📚 A-Level Maths Unit 5 (FP2) Jan 2020 Mark Scheme Common Mistakes | A-Level数学单元五(FP2)2020年1月评分标准易错点总结
The January 2020 A-Level Unit 5 (Further Pure Mathematics 2) paper posed significant challenges, and a careful study of the mark scheme reveals a pattern of recurring, avoidable errors. Many candidates lost marks not because they lacked understanding, but because they rushed through standard procedures, misapplied formulas, or ignored fine details in definitions. This article dissects the most common pitfalls seen that session, providing clear guidance to help you secure the marks that others dropped.
2020年1月A-Level单元五（进阶纯数学2）试卷难度不低，仔细研读评分标准后可以发现，许多失分并非源于知识盲区，而是由于学生匆忙处理常规步骤、误用公式或忽略定义中的细节造成的。这篇文章将逐节剖析该次考试中最典型的易错点，给出具体避错策略，帮助你拿到别人轻易丢掉的分数。
1. Misjudging the Principal Argument | 辐角主值判断失误
In the complex numbers section, a question required students to find all three cube roots of a given complex number and then identify their principal arguments. A large proportion of candidates correctly worked out the roots in modulus‑argument form, but then stated arguments that fell outside the required range (–π, π] or gave inconsistent multiples of 2π. For example, for the root whose true principal argument was –π/6, many wrote 11π/6 instead. According to the mark scheme, this was awarded no marks for the argument part, even if the modulus was correct.
在复数部分，一道试题要求找出给定复数的三个立方根并写出各自主辐角。大量考生虽然正确地算出了模－辐角形式下的根，但最终写出的辐角超出了要求的(–π, π]范围，或者给出了不一致的2π倍数。例如，真实主辐角应为–π/6的那个根，很多人却写成了11π/6。评分标准明确规定，若主辐角错误，即使模正确，辐角部分也无法得分。
- Always check which quadrant the root lies in before finalising the argument.
- Reject angles like 7π/4 in favour of –π/4 if the scheme specifies (–π, π].
- When dividing a full circle for nth roots, start from the principal argument of the original number, not from an arbitrary angle.
- 在确定辐角之前，务必先确认根所在的象限。
- 若评分标准要求 (–π, π]，应将7π/4等角改写为 –π/4。
- 旋转生成 n 次方根时，始终从原复数的主辐角开始等分，不要随意选取起始角。
2. Unnormalised Eigenvectors | 特征向量未归一化
A matrix question asked for a unit eigenvector corresponding to the largest eigenvalue. Examiners noted that many scripts contained the correct direction vector but left it unnormalised, such as (2, –1). The mark scheme explicitly required the vector to have magnitude 1; giving (2, –1) only earned the method mark, while the accuracy mark was reserved for the normalised form (2/√5, –1/√5) or equivalent.
一道矩阵题要求给出对应于最大特征值的单位特征向量。阅卷人发现，大量考卷虽然给出了正确的方向向量，却未进行归一化处理，例如保留为 (2, –1)。评分标准明确要求向量模长为1；仅写出 (2, –1) 最多只能得到方法分，而准确答案 (2/√5, –1/√5) 才能获得全部实分。
- Read the question carefully: if it says ‘unit eigenvector’, normalisation is mandatory.
- Divide each component by √(a² + b²) and simplify the surds.
- Even if the scaling factor is ugly, write it as (1/√14, 3/√14) rather than rounding.
- 仔细读题：只要出现 ‘unit eigenvector’，就必需进行归一化。
- 将每个分量除以 √(a² + b²) 并化简根式。
- 即便因子较复杂，也要保留精确根式形式，例如 (1/√14, 3/√14)，不要四舍五入。
3. Polar Area: Missing the ½ Factor | 极坐标面积缺失1/2因子
The polar coordinates question contained a cardioid r = 4(1 + cos θ) and required the area enclosed. The single most damaging error was omitting the essential ½ from the area formula. Students wrote ∫ r² dθ instead of ½ ∫ r² dθ, and consequently their final answers were exactly double the correct value. Even when the ½ was included, some candidates integrated between 0 and 2π, which doubled the area again; the mark scheme accepted 2 ∫₀π ½ r² dθ as a valid alternative.
