📚 AS Maths Unit 1 Jan 2022: Common Mistakes Summary | AS数学单元1 2022年1月试卷易错点总结
The January 2022 AS Pure Mathematics Unit 1 paper tested core algebraic skills, coordinate geometry, calculus and trigonometry. Although many questions followed familiar patterns, examiners' reports revealed recurring errors that prevented students from achieving top marks. This article summarises the most common pitfalls, explains why they happen and shows how to avoid them in future assessments.
2022年1月的AS纯数单元1试卷覆盖了核心代数技能、坐标几何、微积分和三角学。尽管许多题目遵循熟悉的样式，阅卷报告仍揭示了一些反复出现的错误，使学生未能拿到高分。本文总结最常见的易错点，解释其成因，并展示如何在未来的考试中避免它们。
1. Algebraic Expansion and Sign Errors | 代数展开与符号错误
of −8. Sign errors during expansion cascaded into later parts of questions, costing multiple marks.
当展开如 (3x − 2)(x + 4) 的表达式时，许多考生错误地处理了负项的乘积，将 −2 × 4 写成 +8 而非 −8。展开中的符号错误会连锁影响后续小问，导致多分丢失。
A smaller group also forgot to multiply all terms when expanding binomials, especially when a coefficient appeared outside the bracket. Writing 2(x + 3)(x − 1) as 2(x² + 2x − 3) but then failing to multiply the 2 into every final term was common.
还有一小部分考生在展开多项式时忘记乘以所有项，特别是当括号外有系数时。将 2(x + 3)(x − 1) 写成 2(x² + 2x − 3) 之后没有把 2 乘入每一个终项的错误也很常见。
- Check each term’s sign before combining.
- Multiply all terms inside by any external factor first.
- 合并前检查每一项的符号。
- 首先用外部因子乘以括号内的所有项。
2. Quadratic Discriminant Misunderstandings | 二次判别式的误解
The discriminant b² − 4ac was frequently set up correctly, but students then mishandled the inequality when interpreting results. A common mistake was writing b² − 4ac > 0 for two real roots but then solving the resulting quadratic inequality in the wrong direction.
判别式 b² − 4ac 经常被正确列出，但学生在解释结果时错误地处理了不等式。一个常见错误是对于两个实根写成 b² − 4ac > 0，然后以错误的方向解出二次不等式。
For example, given k² − 4k > 0, some concluded k < 0 and k > 4, which is correct, but when asked for a set of values where the curve lies above the x‑axis, they gave the region where the quadratic was positive rather than relating it back to the original function’s sign.
例如，已知 k² − 4k > 0，一些人得出 k < 0 和 k > 4，这是正确的；但当题目要求曲线在 x 轴上方的取值集合时，他们给出了二次式为正的区域，而没有将其与原函数的符号关联起来。
- Always sketch a quick graph of the quadratic inequality to confirm the regions.
- Re‑read the question to match the condition (real roots, no roots, tangent, etc.) with the inequality.
- 总是快速画一个二次不等式简图以确认区域。
- 重读题目，将条件（实根、无根、相切等）与不等式对应。
3. Inequality Sign Reversals | 不等式符号颠倒
estion required solving −2x ≥ 6, and many lost a mark by writing x ≥ −3 instead of x ≤ −3.
每当不等式两边同乘或除以一个负数时，不等号方向必须改变。在2022年1月的试卷中，有一题需要解 −2x ≥ 6，许多学生因写成 x ≥ −3 而非 x ≤ −3 而丢分。
This error also appeared when simplifying fractional inequalities like (x + 1)⁻¹ > 2, where candidates incorrectly multiplied both sides by (x + 1) without considering sign possibilities.
当化简形如 (x + 1)⁻¹ > 2 的分式不等式时，这一错误也出现了，考生在不考虑符号可能性的情况下错误地将两边同乘以 (x + 1)。
- Whenever multiplying or dividing by a negative, flip the inequality symbol.
- For rational inequalities, multiply by the square of the denominator or consider cases.
- 每当乘以或除以负数时，翻转不等号。
- 对于有理不等式，乘以分母的平方或分类讨论。
4. Misapplication of the Factor Theorem | 因式定理的误用
0. In the exam, candidates often correctly substituted the given value but then made mistakes when performing synthetic division or equating coefficients, especially with cubic polynomials.
因式定理表明，当且仅当 f(a) = 0 时 (x − a) 是 f(x) 的因式。考试中，考生通常能正确代入给定值，但在进行综合除法或比较系数时出错，尤其是对于三次多项式。
A particular error was forgetting to write the remainder as zero when factorising completely, resulting in an incomplete factorisation that could not yield the correct roots. Some also stopped after finding one root, omitting the quadratic factor and its solutions.
一个特定的错误是在完全因式分解时忘记将余数写成零，导致不完整的因式分解，从而无法得出正确的根。还有些学生在找到一个根后就停止了，遗漏了二次因式及其解。
- After division, always write f(x) = (x − a)(quotient) and check by expanding.
