📚 AS-Level Maths Unit 1 Jan19 Mark Scheme: Common Mistakes | AS数学单元1 Jan19评分方案易错点总结
Based on the January 2019 AS Unit 1 Pure Mathematics mark scheme, this article summarises the most frequent errors students made, helping you avoid losing marks in similar ways. Each section pairs English explanation with its Chinese translation, highlighting where candidates slipped up and how to improve.
基于2019年1月AS单元1纯数学评分方案，本文总结了学生最常见的错误，帮助你避免类似的失分。每个部分将英文解释与中文翻译配对，指出考生在哪里出错以及如何改进。
1. Algebraic Simplification & Brackets | 代数化简与括号展开
A common error was failing to distribute negative signs when subtracting expressions. For example, in expanding (x+3)² − (x−2)(x+2), many wrote x²+6x+9 − x² − 4, forgetting to apply the minus to the constant term, obtaining −4 instead of +4.
常见错误是在相减表达式时忘记分配负号。例如，展开 (x+3)² − (x−2)(x+2) 时，许多人写成 x²+6x+9 − x² − 4，忘记将减号应用于常数项，得到 −4 而不是 +4。
Another mistake involved mishandling coefficients when simplifying rational expressions, especially cancelling terms prematurely before factorising completely. The mark scheme noted that candidates lost method marks by cancelling common factors that did not exist across the entire numerator and denominator.
另一个错误涉及在化简有理表达式时错误处理系数，特别是在完全因式分解前过早约项。评分方案指出，考生因约掉在分子和分母中不存在的公因子而失去方法分。
2. Indices & Surds | 指数与根式
lied the power rules, such as writing (a³)² = a⁵ instead of a⁶, or incorrectly simplifying √x⁴ as x² without considering the absolute value when required. In the Jan19 paper, an equation with 2²ˣ saw candidates misuse the rule am+n = am · an, resulting in base errors.
学生经常错误应用幂运算规则，例如写出 (a³)² = a⁵ 而不是 a⁶，或者不正确地将 √x⁴ 化简为 x² 而不考虑必需的绝对值。在Jan19试卷中，一道涉及 2²ˣ 的方程题出现了误用 am+n=am·an 规则，导致底数错误。
Rationalising surds was another stumbling block. Many candidates multiplied numerator and denominator but then simplified √8 to 2√2 incorrectly, writing an extra factor or misplacing the irrational part.
分母有理化是另一个绊脚石。许多考生将分子和分母相乘，但随后错误地将 √8 化简为 2√2，要么多写了一个因子，要么把无理部分放错了位置。
3. Quadratic Functions & Discriminant | 二次函数与判别式
When applying the discriminant b² − 4ac, sign errors in substituting a, b, c were widespread. For instance, if b was negative, candidates might square it and still obtain a negative value. The mark scheme emphasised that b² is always non-negative, yet many lost marks for arithmetic slips.
在应用判别式 b² − 4ac 时，代入 a, b, c 时的符号错误很普遍。例如，如果 b 是负数，考生可能平方后仍得负值。评分方案强调 b² 总是非负的，但许多人因计算错误而失分。
In questions about the nature of roots, some incorrectly concluded 'no real roots' when the discriminant was zero, confusing zero with a negative number. Others misinterpreted 'two distinct real roots' as 'equal roots', losing accuracy marks.
在关于根的性质的问题中，有些人在判别式为零时错误地得出“无实根”，混淆了零和负数。其他人将“两个不等实根”误解为“等根”，因此丢失准确性分。
4. Coordinate Geometry – Straight Lines & Perpendicularity | 坐标几何 – 直线与垂直
A classic mistake when finding the equation of a perpendicular line was forgetting to take the negative reciprocal of the gradient. Students would sometimes just flip the fraction without changing the sign, or use the reciprocal in the wrong order. The Jan19 mark scheme deducted accuracy marks for such errors.
在求垂线方程时，一个典型错误是忘记取斜率的负倒数。学生有时只是翻转分数而不改变符号，或者以错误的顺序使用倒数。Jan19评分方案对于此类错误扣除了准确性分。
often substituted the midpoint coordinates incorrectly into y = mx + c, mixing up x and y.