极坐标部分给出了心形线 r = 4(1 + cos θ)，要求计算其所围面积。杀伤力最大的错误是遗漏公式中必不可少的 ½ 因子。不少学生直接对 r² 进行积分，而不是 ½ ∫ r² dθ，导致最终答案恰好是正确值的两倍。即使写上了 ½，仍有一部分考生在 0 到 2π 上积分，造成面积再次翻倍；评分方案认可 2 ∫₀π ½ r² dθ 为正确替代方案。
½ ∫₀²π r² dθ ⇒ WRONG (generally doubles the area unless symmetric loop handled separately)
Area = 2 × ½ ∫₀π r² dθ for the cardioid ⇒ CORRECT
- Always write the ½ factor immediately – make it the first thing in your integral.
- Sketch the curve to decide whether limits should go from 0 to π, 0 to 2π, or be split.
- For curves symmetrical about the initial line, use the half‑range and double.
- 养成习惯，一写积分就先写下½因子，避免遗忘。
- 先画草图，判断积分限是 0→π、0→2π 还是需要切分。
- 关于极轴对称的曲线，利用半区间积分后再乘2可减少错误。
4. Incorrect Particular Integral Assumption | 特解设定错误
In a second‑order differential equation problem, y” – 3y’ + 2y = 5eˣ appeared. Many candidates correctly recognised that the complementary function involved eˣ and e²ˣ, but when choosing the trial particular integral they set yp = λeˣ. Because eˣ already appears in the complementary function, this trial form fails; the correct choice would have been yp = λx eˣ. The mark scheme penalised the omission of the x multiplier heavily, as it led to an inconsistent equation and zero marks for that part.
在一道二阶微分方程试题中，出现了 y” – 3y’ + 2y = 5eˣ。许多考生正确写出了通解部分含有 eˣ 和 e²ˣ，但在假设特解形式时仍设 yp = λeˣ。由于 eˣ 已经是齐次解的一部分，这种试设法会导致方程矛盾；正确的形式应为 yp = λx eˣ。评分标准对这一疏忽扣分极重，因为后继方程无法成立，整小题可能零分。
| RHS of ODE | Standard trial P.I. | If already in C.F. |
|---|---|---|
| eᵏˣ | λeᵏˣ | λx eᵏˣ |
| sin px or cos px | λ sin px + μ cos px | x(λ sin px + μ cos px) |
| xⁿ | polynomial of degree n | x × polynomial |
- Always start with the complementary function – list its independent terms.
- If a trial form overlaps with any term in the C.F., multiply by x (or x² if necessary).
- Double‑check by mentally substituting your trial yp – you should not get 0 = RHS.
- 解题第一步先写出齐次通解，列出独立的项。
- 若假设的特解与通解中的某项重合，就乘以 x（必要时乘 x²）。
- 通过心中代算检验试设特解是否会导致矛盾，以防白费工夫。
5. Maclaurin Series: Forgotten Factorials | 麦克劳林级数遗忘阶乘
When asked to find the Maclaurin series up to x³ for y = e^(sin x), candidates differentiated correctly but then formed the series as y(0) + y'(0)x + y”(0)x² + y”'(0)x³, completely omitting the 1/2! and 1/3! factors. The mark scheme deducted marks for each missing factorial, as the coefficients were numerically wrong without them. The correct series was 1 + x + x²/2 + 0·x³ (or similar), but many wrote 1 + x + x² – x³, which was unrecognisable.
当要求写出 y = e^(sin x) 精确到 x³ 的麦克劳林级数时，许多考生虽然求导正确，但在构建级数时直接写成 y(0) + y'(0)x + y”(0)x² + y”'(0)x³，彻底遗漏了 1/2! 与 1/3! 因子。评分标准对每个遗漏的阶乘都扣分，因为缺失阶乘后系数数值完全错误。正确的级数为 1 + x + x²/2 + …，但常见错误答案 1 + x + x² – x³ 令人啼笑皆非。
Maclaurin: f(x) ≈ f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …
- Write the general formula at the top of your page so you don’t forget the division
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