- Continue to factorise the quotient fully; do not stop at the linear factor.
- 除法之后，总是写出 f(x) = (x − a)(商式) 并通过展开检验。
- 继续完全分解商式；不要止步于线性因式。
5. Graph Transformation Confusions | 图形变换混淆
be the transformation from y = f(x) to y = 3f(2x). Many incorrectly described a vertical stretch of scale factor 3 and a horizontal stretch of scale factor 2, forgetting that f(2x) is a horizontal compression by 1/2, not a stretch.
形如 y = f(x) + a、y = f(x + a)、y = af(x) 与 y = f(ax) 的变换经常被混淆。在2022年1月的一道题中，学生被要求描述从 y = f(x) 到 y = 3f(2x) 的变换。许多人错误地描述了垂直拉伸比例因子3和水平拉伸比例因子2，忘记了 f(2x) 是水平压缩 1/2 而不是拉伸。
When combining transformations, the order also caused errors. Describing a horizontal shift and a horizontal stretch on the same function without applying the correct sequence led to wrong coordinates for turning points.
当组合变换时，顺序也导致错误。在同一个函数上描述水平平移和水平拉伸而未采用正确顺序，会导致转折点坐标错误。
- f(ax) with a > 1 is a horizontal compression, scale factor 1/a.
- Apply transformations inside the bracket first (horizontal), then outside (vertical).
- f(ax) 当 a > 1 时为水平压缩，比例因子 1/a。
- 先应用括号内的变换（水平），再应用括号外的（垂直）。
6. Differentiation of Negative and Fractional Powers | 负指数和分数指数求导错误
When differentiating terms like 1/x² or √x correct, but others wrote −2/x³ incorrectly as −2/x² or forgot the negative sign entirely.
在对 1/x² 或 √x 之类的项求导时，学生常常将它们重写成 x⁻² 或 x¹⁄²，但在应用幂法则时出错。对于 y = 1/x²，一些人给出 dy/dx = −2x⁻³ 是正确的，但另一些人错误地将 −2/x³ 写成 −2/x²，或者完全忘记了负号。
A major issue was differentiation of surds: √x correctly becomes (1/2)x⁻¹⁄², but the constant multiple was frequently mishandled, e.g. 3√x becoming 3/(2√x) instead of (3/2)x⁻¹⁄² . The final answer format also mattered – leaving the derivative with a negative power instead of writing it as a fraction lost simplification marks.
一个主要问题是无理式的求导：√x 正确化为 (1/2)x⁻¹⁄²，但常数倍数经常处理不当，如 3√x 变为 3/(2√x) 而非 (3/2)x⁻¹⁄²。最终答案的形式也很重要——将导数保留为负指数形式而不写成分数会失去化简分。
- Rewrite all terms as xⁿ before differentiating.
- After differentiation, convert back to a positive exponent fraction if required.
- 求导前将所有项改写为 xⁿ 形式。
- 求导后，按要求转回正指数分数的形式。
7. Integration Constant Omission | 遗漏积分常数
In indefinite integration, the constant of integration '+ C' was forgotten by a significant number of candidates. Even when it was written, students often missed the fact that a constant term in the integrand integrates to a linear term with a constant.
在不定积分中，大量考生忘记了积分常数 '+ C'。即使写了，学生也经常忽略被积函数中的常数项会积分出一个带常数的线性项这一事实。
For example, integrating 4x + 3 gave 2x² + 3x, but the +C was left off. When integrating a derivative given in context, failing to include +C meant the equation for a curve could not be correctly matched to a given point, resulting in a wrong final equation.
例如，积分 4x + 3 得到 2x² + 3x，但漏掉了 +C。当在上下文中对一个导数积分时，未能包含 +C 意味着曲线的方程无法正确地与给定点匹配，导致方程错误。
- Always end an indefinite integral with '+ C'.
- If a point is given to find C, substitute immediately after integration.
- 始终在不定积分末尾加上 '+ C'。
- 如果给定点以求 C，积分后立即代入。
8. Coordinate Geometry: Line Intersections | 坐标几何：直线交点
other line derived from a perpendicular gradient, students often substituted the expression incorrectly or solved the linear equation with arithmetic mistakes.
通过联立方程求两条直线的交点时常常出现代数失误。在一道涉及 y = 2x − 5 和另一条由垂直斜率导出的直线交点的问题中，学生经常错误地代入表达式，或在解线性方程时出现算术错误。
Another common mistake was using the midpoint formula instead of the intersection when the question asked for a point where the diagonals of a parallelogram cross. Reviewing the difference between midpoint and intersection in contextual problems is essential.
另一个常见错误是当题目要求找出平行四边形对角线交叉的点时，使用了中点公式而不是交点公式。在具体情境中回顾中点与交点的区别至关重要。
- Set equations equal carefully and move terms methodically.
- Draw a diagram to distinguish between midpoint and intersection.
