A-Level Maths Question Paper Unit 3 Jan22: Common Mistakes Summary | A-Level 数学试卷第三单元2022年1月易错点总结

📚 A-Level Maths Question Paper Unit 3 Jan22: Common Mistakes Summary | A-Level 数学试卷第三单元2022年1月易错点总结

The January 2022 Unit 3 question paper for A-Level Mathematics often tests core pure topics such as algebra, trigonometry, exponentials, calculus, vectors, and numerical methods. Analysing common student mistakes can help you avoid losing marks unnecessarily. This article summarises the most frequent errors observed in that paper, with clear explanations in both English and Chinese.

2022年1月的A-Level 数学第三单元试卷主要考查代数、三角、指数对数、微积分、向量和数值方法等核心纯数学内容。分析学生的常见错误，能帮你避免不必要的失分。本文总结了该试卷中出现频率最高的错误，并附上中英文清晰讲解。

1. Cancelling Terms Instead of Factors | 误约项而非因式

When simplifying rational expressions such as (x² – 4)/(x – 2), many students incorrectly cancel individual terms. They might cross out the ‘x²’ and ‘x’ to leave x – 4, or treat the expression as x – 2 directly. The correct method is to fully factorise the numerator as (x – 2)(x + 2) and then cancel the common factor, giving x + 2 (for x ≠ 2). Always remember: you can only cancel factors that multiply the entire numerator and denominator, never isolated terms.

在化简如 (x² – 4)/(x – 2) 的有理式时，很多学生错误地直接约去个别项。他们可能划掉 x² 和 x，得到 x – 4，或者直接认为结果是 x – 2。正确方法是先将分子完全分解因式为 (x – 2)(x + 2)，然后约去公因式，得到 x + 2 (x ≠ 2)。请始终牢记：只能约去分子与分母整体相乘的公因式，绝不能单独约去加减项。

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

A common pitfall in solving sinθ = 0.5 for 0° ≤ θ ≤ 360° is writing only θ = 30°. Using the CAST diagram or general solution, we find θ = 30° and 150°. Similarly, for cosθ = –√3/2, some give only 150°, forgetting 210°. Another mistake arises when solving sin(2θ) = 0.5: students solve 2θ = 30°, 150° then divide by 2 to get θ = 15°, 75°, but fail to add 360° to each 2θ solution to find further solutions within the range, such as 2θ = 390°, 510° giving θ = 195°, 255°. Always check the interval and account for the multiplier inside the trig function.

解 sinθ = 0.5，θ 在 0° 到 360° 之间时，常见错误是只写出 θ = 30°。使用 CAST 图或通解公式，我们会得到 θ = 30° 和 150°。同理，解 cosθ = –√3/2，有人只给出 150°，忘记 210°。另一个错误发生在解 sin(2θ) = 0.5 时：学生解出 2θ = 30°, 150°，然后除以 2 得 θ = 15°, 75°，但忘记将每个 2θ 的解加上 360° 的整数倍来寻找范围内的其他解，比如 2θ = 390°, 510° 可得到 θ = 195°, 255°。务必检查给定区间，并考虑三角函数内倍角带来的周期影响。

3. Ignoring Domain in Log Equations | 忽略对数方程定义域

When solving equations like log₂(x – 3) + log₂(x) = 2, students often combine logs as log₂(x(x – 3)) = 2, then solve x² – 3x – 4 = 0 obtaining x = 4 and x = –1. They may discard the negative root but sometimes forget to check the original domain: both x – 3 > 0 and x > 0, so x > 3. Therefore x = –1 is invalid even if algebraically it emerges. Another error is misapplying log rules, such as writing log(a + b) = log a + log b. Remember, log(a + b) cannot be split.

解方程 log₂(x – 3) + log₂(x) = 2 时，学生常合并为 log₂(x(x – 3)) = 2，然后解 x² – 3x – 4 = 0 得到 x = 4 和 x = –1。他们可能会舍去负根，但有时忘记检查原方程的定义域：需要 x – 3 > 0 且 x > 0，即 x > 3。因此 x = –1 无效，即使代数运算得出了它。另一个错误是错误使用对数法则，例如写成 log(a + b) = log a + log b。请记住，log(a + b) 不能拆分。

4. Chain Rule Misapplication | 链式法则误用

Differentiating composite functions like y = (3x² + 1)⁵ requires the chain rule: dy/dx = 5(3x² + 1)⁴ × 6x. A frequent mistake is to omit the derivative of the inner function, writing only 5(3x² + 1)⁴, or forgetting to multiply by 6x. With trigonometric functions, d/dx sin(2x) = 2 cos(2x), but some write cos(2x). For exponential functions, d/dx e^(4x) = 4e^(4x). Always differentiate the outer function, then multiply by the derivative of the inner function.