数。
Standard integrals to memorise: ∫ eˣ dx = eˣ + C, ∫ cos x dx = sin x + C, ∫ sin x dx = -cos x + C, ∫ sec² x dx = tan x + C. With linear arguments, divide by the coefficient, e.g., ∫ e²ˣ dx = (1/2) e²ˣ + C.
需要熟记的标准积分：∫ eˣ dx = eˣ + C，∫ cos x dx = sin x + C，∫ sin x dx = -cos x + C，∫ sec² x dx = tan x + C。若被积函数为线性函数，记得除以系数，如 ∫ e²ˣ dx = (1/2) e²ˣ + C。
For definite integrals ∫ₐᵇ f(x) dx, evaluate the antiderivative at the upper and lower limits and subtract: F(b) – F(a). The result represents the net signed area between the curve and the x-axis.
定积分 ∫ₐᵇ f(x) dx 的计算是先求原函数，再用上限值减下限值：F(b) – F(a)。结果表示曲线与 x 轴之间的净有向面积。
The area between two curves y = f(x) and y = g(x) from x=a to x=b is ∫ₐᵇ |f(x) – g(x)| dx. If the upper and lower curves swap, split the interval to avoid negative outputs.
两条曲线 y = f(x) 与 y = g(x) 之间从 x=a 到 x=b 的面积为 ∫ₐᵇ |f(x) – g(x)| dx。如果上下位置交换，需分段积分以避免负面积。
When an exact integral is hard, the trapezium rule gives an estimate: Area ≈ (h/2)[y₀ + 2(y₁+y₂+…+yₙ₋₁) + yₙ], where h = (b-a)/n. This is often asked in WJEC and a sketch or table of values is usually provided.
难以直接积分时，梯形法则可给出近似面积：面积 ≈ (h/2)[y₀ + 2(y₁+y₂+…+yₙ₋₁) + yₙ]，其中 h = (b-a)/n。WJEC 常给出表格或函数值让你计算。
Integration by substitution is used when you spot a function and its derivative. For ∫ f(g(x)) g'(x) dx, let u = g(x) so that du = g'(x) dx, transforming the integral into ∫ f(u) du. Integration by parts, ∫ u dv = uv – ∫ v du, is essential for products such as x eˣ or x ln x.
当看见函数与其导数共存时使用换元积分法。对于 ∫ f(g(x)) g'(x) dx，令 u = g(x)，则 du = g'(x) dx，转化成 ∫ f(u) du。分部积分法 ∫ u dv = uv – ∫ v du 处理像 x eˣ 或 x ln x 类型的乘积。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1
5. Trigonometry – Identities and Equations | 三角恒等式与方程
Use the CAST diagram or a graphical approach to find all solutions of trig equations in a given interval. For sin, cos, tan remember the principal values and the periodic properties: sin(θ) = sin(180°-θ), cos(θ) = cos(-θ) = cos(360°-θ), tan(θ) = tan(θ+180°).
在给定区间内解三角方程时用 CAST 图或图像法。牢记正弦、余弦、正切的主值及周期性：sin(θ)=sin(180°-θ)，cos(θ)=cos(-θ)=cos(360°-θ)，tan(θ)=tan(θ+180°)。
WJEC expects you to know exact values for 0°, 30°, 45°, 60°, 90° in surd form. For example, sin 30° = 1/2, cos 45° = √2/2, tan 60° = √3. Practice recognising them quickly to save time.
WJEC 要求熟记 0°、30°、45°、60°、90° 的精确根式值。例如 sin 30°=1/2，cos 45°=√2/2，tan 60°=√3。熟练识别这些值能为考试节省时间。
The compound angle formulas are given in the formula booklet, but you must know how to apply them: sin(A±B) = sinA cosB ± cosA sinB, cos(A±B) = cosA cosB ∓ sinA sinB. For tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB).
和角公式在公式表中可查，但必须会灵活运用：sin(A±B)=sinA cosB ± cosA sinB，cos(A±B)=cosA cosB ∓ sinA sinB，tan(A±B)=(tanA ± tanB)/(1 ∓ tanA tanB)。
Double angle formulas: sin 2θ = 2 sinθ cosθ, cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ. These are especially important in integration and when proving identities.
倍角公式：sin 2θ = 2 sinθ cosθ，cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ。在积分和恒等式证明时极为重要。
To solve equations of the form a sinθ ± b cosθ = c, rewrite it as R sin(θ ± α) or R cos(θ ∓ α). Find R = √(a²+b²) and α using tan α = b/a (check quadrant). This reduces the equation to a single trig function, which can then be solved easily.
对于 a sinθ ± b cosθ = c 型的方程，改写为 R sin(θ ± α) 或 R cos(θ ∓ α)。其中 R = √(a²+b²)，α 满足 tan α = b/a（注意象限）。这样简化为单一三角函数即可求解。
sin²θ + cos²θ = 1
6. Sequences, Series and Binomial Expansion | 数列、级数与二项展开
An arithmetic sequence has common difference d. nth term: uₙ = a + (n-1)d. Sum of first n terms: Sₙ = n/2 [2a + (n-1)d] or n/2 (a + l), where l is the last term. Check whether a term is given or the sum before selecting the right formula.
等差数列公差为 d。第 n 项：uₙ = a + (n-1)d。前 n 项和：Sₙ = n/2 [2a + (n-1)d] 或 n/2 (a + l)，其中 l 为末项。先看清给的是项还是和，再选对公式。
A geometric sequence has common ratio r, with uₙ = a rⁿ⁻¹. The sum Sₙ = a(1 – rⁿ)/(1 – r), valid for r ≠ 1. If |r| < 1, the sum to infinity exists: S∞ = a/(1 - r). Questions often ask for the sum to infinity or the condition for convergence.
等比数列的公比为 r，第 n 项 uₙ = a rⁿ⁻¹。前 n 项和 Sₙ = a(1 – rⁿ)/(1 – r)，r ≠ 1。若 |r| < 1，则无穷级数和为 S∞ = a/(1 - r)。题目常考收敛条件或求无穷和。
Sigma notation Σ requires careful index handling. Σ (arᵏ) from k=1 to n should be interpreted correctly. Convert to standard forms before applying sum formulas.
Σ 求和符号要求仔细处理下标。例如从 k=1 到 n 的 Σ (arᵏ) 要正确展开，先化为标准形式再套用求和公式。
The binomial expansion for (1 + x)ⁿ where n is rational is 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + … and is valid for |x| < 1. For (a + bx)ⁿ, factor out aⁿ to get the form (1 + (b/a)x)ⁿ. Expand only up to the required term; state the range of validity clearly.
有理指数 n 的二项展开 (1
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