📚 Common Mistakes in AS Maths Unit 1 (Jan 21 Mark Scheme) | AS 数学第一单元（2021年1月评分方案）常见错误总结
Analysing the January 2021 mark scheme for AS Mathematics Unit 1 (Pure Mathematics) reveals a set of recurring errors that cost students valuable marks. This article explores those common pitfalls, explains the correct reasoning, and shows how to avoid them in future exams. Whether you are revising for a retake or preparing for a fresh attempt, mastering these areas will sharpen your accuracy and boost your confidence.
分析 2021 年 1 月 AS 数学第一单元（纯数学）的评分方案，可以发现一系列反复出现、导致学生丢分的典型错误。本文将深入剖析这些常见陷阱，讲解正确的推理过程，并告诉大家如何在未来的考试中避开它们。无论你是在准备补考还是首次应考，掌握这些要点都将提升你的解题准确度和信心。
1. Algebraic Simplification | 代数化简错误
Many candidates lost marks by mishandling signs when expanding brackets, especially with expressions like −(2x − 3). A typical mistake was writing −2x − 3 instead of −2x + 3. The mark scheme consistently penalised any sign error that led to an incorrect result, even if the rest of the method was correct.
许多考生在去括号时符号处理出错，特别是遇到 −(2x − 3) 这类表达式。典型的错误是写成 −2x − 3，而不是正确的 −2x + 3。评分方案对任何导致最终结果错误的符号错误都持续扣分，即便其余步骤正确也不放过。
When simplifying rational expressions such as (x² − 4)/(x + 2), students frequently cancelled without factorising fully, or cancelled terms rather than factors. The correct simplification is (x − 2)(x + 2)/(x + 2) = x − 2, provided x ≠ −2. Writing ‘= x − 3’ after an incorrect factorisation was a common slip.
在化简有理式如 (x² − 4)/(x + 2) 时，学生经常在没有完全因式分解的情况下约分，或者约去的是项而不是因式。正确的化简是 (x − 2)(x + 2)/(x + 2) = x − 2，且需注明 x ≠ −2。由于因式分解错误而得出 ‘= x − 3’ 是常见失误。
2. Solving Quadratic Equations | 二次方程求解失误
A frequent error occurred when solving quadratics by factorisation. Students would correctly write (2x + 1)(x − 3) = 0 but then state x = −1/2 and x = 3, but sometimes write x = 1/2 for the first root, forgetting the sign. The mark scheme required both roots to be correct and clearly stated.
用因式分解法解二次方程时，一个常见错误是：学生能正确写出 (2x + 1)(x − 3) = 0，但在给出根时第一个根写成 x = 1/2，忘记了负号。评分方案要求两个根都正确且表达清晰。
Using the quadratic formula x = [−b ± √(b² − 4ac)]/(2a), many misapplied the formula by dropping the denominator for only one term. For example, they wrote −b ± √(b² − 4ac)/2a, which was interpreted as −b ± [√(b² − 4ac)/(2a)], losing the grouping. Brackets must be used correctly in calculator work as well.
在使用求根公式 x = [−b ± √(b² − 4ac)]/(2a) 时，很多人用错，只把一个项除以分母，写成 −b ± √(b² − 4ac)/2a，这会被解读为 −b ± [√(b² − 4ac)/(2a)]，丢失了括号分组。计算器输入时也必须正确使用括号。
Completing the square was another area where errors crept in. For x² + 6x + 5, candidates often wrote (x + 3)² − 9 + 5 = (x + 3)² − 4, but then incorrectly expanded back to x² + 6x + 9 − 4, forgetting the +9 step. Practising both directions helps cement the identity.
配方法也是易出错的地方。对于 x² + 6x + 5，考生常写成 (x + 3)² − 9 + 5 = (x + 3)² − 4，但在回代验证时却错误地展开为 x² + 6x + 9 − 4，漏掉了 +9 这一步。双向练习有助于巩固恒等关系。
3. Inequalities and Set Notation | 不等式与集合符号
Solving linear inequalities caused problems when multiplying or dividing by a negative number. For −2x < 8, some left the direction unchanged and wrote x < −4. The correct step is x > −4. The mark scheme explicitly required the inequality sign to be reversed.
解一元一次不等式时，当两边同乘以或除以负数时容易出问题。对于 −2x < 8，有些人保持不等号方向不变，写成 x < −4。正确的步骤是 x > −4。评分方案明确要求不等号必须反向。
Quadratic inequalities like (x − 1)(x + 3) > 0 were often solved by simply listing x > 1 and x > −3, ignoring the region outside the roots. A sketch graph or sign diagram is essential to obtain x < −3 or x > 1. Candidates who wrote ‘x > 1’ and ‘x < −3' but joined them with 'and' instead of 'or' lost marks.
