📚 Pure Maths Paper 2 Mark Scheme: How to Read It and Score Higher | 纯数学卷二评分方案：如何解读并取得更高分
Pure Mathematics Paper 2 tests the core techniques that separate a confident mathematician from a nervous calculator. The mark scheme reveals exactly where credit is given, so learning to read it can raise your grade without learning any new content.
纯数学卷二考查的是那些将真正自信的数学学习者与只会按计算器的学生区分开来的核心技能。评分方案明确说明了分数在哪里给出，因此学会解读评分方案，即使不学新知识，也能提升你的成绩。
1. What the Pure Paper 2 Mark Scheme Rewards | 纯数学卷二评分方案奖励什么
The mark scheme splits credit into method marks, accuracy marks, and independent marks. A correct final answer alone is rarely enough if the working is missing because the examiner cannot see how you arrived at it.
评分方案将分数分为方法分、准确分和独立分。如果缺少解题过程，仅有一个正确的最终答案通常是不够的，因为阅卷人无法看出你是如何得到答案的。
ing into the quadratic formula earns a method mark even if a later arithmetic slip costs the final accuracy mark.
在纯数学卷二中，解题过程就是证据。例如，在解二次方程时，写出 f(x) = 0 然后代入求根公式，即使后面的算术错误导致最终准确分丢失，方法分仍然会给出。
2. Method Marks vs Accuracy Marks | 方法分与准确分
Method marks, shown as M1, reward a valid process such as differentiating, integrating, using a correct identity, or setting up an equation. Accuracy marks, shown as A1, reward a correct answer after a valid method.
方法分，通常标记为 M1，奖励的是有效的过程，例如微分、积分、使用正确的恒等式或建立方程。准确分，标记为 A1，奖励的是有效方法之后的正确答案。
ed for statements that do not need a method, such as writing the exact value of sin(π/3) = √3/2. A dependent method mark, dM1, requires a previous method mark to have been earned.
独立分，标记为 B1，给的是不需要展示方法的陈述，例如写出 sin(π/3) = √3/2 的精确值。依赖方法分 dM1 则要求之前的某个方法分已经获得。
| Mark type | What it rewards | 中文说明 |
|---|---|---|
| M1 | A valid method shown | 展示有效方法 |
| A1 | Accurate result after method | 方法后的准确结果 |
| B1 | Independent correct statement | 独立的正确陈述 |
| dM1 | Dependent method after earlier M1 | 依赖之前 M1 的方法 |
| A1 ft | Follow-through accuracy after an error | 错误后继续传递的准确分 |
Understanding this language helps you self-mark past papers. You can see where you lost method marks because a step was skipped, and where an A1 ft would have saved you despite an earlier numerical error.
理解这些标记语言能帮助你自我批改历年真题。你能看出哪里因为跳步而丢了方法分，也能明白即使前面出现数值错误，A1 ft 又可能在哪些地方替你保留分数。
3. Algebra and Proof: Showing Sufficient Working | 代数与证明：展示充分步骤
For proof by contradiction, state the assumption clearly at the start. Then derive a contradiction, and finish with a conclusion that the original statement must be true.
使用反证法时，要在一开始清楚地写出假设。然后推出矛盾，最后给出原命题必定成立的结论。
y writing the factorised form without this check can lose marks in a proof-style question.
在因式分解三次或二次多项式时，要展示因式定理的步骤：求出 f(2) = 0，所以 (x - 2) 是一个因式。在证明类问题中，只写出因式分解形式而不展示这个检验过程，可能会丢分。
For solving inequalities such as x² - 5x + 6 > 0, sketch the parabola or use a sign table. Answers without a method showing how the intervals were chosen often earn only partial credit.
对于解不等式，例如 x² - 5x + 6 > 0，可以画出抛物线草图或使用符号表。如果没有方法展示区间是如何选出的，答案通常只能得到部分分数。
4. Functions and Graphs: Domain, Range, Transformations | 函数与图像：定义域、值域、变换
Write the domain and range using correct set notation. For example, for f(x) = √x - 2, the domain is x ≥ 2 and the range is f(x) ≥ 0.
要用正确的集合符号写出定义域和值域。例如，对于 f(x) = √x - 2，定义域为 x ≥ 2，值域为 f(x) ≥ 0。
swap x and y, and rearrange to y = f⁻¹(x), stating the domain of the inverse if required.