- 仔细地令方程相等并有条理地移项。
- 画图以区分中点与交点。
9. Trigonometry Equation Domain Oversights | 三角方程定义域疏忽
t the second solution entirely. Quadrant rule (ASTC) was frequently misapplied for negative values like cosθ = −0.3.
在 0° ≤ θ ≤ 360° 范围内解 sinθ = 0.4，会产生第一和第二象限的值，但许多考生只给出了锐角主值，或完全忘记了第二个解。象限法则 (ASTC) 经常在负值如 cosθ = −0.3 中被错误应用。
In equations involving sin²θ, the step of taking square roots gave both positive and negative roots, but the negative root solutions were sometimes discarded without considering the domain. Also, not converting degrees to radians when the domain was given in radians was a frequent avoidable error.
在涉及 sin²θ 的方程中，开平方根步骤会得到正负两根，但负根的解有时被舍弃而未考虑定义域。此外，当定义域以弧度给出时没有将度转换为弧度，是一个常见的可避免错误。
- Use the quadrant diagram for all solutions within the given interval.
- Check whether the domain is in degrees or radians and set calculator mode accordingly.
- 使用象限图求出给定区间内的所有解。
- 检查定义域是角度制还是弧度制，并相应设置计算器模式。
10. Binomial Expansion Coefficient Mistakes | 二项式展开系数错误
The binomial expansion of (1 + ax)ⁿ was mostly applied correctly, but errors appeared when the expansion was asked in the form (a + bx)ⁿ. Students forgot to factor out the 'a' first, leading to incorrect binomial coefficients. For (2 + 3x)⁵, the correct approach is to write as 2⁵[1 + (3x/2)]⁵, but many directly expanded without factoring, mixing up coefficients.
对 (1 + ax)ⁿ 的二项式展开大多正确，但当题目要求展开 (a + bx)ⁿ 时出现错误。学生忘记先提取因式 'a'，导致二项式系数错误。对于 (2 + 3x)⁵，正确的做法是写成 2⁵[1 + (3x/2)]⁵，但许多人没有因式分解直接展开，混淆了系数。
Combining the constant factor 2⁵ with the binomial coefficient nCr also caused slips, as did simplifying powers of constants when there were multiple terms in the bracket.
将常数因子 2⁵ 与二项式系数 nCr 合并也造成了失误，括号中含有多项时化简常数的幂也容易出错。
- Always rewrite (a + bx)ⁿ as aⁿ(1 + (b/a)x)ⁿ before expanding.
- Multiply the binomial coefficients by the appropriate powers of a.
- 展开前始终将 (a + bx)ⁿ 重写成 aⁿ(1 + (b/a)x)ⁿ。
- 用 a 的适当次幂乘以二项式系数。
11. Hidden Quadratics and Substitution | 隐藏二次方程与代换法
Equations like 4x − 3√x − 1 = 0 were meant to be solved by substituting u = √x, giving a quadratic in u. Many candidates attempted to square terms without isolating the radical first, creating extraneous solutions and wasting time.
形如 4x − 3√x − 1 = 0 的方程需要通过设 u = √x 的方法求解，得到一个关于 u 的二次方程。许多考生尝试不先隔离根式就直接平方各项，产生了增根并浪费了时间。
After solving for u, some forgot to back‑ or −1/4 instead of x = 1² = 1 and discarding the negative root for the surd substitution.
解出 u 之后，一些人忘记回代求 x，给出 u = 1 或 u = −1/4 后却得出结论 x = 1 或 −1/4，而实际上应当是 x = 1² = 1，并根据无理代换舍去负根。
- Recognise when a substitution will reduce the equation to a quadratic.
- Reverse substitute and check answers in the original equation.
- 识别何时代换法能将方程化简为二次方程。
- 回代并在原方程中检验答案。
12. Modelling and Units | 建模与单位
In the final contextual problem involving the volume of a solid, students differentiated correctly to find a maximum but then failed to confirm it was a maximum via second derivative or a sign test. Simply stating 'by second derivative it is a maximum' without showing the calculation did not gain full marks.
在涉及固体体积的最终情境问题中，学生正确求导找到了最大值点，但未能通过二阶导数或符号检验确认它是最大值。仅声称“由二阶导数可知其为最大值”而不展示计算过程，没有得到全部分数。
Another repeated mistake was giving the answer without correct units or to an inappropriate degree of accuracy. The question often specified 'give your answer to 3 significant figures' but answers were given to 2 significant figures or left in terms of π when a decimal approximation was required.
另一个重复出现的错误是给出答案时没有正确单位或精度不当。题目常要求“给出答案保留3位有效数字”，但答案却只保留了2位有效数字，或者在要求小数近似时仍保留含有 π 的形式。
- Always perform the second derivative test or gradient sign check in optimisation.
- Check the required units and rounding instructions before writing the final answer.
- 在优化问题中始终执行二阶导数判定或梯度符号检验。
- 在写下最终答案前检查要求的单位和取整指示。
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