二次不等式如 (x − 1)(x + 3) > 0 经常被错误地解为 x > 1 和 x > −3，忽略了根两侧的区域。画个草图或符号表是必要的，正确答案是 x < −3 或 x > 1。若考生写了 ‘x > 1’ 和 ‘x < −3'，却用 'and' 而非 'or' 连接，也会被扣分。
Set notation and interval notation were often mixed up. {x: x < 5} ∩ {x: x > 2} was sometimes given as (2,5) without curly braces, or {x: 2 < x < 5}, but a few wrote {x: x < 5 and x > 2} without the correct set format. Using both forms correctly is vital.
集合符号和区间符号经常被混淆。{x: x < 5} ∩ {x: x > 2} 的答案有时写成 (2,5) 但没有花括号，或者写成 {x: 2 < x < 5}，而有些人写 {x: x < 5 and x > 2} 却没有遵循正确的集合格式。正确使用两种形式至关重要。
4. Graphs and Transformations | 函数图像与变换
Candidates struggled with describing transformations in words, mixing up ‘translation’ and ‘stretch’. For y = f(x) → y = 3f(x), the correct description is ‘stretch parallel to the y-axis by scale factor 3’. Many wrote ‘stretch in the x-axis’ or ‘translation in the y-direction’. Precision in language is required.
考生在口头描述图像变换时感到困难，混淆了 ‘平移’ 和 ‘伸缩’。对于 y = f(x) → y = 3f(x)，正确的描述是 ‘沿 y 轴方向伸缩，比例系数为 3’。很多人写成 ‘沿 x 轴伸缩’ 或 ‘y 方向平移’。语言必须准确。
When sketching transformed graphs, errors included shifting in the wrong direction. For f(x + 2), the graph should move left by 2 units, but many shifted it right. Similarly, f(−x) reflects in the y-axis, but candidates often reflected in the x-axis or rotated the shape.
在绘制变换后的图像时，错误包括方向弄反。对于 f(x + 2)，图像应向左平移 2 个单位，但很多人向右移。同样，f(−x) 是关于 y 轴反射，但考生常常进行关于 x 轴的反射，或者搞错旋转方向。
Finding intersection points of two curves from a sketch was another weak spot. The mark scheme accepted exact coordinates read from the graph if clearly labelled, but if candidates attempted algebraic verification they often made arithmetic slips. A quick check is always wise.
从草图求两条曲线的交点坐标是另一个薄弱环节。评分方案接受从图上读取的精确坐标，但要求标注清楚。如果考生尝试用代数方法验证，则常出现计算错误。快速验算一下总是明智的。
5. Differentiation Basics | 求导基础
Differentiating simple powers like x⁻² was a problem. Many wrote the derivative as −2x⁻³, which is correct, but then lost marks when simplifying expressions like 3/∛x because they rewrote it as 3x⁻¹/² instead of 3x⁻¹/³. This root-to-power conversion error affected the entire derivative.
对简单幂函数例如 x⁻² 求导容易出问题。很多人写出导数 −2x⁻³（正确），但在化简表达式如 3/∛x 时丢分，因为他们把它改写成了 3x⁻¹/²，而不是正确的 3x⁻¹/³。这种根式转幂的转换错误影响了整个求导过程。
The product and quotient rules were occasionally applied incorrectly. For y = (x² + 1)(2x − 3), some attempted to ‘differentiate each factor separately and multiply’, which is not valid. The correct approach is to expand first or use the product rule: u’v + uv’.
乘法法则和除法法则偶尔被错误应用。对于 y = (x² + 1)(2x − 3)，有人试图 ‘分别对每个因式求导再相乘’，这是无效的。正确做法是先展开，或者使用乘法法则：u’v + uv’。
Finding the equation of a tangent or normal to a curve was a major topic. A typical error was mixing up the gradient of the tangent (dy/dx) with that of the normal (−1/(dy/dx)). Some used the tangent gradient to write the normal equation, losing a vital mark. Others forgot to find the y-coordinate at the point of contact before forming the line equation.
求曲线的切线或法线方程是一个大主题。典型错误是把切线斜率 (dy/dx) 和法线斜率 (−1/(dy/dx)) 弄混。有些人用切线斜率去写法线方程，丢掉关键分数。还有人在列直线方程前忘记先求切点处的 y 坐标。
6. Integration and Area | 积分与面积
When integrating expressions like 4/x, candidates often wrote 4 ln x + c, which is correct, but then when working with definite integrals they forgot to apply the limits correctly. For ∫₁² 4/x dx, they wrote [4 ln x]₁² but then evaluated 4 ln 2 − 4 ln 1 as 4 ln 2 − 4, mistaking ln 1 = 1 instead of 0.
在积分 4/x 这样的表达式时，考生常正确写出 4 ln x + c，但在处理定积分时却忘记正确代入上下限。对于 ∫₁² 4/x dx，他们写出 [4 ln x]₁²，但把 4 ln 2 − 4 ln 1 算成 4 ln 2 − 4，错把 ln 1 当成 1，实际应为 0。
Finding the area between a curve and the x-axis required careful splitting when the curve crossed the axis. If y = x(x − 1)(x + 2) was negative over part of the interval, simply integrating without splitting gave the net area rather than the total area. The mark scheme expected separate integrals with signs swapped as needed, or use of absolute value.
