Common Mistakes in AS Mathematics Unit 2 (Jan 2022 Report) | AS 数学 Unit 2 常见错误总结（2022年1月考试报告）

📚 Common Mistakes in AS Mathematics Unit 2 (Jan 2022 Report) | AS 数学 Unit 2 常见错误总结（2022年1月考试报告）

In the January 2022 examination session for AS Mathematics Unit 2 (Pure Mathematics 2), examiners identified a range of common errors that prevented candidates from achieving higher marks. This article summarises those key pitfalls and offers clear guidance on how to avoid them. By understanding these recurring mistakes, students can refine their exam technique and deepen their conceptual understanding.

在2022年1月的AS数学第二单元（纯数学2）考试中，考官总结了一系列常见的错误，这些错误阻碍了考生获得更高的分数。本文汇总了这些关键易错点，并提供清晰的改进指导。通过理解这些反复出现的错误，学生可以完善应考技巧并加深概念理解。

1. Algebraic Simplification and Cancelling Errors | 代数化简与约分错误

Many candidates lost marks by incorrectly cancelling terms in rational expressions. A common mistake was to cancel individual terms in a numerator that was a sum, e.g. simplifying (x+3)/x to 3. Correct cancelling requires a factor common to all terms. Always factorise first: (x+3)/x cannot be simplified unless x is a factor of both terms. Another frequent error was misapplying the rules of indices, such as writing √(x²+9) as x+3. Recognising the difference between √(x²) = |x| and √(x²+9) ≠ x+3 is crucial.

许多考生因错误约分有理式而丢分。常见错误是将分子为多项和的其中一项约掉，例如将 (x+3)/x 简化为 3。正确的约分需要各项有公因子。一定要先分解因式：(x+3)/x 无法化简，除非 x 是两项的公因式。另一个常见错误是误用指数法则，比如将 √(x²+9) 写成 x+3。区分 √(x²)=|x| 与 √(x²+9)≠x+3 至关重要。

Errors also occurred when simplifying expressions like (2x⁴ – 8x²) / (2x²). Some candidates wrote x² – 4x², failing to divide each term correctly. The proper simplification is to factor 2x²: 2x²(x² – 4) / (2x²) = x² – 4, for x ≠ 0.

学生在化简 (2x⁴ – 8x²)/(2x²) 时也出现错误。有考生写成 x² – 4x²，未能正确逐项相除。正确的做法是提取公因子 2x²：2x²(x² – 4)/(2x²) = x² – 4，且 x ≠ 0。

2. Misuse of Logarithm Properties | 对数性质误用

A very common mistake was confusing the laws of logarithms. Candidates would incorrectly state that log(a+b) = log a + log b, or that log(a)/log(b) equals log(a/b). The correct laws are log(ab) = log a + log b, log(a/b) = log a – log b, and log(aⁿ) = n log a. The change-of-base formula is log_a b = (log b)/(log a). Examiners also saw errors when solving equations like 2log₃ x = log₃(2x+1). Many candidates dropped the logs too hastily, writing 2x = 2x+1, forgetting that 2log₃ x = log₃(x²). The correct approach leads to x² = 2x+1, yielding x = 1+√2 (rejecting the negative solution as log of a negative is undefined).

一个非常普遍的错误是混淆对数的运算律。考生常错误地认为 log(a+b)=log a+log b，或者 log(a)/log(b)=log(a/b)。正确的法则为：log(ab)=log a+log b，log(a/b)=log a-log b，log(aⁿ)=n log a。换底公式是 log_a b=(log b)/(log a)。考官还发现，在解类似 2log₃ x=log₃(2x+1) 的方程时，许多考生过早去掉对数符号，错误地写成 2x=2x+1，忘记了 2

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