📚 A-Level Maths Unit 3 (WMA13) Jan 2020 Mark Scheme: Common Error Analysis | A-Level 数学单元3 (WMA13) 2020年1月评分方案易错点总结
This article summarises the most frequent mistakes candidates made in the A-Level Maths Unit 3 (WMA13) January 2020 examination, based on the official mark scheme. By reviewing these common pitfalls in pure mathematics, students can sharpen their exam technique and avoid losing marks on topics such as functions, trigonometry, exponentials, calculus, and numerical methods.
本文基于官方评分方案，总结了学生在 A-Level 数学单元3 (WMA13) 2020年1月考试中最常见的错误。通过回顾纯数学中的这些常见陷阱，学生可以完善考试技巧，避免在函数、三角学、指数、微积分和数值方法等主题上失分。
1. Domain of Inverse Functions and Range Checks | 反函数的定义域与值域检查
Many candidates found the inverse function correctly but forgot to state the domain, or wrote the domain incorrectly. The mark scheme requires the domain to match the range of the original function. For example, if f(x) = e²ˣ + 1, then the range is f(x) > 1, so the inverse’s domain must be x > 1. Simply writing x > 0 or all real numbers was a common error.
许多考生正确地求出了反函数，但忘记写定义域，或者写错了定义域。评分方案要求定义域必须与原函数的值域一致。例如，若 f(x) = e²ˣ + 1，值域为 f(x) > 1，因此反函数的定义域必须为 x > 1。简单地写成 x > 0 或全部实数是常见错误。
Another related mistake involved using the wrong variable when swapping x and y. Some students wrote f⁻¹(x) = … but then stated the domain in terms of y, which was not accepted.
另一个相关错误是在交换 x 和 y 时使用了错误的变量。有些学生写出了 f⁻¹(x) = …，但随后用 y 来表示定义域，这不会被接受。
2. Chain Rule with Trigonometric and Exponential Functions | 三角函数与指数函数的链式法则
When differentiating expressions like sin³(2x) or e⁻ˣ², many candidates omitted the derivative of the inner function or misapplied the power. For d/dx sin³(2x), the correct application is 3 sin²(2x) * cos(2x) * 2, but many forgot the factor 2 or wrote cos(2x) incorrectly. The mark scheme penalises missing inner derivatives heavily.
在对 sin³(2x) 或 e⁻ˣ² 这类表达式求导时，许多考生漏掉了内层函数的导数，或者误用了幂次。对于 d/dx sin³(2x)，正确应用是 3 sin²(2x) * cos(2x) * 2，但许多人忘记了因子 2，或者把 cos(2x) 写错。评分方案对漏掉内层导数惩罚很重。
Similarly, for e⁻ˣ², the derivative is −2x e⁻ˣ², but some wrote −x e⁻ˣ² or simply e⁻ˣ², neglecting the chain rule for the exponent. This error was observed repeatedly in implicit differentiation and related rates questions.
同样，对于 e⁻ˣ²，导数是 −2x e⁻ˣ²，但有些人写成了 −x e⁻ˣ² 或仅仅 e⁻ˣ²，忽略了对指数部分的链式法则。这种错误在隐函数求导和相关速率问题中屡见不鲜。
3. Solving Equations with Natural Logarithms | 含自然对数的方程求解
Candidates often mishandled the equation when terms like ln(x+1) − ln(x−2) = 2 was given. A frequent mistake was applying ln(a) − ln(b) = ln(a/b) correctly but then exponentiating too early without isolating the log term. Some wrote (x+1)/(x−2) = 2, forgetting to raise e to both sides. The correct step is e² = (x+1)/(x−2).
考生在处理如 ln(x+1) − ln(x−2) = 2 这样的方程时经常出错。一个常见错误是正确应用了 ln(a) − ln(b) = ln(a/b)，但随后在不隔离对数项的情况下过早指数化。有些人写成了 (x+1)/(x−2) = 2，忘记了将两边作为 e 的指数。正确步骤是 e² = (x+1)/(x−2)。
Also, when exponentiating, candidates must ensure the argument stays positive; some gave negative solutions that made the original log undefined. Checking domain restrictions was frequently omitted.
另外，在进行指数化时，考生必须保证真数为正；有些人给出了使原对数无定义的负数解。检查定义域限制的步骤经常被忽略。
4. Partial Fractions with Improper or Repeated Factors | 假分式或重因子的部分分式
In the Jan 2020 paper, a partial fractions question involved a repeated linear denominator, e.g., (x − 1)². Many students incorrectly set up the form as A/(x−1) + B/(x−1) instead of A/(x−1) + B/(x−1)². The mark scheme strictly required the correct decomposition, and marks were lost for missing the squared term.
