📚 Top Mistakes in AS Maths Unit 1 January 2021 Paper | AS数学单元1 2021年1月真题易错点总结
The January 2021 AS Mathematics Unit 1 paper tested core pure maths skills, and examiners’ reports revealed a series of recurring mistakes that cost students easy marks. Understanding these pitfalls is essential for future candidates aiming to secure a high grade. This article summarises the most common errors found in that sitting, with detailed bilingual explanations to help you avoid them.
2021年1月的AS数学单元1试卷考察了纯数核心技能，考官报告揭示了学生反复犯下的一系列错误，这些错误白白丢掉了容易得到的分数。了解这些陷阱对于未来希望获得高分的考生至关重要。本文总结了该场考试中最常见的错误，并附上详细的中英双语解释，帮助你避免这些错误。
1. Misusing the Discriminant for Quadratics | 误用二次方程的判别式
In the question involving real and distinct roots, many candidates wrote down the correct condition Δ > 0 but then computed b² – 4ac incorrectly. A typical mistake was misidentifying coefficients after rearranging the equation, for example treating a constant term as zero when it wasn’t. Some also forgot to reverse an inequality sign when multiplying through by –1.
在涉及实根和不等根的问题中，许多考生写下了正确的条件 Δ > 0，但随后错误地计算了 b² – 4ac。典型的错误是在重新整理方程后错误地确定系数，例如把非常数项当作零。有些人还在乘以 -1 时忘记反转不等号的方向。
2. Losing Solutions in Logarithmic Equations | 对数方程中丢失解
When solving equations like logₘ(x) + logₘ(x – 3) = 1, students frequently combined logs correctly but then failed to state the domain restrictions x > 0 and x > 3 before squaring or solving. This led to extraneous answers being accepted without checking. Remember that the argument of a logarithm must always be positive.
在求解类似 logₘ(x) + logₘ(x – 3) = 1 的方程时，学生们常常正确合并对数，但在平方或求解之前没有说明定义域限制 x > 0 和 x > 3。这导致没有经过检验就把增根当作解接受下来。请记住，对数的真数必须始终为正数。
3. Chain Rule Errors in Differentiation | 求导时链式法则错误
A very common slip occurred when differentiating composite functions like (3x² – 5)⁴. Students would write 4(3x² – 5)³ but forget to multiply by the derivative of the inner function (6x). Similarly, in e⁴ˣ type problems, some wrote 4e⁴ˣ correctly while others left it as e⁴ˣ alone. Pay attention to the ‘multiply by the derivative of the inside’ step.
一个非常常见的失误发生在对类似 (3x² – 5)⁴ 的复合函数求导时。学生们会写出 4(3x² – 5)³，却忘记乘上内层函数的导数 (6x)。类似地，在 e⁴ˣ 型问题中，有些人正确地写出了 4e⁴ˣ，而另一些人则只写出 e⁴ˣ。要特别留意“乘上内层导数”这一步。
4. Omitting the Constant of Integration | 漏掉积分常数
Indefinite integration questions often saw final answers missing ‘+ C’. Even when students wrote it initially, they sometimes cancelled it after substituting limits in a definite integral problem because they got confused. In a ‘find the equation of a curve given the derivative and a point’ problem, omitting C meant unable to determine the specific equation, losing two marks unnecessarily.
不定积分题目的最终答案经常缺少“+ C”。即使最初写了，有时学生也会在定积分问题代入上下限后因混淆而删去常数。在“给定导数和一个点，求曲线方程”的题目中，漏掉 C 意味着无法确定特定方程，白白丢掉两分。
5. Trigonometric Equation Solution Sets | 三角方程的解集问题
In the January 2021 paper, a trig equation on the interval 0° ≤ θ ≤ 360° was given. A minority of candidates used the graph or CAST diagram correctly for the primary solutions but forgot to add 360° or 180° shifts for the second angle. In particular, sin θ = k gives two principal values, and failing to find the obtuse angle was a common cause of incomplete answers.
在2021年1月的试卷中，有一道三角方程题，区间为 0° ≤ θ ≤ 360°。少数考生正确使用图形或CAST图求出主解，但忘记为第二个角加上 360° 或 180° 的周期变换。特别是 sin θ = k 有两个主值，未能找出钝角是导致答案不完整的常见原因。
6. Arithmetic vs Geometric Sequence Confusion | 等差数列与等比数列的混淆
Several candidates misapplied formulas, using a + (n-1)d for a geometric sequence or a × rⁿ⁻¹ for an arithmetic one. The question clearly stated ‘common ratio’, yet some still used the sum formula for an arithmetic series. Carefully read the type of sequence before selecting the formula set.