求反函数时，首先要检查函数在给定定义域上是否一一对应。然后设 y = f(x)，交换 x 和 y，再整理为 y = f⁻¹(x)，如有要求还要写出反函数的定义域。
(2x) is a horizontal stretch by scale factor 1/2. Mixing these is a common accuracy error.
图像变换的描述必须准确。y = 2f(x) 是竖直方向拉伸为原来的 2 倍，而 y = f(2x) 是水平方向压缩为原来的 1/2。混淆这两者是常见的准确分错误。
5. Trigonometry: Exact Values and Identities | 三角学：精确值与恒等式
Know the exact trigonometric values without a calculator: sin(30°) = 1/2, cos(45°) = 1/√2, tan(60°) = √3, and their radian equivalents. In Pure Paper 2, exact values are expected unless the question says otherwise.
要能在不使用计算器的情况下写出三角函数的精确值：sin(30°) = 1/2，cos(45°) = 1/√2，tan(60°) = √3，以及它们的弧度制对应值。在纯数学卷二中，除非题目另有说明，否则都要求给出精确值。
Use identities correctly: sin²θ + cos²θ ≡ 1, tan θ ≡ rms such as sin 2θ = 2 sin θ cos θ. Show which identity you are using before substituting.
正确使用恒等式：sin²θ + cos²θ ≡ 1，tan θ ≡ sin θ / cos θ，以及倍角公式，如 sin 2θ = 2 sin θ cos θ。在代入之前要写出你正在使用的是哪一个恒等式。
When solving trigonometric equations, find all solutions in the given interval. Draw a CAST diagram or sketch the graph, and do not stop at the first calculator value.
解三角方程时，要找出给定区间内的所有解。画出 CAST 图或函数草图，不要只停留在计算器给出的第一个值。
6. Differentiation: Techniques and Common Errors | 微分：技巧与常见错误
y/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴, showing the multiplier from the inner derivative.
准确应用链式法则、乘法法则和除法法则。对于 y = (3x² + 1)⁵，要写出 dy/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴，展示来自内层函数导数的乘数。
Know the derivatives of eˣ, ln x, sin x, cos x, and tan x. A common error is writing the derivative of ln x as x⁻¹ instead of 1/x, or forgetting the negative sign in the derivative of cos x.
掌握 eˣ、ln x、sin x、cos x 和 tan x 的导数。常见错误包括把 ln x 的导数写成 x⁻¹ 而不是 1/x，或者在 cos x 的导数中忘记负号。
In optimisation problems, show that the derivative equals zero and justify whether the stationary point is a maximum or minimum, using the second derivative or a sign change test.
在优化问题中，要写出导数等于零的步骤，并说明驻点是最大值还是最小值，可以使用二阶导数检验或符号变化检验。
7. Integration: Constants, Limits and Area | 积分：常数、上下限与面积
ing ∫ 3x² dx = x³ + C is correct; missing + C loses an accuracy mark in many mark schemes.
对于不定积分，一定要加上积分常数。写出 ∫ 3x² dx = x³ + C 才是正确的；很多评分方案中，缺少 + C 会丢失准确分。
mits carefully and show the evaluation. For example, ∫₀¹ x² dx = [x³/3]₀¹ = 1/3 - 0 = 1/3.
对于定积分，要仔细代入上下限并展示计算过程。例如，∫₀¹ x² dx = [x³/3]₀¹ = 1/3 - 0 = 1/3。
rval. Use ∫ (top - bottom) dx, and take the absolute value if the area crosses the x-axis.
求两条曲线之间的面积时，要先判断在区间内哪条曲线在上方。使用 ∫ (上方函数 - 下方函数) dx，如果面积跨越 x 轴，则要取绝对值。
The trapezium rule requires the full table of y-values and the correct multiplier h/2. Showing the table is often part of the method mark.
梯形法则要求完整的 y 值表和正确的乘数 h/2。展示表格通常是方法分的一部分。
8. Sequences and Series: Arithmetic and Geometric | 数列与级数：等差与等比
(n - 1)d]. State the values of a, d, and n before substituting to make your method clear.
使用等差数列公式 uₙ = a + (n - 1)d 和 Sₙ = n/2 [2a + (n - 1)d]。在代入之前先写出 a、d 和 n 的值，使你的方法更加清晰。
ₙ = arⁿ⁻¹, Sₙ = a(1 - rⁿ)/(1 - r), and the sum to infinity S∞ = a/(1 - r), valid only when |r| < 1.