在2020年1月的试卷中，一道部分分式题包含了一个重复线性分母，例如 (x − 1)²。许多学生错误地设成 A/(x−1) + B/(x−1)，而不是 A/(x−1) + B/(x−1)²。评分方案严格要求正确的分解形式，因漏掉平方项而失分。
For improper fractions where the numerator degree equals or exceeds the denominator, candidates often forgot to perform polynomial division first. The mark scheme awards marks for the whole process, so starting with division was essential to gain full credit.
对于分子次数等于或超过分母的假分式，考生常常忘记先进行多项式除法。评分方案对整个过程给分，因此先做除法对获得满分至关重要。
5. Integration by Substitution: Handling Limits and Constants | 换元积分：上下限和常数的处理
When evaluating a definite integral using substitution, many candidates either forgot to change the limits or changed them incorrectly. For instance, substituting u = x² + 1, with original limits x=0 and x=2, new limits should be u=1 and u=5, but some kept x=0 and x=2 or mixed signs. This caused wrong final answers and loss of method marks.
在使用换元积分计算定积分时，许多考生要么忘记改变上下限，要么换错了。例如，代换 u = x² + 1，原上下限 x=0 和 x=2，新的上下限应为 u=1 和 u=5，但有些人保留了 x=0 和 x=2，或者正负号混淆。这导致最终答案错误，并且丢失方法分。
Another common slip was omitting the constant multiple when relating dx and du. If du/dx = 2x, then dx = du/(2x), but candidates sometimes wrote dx = du and lost the factor. This was particularly evident when the substitution involved trigonometric functions.
另一个常见疏忽是在关联 dx 和 du 时遗漏常数倍数。如果 du/dx = 2x，则 dx = du/(2x)，但考生有时写成 dx = du，丢掉了因子。这在涉及三角函数的代换中尤为明显。
6. Binomial Expansion Validity and Range of x | 二项式展开的有效性及x范围
In the binomial expansion part, the question usually asks for the expansion in ascending powers of x and the range of validity. A typical mistake was stating the validity as |x| < 1 without checking the original expression. For (3 − 2x)⁻¹, the condition is |2x/3| < 1, i.e., |x| < 3/2. Many lost a mark by writing |x| < 1/2 or |x| < 1.
在二项式展开部分，题目通常要求按 x 的升幂展开，并给出有效范围。一个典型错误是没有检查原表达式，直接写上 |x| < 1。对于 (3 − 2x)⁻¹，条件是 |2x/3| < 1，即 |x| < 3/2。许多人因写成 |x| < 1/2 或 |x| < 1 而丢分。
Also, when the expansion involved a fraction like (1 + kx)ⁿ, students sometimes forgot to factor out the constant a from (a + bx)ⁿ, leading to an incorrect series. The mark scheme insists on the correct factored form first.
此外，当展开式涉及像 (1 + kx)ⁿ 这样的分数时，学生有时忘记从 (a + bx)ⁿ 中提取常数 a，从而导致级数错误。评分方案坚持要求先写出正确的提取因子形式。
7. Iteration and Convergence Conditions | 迭代与收敛条件
The numerical methods section featured an iterative formula, and candidates were asked to show the root lies in an interval and later to use iteration. A common error was not showing sufficient working to prove the sign change in the interval. The mark scheme required clear substitution into f(x); simply stating ‘sign change’ without values gave no marks.
数值方法部分给出了一个迭代公式，要求考生证明根在某个区间内，随后使用迭代。常见错误是没有展示足够的过程来证明区间上的符号变化。评分方案要求清晰地代入 f(x) 计算；仅仅说’符号变化’而不写数值是不得分的。
When discussing convergence of the iteration, candidates often incorrectly stated that any iteration will converge if the derivative is less than 1, but forgot to check the interval. The specific condition is |g'(x)| < 1 in the neighbourhood of the root, and many failed to apply this to the given function g(x).
在讨论迭代收敛性时，考生经常错误地声称只要导数小于1任何迭代都收敛，而忘记检查区间。具体条件是 |g'(x)| < 1 在根的邻域内，很多人未能将此应用于给定的函数 g(x)。
8. Modulus Equations and Inequalities: Squaring Method Risks | 绝对值
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