不少考生用错了公式，对等比数列使用了 a + (n-1)d，或者对等差数列使用了 a × rⁿ⁻¹。题目明确写明了“公比”，但仍有人使用了等差级数求和公式。在选择公式之前，请仔细阅读数列的类型。
7. Mishandling the Binomial Expansion Coefficients | 二项展开式的系数处理不当
The binomial expansion of (2 + 3x)⁵ caused difficulty. While most knew the rows of Pascal’s triangle, they mishandled the coefficient of the variable part. For example, for the term in x², the correct combination is ⁵C₂ × 2³ × (3x)² = 10 × 8 × 9x² = 720x². Errors like forgetting to raise the 3 to the power of 2 or using 2⁵ were frequent.
(2 + 3x)⁵ 的二项展开给考生带来了困难。虽然大多数人都知道帕斯卡三角形的各行，但他们在处理变量部分的系数时出现了错误。例如，对于 x² 项，正确的组合是 ⁵C₂ × 2³ × (3x)² = 10 × 8 × 9x² = 720x²。忘记将3平方或直接使用 2⁵ 之类的错误十分常见。
8. Algebraic Fraction Simplification Mistakes | 代数分式化简错误
A question involving 2/(x+1) + 3/(x-1) led to widespread mistakes when combining into a single fraction. Students often wrote the numerator as 2(x-1) + 3(x+1) but then expanded incorrectly, e.g., writing 2x – 1 instead of 2x – 2. Another slip was forgetting to find a common denominator and simply adding numerators and denominators.
一道涉及 2/(x+1) + 3/(x-1) 的题目在合并为单个分式时出现了大量错误。学生通常将分子写为 2(x-1) + 3(x+1)，但展开时出错，例如写成 2x – 1 而不是 2x – 2。另一个疏漏是忘记找公分母，直接将分子和分母相加。
9. Inequality Direction When Dividing by Negatives | 除以负数时不等号方向错误
While solving linear inequalities, students routinely forgot to flip the inequality sign when dividing or multiplying by a negative coefficient. For example, –2x < 6 becomes x > –3, but many left it as x < –3. This mistake was particularly prevalent in questions where the inequality was part of a larger problem, such as finding the domain of a function.
在求解线性不等式时，学生经常在除以或乘以负系数时忘记反转不等号。例如，–2x < 6 应变为 x > –3，但很多人写成 x < –3。当不等式是更大问题的一部分（例如求函数定义域）时，这种错误尤为普遍。
10. Misreading Coordinate Geometry Conditions | 读错坐标几何条件
In a problem about perpendicular bisectors, candidates sometimes found the gradient of the original line correctly, but then used that same gradient for the perpendicular line instead of taking the negative reciprocal. Others correctly found the midpoint but substituted coordinates into the wrong equation. A quick sketch would have revealed obvious inconsistencies.
在有关垂直平分线的问题中，考生有时正确地求出了原直线的斜率，但随后却用同一个斜率作为垂直线的斜率，而没有取负倒数。其他人正确求出了中点，但将坐标代入了错误的方程。如果快速画个草图，就能发现明显的不一致之处。
11. Indices and Rational Exponents | 指数与有理指数
When simplifying expressions like (16x⁸)⁻³/⁴, students often struggled with the fractional negative exponent. Common errors included applying the power –³/⁴ to 16 only, ignoring the x term, or writing 16⁻³/⁴ as 1/(16³) and then forgetting to take the fourth root first. The correct simplification is (1/(16³))¹/⁴ × (x⁸)⁻³/⁴, which yields 1/8 × x⁻⁶ = 1/(8x⁶).
在化简像 (16x⁸)⁻³/⁴ 这样的表达式时，学生经常在处理负分数指数时遇到困难。常见错误包括只对16应用 –³/⁴ 次方而忽略了 x 项，或者将 16⁻³/⁴ 写成 1/(16³) 然后忘记先开四次方根。正确的化简是 (1/(16³))¹/⁴ × (x⁸)⁻³/⁴，结果为 1/8 × x⁻⁶ = 1/(8x⁶)。
12. Proof Questions: Insufficient Reasoning | 证明题：推理不充分
A simple proof question asked students to show that the sum of three consecutive integers is a multiple of 3. Some candidates just tested with numbers 1, 2, 3 and concluded the statement was true. A mathematical proof requires algebraic representation: n + (n+1) + (n+2) = 3n + 3 = 3(n+1), which is clearly a multiple of 3. Empirical testing does not constitute a proof.
一道简单的证明题要求学生证明三个连续整数之和是3的倍数。有些考生只用数字 1, 2, 3 测试，然后得出命题为真。数学证明需要代数表述：n + (n+1) + (n+2) = 3n + 3 = 3(n+1)，显然为3的倍数。经验性的测试不能构成证明。
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