对于等比数列，使用 uₙ = arⁿ⁻¹、Sₙ = a(1 - rⁿ)/(1 - r) 以及无穷级数和 S∞ = a/(1 - r)，后者仅在 |r| < 1 时成立。
questions, check whether the context is arithmetic or geometric. A car depreciating by 15% per year is geometric with r = 0.85, not arithmetic with d = -15.
在建模题中，要判断背景是等差还是等比。一辆汽车每年贬值 15% 是等比模型，公比 r = 0.85，而不是公差 d = -15 的等差模型。
9. Exponentials and Logarithms: Laws and Modelling | 指数与对数：法则与建模
x - logₐy, and logₐ(xⁿ) = n logₐx. Misusing these laws is a frequent source of lost marks.
正确应用对数法则：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx - logₐy，以及 logₐ(xⁿ) = n logₐx。误用这些法则是常见的失分原因。
To solve e²ˣ = (ln 5)/2. Writing x = ln(5/2) or ln 5 / ln 2 is incorrect and would lose accuracy marks.
解 e²ˣ = 5 时，两边取自然对数：2x = ln 5，因此 x = (ln 5)/2。写成 x = ln(5/2) 或 ln 5 / ln 2 都是错误的，会丢失准确分。
P₀eᵏᵗ, substitute known values to find k, then use the model to answer the question. Keep all working to at least three significant figures to avoid premature rounding errors.
对于增长与衰减模型，例如 P = P₀eᵏᵗ，代入已知值求出 k，然后用该模型回答问题。所有中间计算至少保留三位有效数字，以避免过早取整带来的误差。
10. Vectors and Parametric Equations: Setting Out Clearly | 向量与参数方程：清晰书写
Write vectors as column vectors or using i, j notation consistently. For magnitude, use |v| = √x² + y², and for a unit vector, divide by the magnitude.
向量可以用列向量或 i, j 记号一致地书写。求模长使用 |v| = √x² + y²，求单位向量则用向量除以它的模长。
When using the scalar product a · b = |a||b| es separately. This makes it easier to earn method marks even if the final angle is wrong.
使用数量积 a · b = |a||b| cos θ 时，要分别写出点积计算和模长计算。这样即使最终角度错误，也更容易获得方法分。
+ 1 and y = t³ - 2t, then dy/dx = (3t² - 2)/(2t). State the condition dx/dt ≠ 0 if needed.
对于参数方程求导，使用 dy/dx = (dy/dt)/(dx/dt)，并进行化简。如果 x = t² + 1，y = t³ - 2t，则 dy/dx = (3t² - 2)/(2t)。如有需要，说明 dx/dt ≠ 0 的条件。
11. Numerical Methods and Iteration | 数值方法与迭代
ere is a root in the interval [1, 2]. The function must be continuous for this argument to hold.
要证明根位于两个值之间，需要展示 f(x) 的符号变化。例如，f(1) = -2，f(2) = 3，因此区间 [1, 2] 内存在一个根。该论证要求函数连续。
arly, giving each value to the required accuracy. Mark schemes usually require at least four iterations and a consistent final root.
使用迭代法时，写出公式 xₙ₊₁ = g(xₙ)，并清晰地展示每一次迭代，每个值都按要求的精度给出。评分方案通常要求至少四次迭代，并给出稳定的最终根。
When using the Newton-Raphson method, use xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ). Show the derivative f'(x) first; omitting it loses a method mark.
使用牛顿-拉弗森法时，使用 xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)。先写出导数 f'(x)；省略它会丢失方法分。
12. Exam Strategy and Mark Scheme Language | 考试策略与评分方案语言
Read every question part before calculating, and note the marks available. A two-mark question usually has one method mark and one accuracy mark, so show one clear step and a correct answer.
在开始计算前阅读每个小问，并注意题目的分值。两分题通常包含一个方法分和一个准确分，因此要展示一个清晰的步骤并给出正确答案。
Use the mark scheme to self-mark with abbreviations: 'oe' means or equivalent, 'isw' means ignore subsequent working, and 'cao' means correct answer only. Learning these terms prevents you from marking your own work incorrectly.
使用评分方案自我批改时要注意缩写：'oe' 表示或等价写法，'isw' 表示忽略后续计算，'cao' 表示只有正确答案才得分。了解这些术语可以避免错误地批改自己的答题。
Finally, practise writing as if every line is being marked. Clear notation, labelled angles, ordered coordinates, and stated substitutions do not take much time, but they protect method marks across the whole paper.
最后，练习把每一行都当作会被评分来书写。清晰的符号、标注的角度、有序的坐标和写明的代换不需要花太多时间，但它们能在整份试卷中保护你的方法分。
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