IB & AQA Maths: Common Pitfalls Explained | IB AQA 数学：易错题精讲

📚 IB & AQA Maths: Common Pitfalls Explained | IB AQA 数学：易错题精讲

Mistakes in mathematics exams often arise from misunderstanding key concepts or applying oversimplified rules. This article highlights typical pitfalls encountered in IB and AQA mathematics, covering algebra, trigonometry, calculus, probability, vectors, complex numbers, logarithms, functions, and hidden quadratics. Each section provides a common error along with the correct approach, helping you avoid losing marks on exam day.

在数学考试中，错误往往源于对关键概念的误解或应用过于简化的规则。本文重点介绍 IB 和 AQA 数学中典型的易错点，涵盖代数、三角、微积分、概率、向量、复数、对数、函数以及隐藏二次方程等内容。每节展示常见错误及正确解法，帮助你在考试中避免失分。

1. Cancelling Without Factoring | 忘记因式分解直接约分

Question: Simplify the expression (x² – 4)/(x – 2).

题目：化简表达式 (x² – 4)/(x – 2)。

Common mistake: Students often cancel the x terms directly, writing (x² – 4)/(x – 2) = x – 2. This is algebraically invalid because it treats the numerator as x² divided by x minus 4 divided by 2, which is not legal.

常见错误：学生经常直接约去 x 项，得出 (x² – 4)/(x – 2) = x – 2。这在代数上是不合法的，因为它将分子视为 x² 除以 x 再减去 4 除以 2，不能这样操作。

Correct solution: Factor the numerator as a difference of squares: x² – 4 = (x – 2)(x + 2). Then the fraction becomes (x – 2)(x + 2)/(x – 2). For x ≠ 2, we can cancel the common factor (x – 2) and obtain x + 2. Always state the restriction x ≠ 2, otherwise the original expression is undefined.

正确解法：将分子分解为平方差：x² – 4 = (x – 2)(x + 2)。于是分式变为 (x – 2)(x + 2)/(x – 2)。当 x ≠ 2 时，可约去公因式 (x – 2) 得到 x + 2。务必注明 x ≠ 2 的条件，否则原式无定义。

2. Losing Solutions in Trigonometric Equations | 解三角方程漏解

Problem: Solve sin 2x = 1/2 for 0 ≤ x ≤ 2π.

问题：在 0 ≤ x ≤ 2π 范围内解方程 sin 2x = 1/2。

Common pitfall: Students set 2x = π/6 and 5π/6, then divide by 2 to get x = π/12 and 5π/12, forgetting that the sine function has period 2π, so additional solutions arise from adding multiples of 2π before dividing.

常见陷阱：学生令 2x = π/6 和 5π/6，然后除以 2 得到 x = π/12 和 5π/12，忘记了正弦函数的周期为 2π，需要加上 2π 的整数倍再除以 2 才能得到所有解。

Correct method: The general solutions for 2x are 2x = π/6 + 2

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