求中点通常比较稳当，但在利用它确定垂直平分线时，考生常常把中点坐标错误地代入 y = mx + c，将 x 和 y 混淆。
5. Differentiation – Power Rule & Tangents | 微分 – 幂法则与切线
Errors in the power rule included reducing the exponent by 1 but forgetting to multiply by the original exponent, or differentiating constants incorrectly. In one Jan19 question, dy/dx of 3x⁻² was frequently given as −6x⁻¹ instead of the correct −6x⁻³.
幂法则的错误包括将指数减1但忘记乘以原指数，或者常数求导错误。在Jan19的一道题中，3x⁻² 的导数经常被写成 −6x⁻¹，而不是正确的 −6x⁻³。
tangent, many students correctly computed the gradient but then used the original function' te where x should go. The mark scheme required clear substitution into y − y₁ = m(x − x₁).
在求切线方程时，许多学生正确计算了斜率，但随后错误地使用了原函数的坐标，将 y 坐标置于 x 的位置。评分方案要求清晰地代入 y − y₁ = m(x − x₁)。
6. Integration – Indefinite Integrals & Constant of Integration | 积分 – 不定积分与积分常数
The most penalised error was omitting the '+ C' in indefinite integration. Even if the rest of the solution was correct, marks for the final answer were lost. Additionally, when integrating 1/x, some wrote ln|x| + C but failed to consider the domain correctly.
被扣分最多的错误是在不定积分中漏掉 '+ C'。即使其余解答正确，最终答案的分数也会丢失。此外，在积分 1/x 时，一些人虽然写出 ln|x| + C，但未能正确考虑定义域。
In definite integration for area, sign errors when evaluating at lower limits were common. Students frequently subtracted the value at the upper limit from the lower limit instead of the other way around, yielding a negative area without taking absolute value when required.
在定积分求面积时，计算下限时的符号错误也很常见。学生经常用下限值减去上限值，而不是反过来，导致面积为负，并且在需要绝对值时没有取绝对值。
7. Exponential & Logarithmic Equations | 指数与对数方程
When solving e²ˣ e coefficient: they wrote 2x = ln 5, which is correct, but some then attempted x = ln(5/2) – a fundamental misunderstanding of log properties.
在解 e²ˣ = 5 时，许多考生正确应用了 ln，但随后处理系数出错：他们写出 2x = ln 5，这是正确的，但有些人之后却错误地写成 x = ln(5/2) —— 这是对对数性质的根本性误解。
tween exponential and logarithmic forms also tripped up some, with ln a = b mistaken for eb = a but writing b = ea. The Jan19 paper tested this explicitly, and mark scheme notes revealed frequent confusion.
指数形式与对数形式之间的转换也难倒了一些人，将 ln a = b 误认为 eb = a，却写成 b = ea。Jan19试卷明确考验了这一点，评分方案笔记显示经常混淆。
8. Trigonometric Equations & CAST Diagram | 三角方程与CAST图
60°, many gave θ = 30° only, missing θ = 150°. The mark scheme requires all solutions in the range.
一个反复出现的问题是只找到主值而忽略给定区间内的第二个解。对于 sin θ = 0.5 在 0° 到 360° 之间，很多人只给出 θ = 30°，漏掉了 θ = 150°。评分方案要求区间内的所有解。
Using the CAST diagram, students often incorrectly labelled the quadrants where cosine is positive, leading to wrong reflection angles. Some misremembered that cosine is positive in the third quadrant, which lost them accuracy marks.
在使用 CAST 图时，学生常常错误标记余弦为正的象限，导致选择错误的翻转角。有些人记错认为余弦在第三象限为正，这让他们丢掉了准确性分。
9. Polynomials & Factor Theorem | 多项式与因式定理
but then incorrectly compute f(a) due to bracket errors. For a cubic with negative coefficient, dropping a negative sign led to f(a) ≠ 0, yet they claimed a factor. The mark scheme penalised this as an arithmetic slip.
在使用因式定理时，考生会将 x = a 代入 f(x)，但因括号错误导致计算 f(a) 出错。对于负系数的三次多项式，丢掉负号导致 f(a) ≠ 0，却声称是因式。评分方案将此作为
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