求曲线与 x 轴围成的面积时，如果曲线穿过轴线就需要仔细分段。若 y = x(x − 1)(x + 2) 在部分区间为负，不拆分直接积分会得到净面积而非总面积。评分方案期望考生根据需要分段积分并调整符号，或者使用绝对值。
Indefinite integration often missed the constant ‘+c’. In a differential equation context or when finding f(x) from f'(x), the omission cost a mark. Even if the constant was later determined, leaving ‘+c’ out in the initial step was still penalised if the working was shown.
不定积分常漏掉常数 ‘+c’。在微分方程背景下或由 f'(x) 求 f(x) 时，这个遗漏会导致丢分。即使常数后来被求出，如果解题步骤显示最初没写 ‘+c’，仍会被扣分。
7. Trigonometric Equations | 三角方程
Solving sin θ = 0.5 in the range 0° ≤ θ ≤ 360° often yielded only θ = 30°, missing the second solution θ = 150°. The mark scheme required all solutions within the given interval, and candidates who did not use the CAST diagram or symmetry of the sine curve frequently left out one value.
解 sin θ = 0.5 在 0° ≤ θ ≤ 360° 范围内的方程时，很多人只得出 θ = 30°，漏掉了第二个解 θ = 150°。评分方案要求给出区间内的所有解，不使用 CAST 图或正弦曲线对称性的考生经常漏掉一个值。
Equations involving tan θ caused trouble when rearranged to tan θ = k and candidates gave principal values only. For tan θ = −1, they wrote θ = −45°, but the required range was 0° to 360°, so the correct answers were 135° and 315°. Understanding periodicity is essential.
涉及 tan θ 的方程在化为 tan θ = k 后造成困难，考生往往只给出主值。对于 tan θ = −1，他们写 θ = −45°，但题目要求范围是 0° 到 360°，所以正确答案应为 135° 和 315°。理解周期性至关重要。
Trig identities were sometimes misapplied. For example, using sin²θ + cos²θ = 1 to replace sin²θ, students wrote sin²θ = 1 − cos²θ but then incorrectly substituted 1 − cos θ instead of 1 − cos²θ. The square over the cosine is vital and was frequently dropped.
三角恒等式有时被误用。例如，用 sin²θ + cos²θ = 1 替换 sin²θ 时，学生正确地写出 sin²θ = 1 − cos²θ，但在代入时却错误地写成 1 − cos θ 而不是 1 − cos²θ。余弦上的平方至关重要，却经常被漏掉。
8. Coordinate Geometry | 坐标几何
Finding the midpoint and length of a line segment was generally well done, but errors surfaced when using the gradient formula (y₂ − y₁)/(x₂ − x₁) with negative coordinates. For A(−3, 4) and B(1, −2), some wrote gradient = (−2 − 4)/(1 − −3) which simplifies to −6/4 = −3/2, but they computed −2 − 4 as −2, yielding an incorrect gradient. Double signs must be handled carefully.
求线段的中点和长度普遍完成得不错，但在使用斜率公式 (y₂ − y₁)/(x₂ − x₁) 且坐标为负时会出现错误。对于 A(−3, 4) 和 B(1, −2)，有人写作 (−2 − 4)/(1 − −3) 化简为 −6/4 = −3/2，但他们把 −2 − 4 算成了 −2，得出错误斜率。双重符号必须小心处理。
The equation of a perpendicular bisector involved both the midpoint and the negative reciprocal gradient. A common mistake was to forget to take the negative reciprocal, using the same gradient as the original line instead. Another was to use the midpoint as the point on the original line, but mis-calculate its coordinates.
求垂直平分线方程需要同时用到中点和负倒数斜率。常见错误是忘记取负倒数，而直接使用原直线的斜率。另一个错误是把中点当作原直线上的点，却算错了中点坐标。
Intersection of two lines often led to algebraic slips when solving simultaneous equations. If the equations were 2x + 3y = 12 and y = 2x − 1, substitution was straightforward, but candidates mixed up x and y after finding one variable. Clearly labelling which variable has been solved helps avoid this.
求两直线交点时，解联立方程常产生代数错误。如果方程是 2x + 3y = 12 和 y = 2x − 1，代入法很简单，但考生在求出一个变量后会把 x 和 y 搞混。清楚地标注已解出的变量有助于避免这种情况。
9. Exponential and Logarithmic Equations | 指数与对数方程
When solving equations like 2ˣ = 5, candidates correctly took logs of both sides: x log 2 = log 5, but then divided incorrectly, writing x = log 5 / log 2, which is right, but some typed it into calculators as log(5/2) by mistake, obtaining a completely different value. The mark scheme required the correct decimal to at least 2 d.p. unless exact form was asked.
在解方程 2ˣ = 5 时，考生正确地对两边取对数：x log 2 = log 5，但在除法时出错，虽然写成 x = log 5 / log 2 是对的，但有些人在计算器里错误地输入成 log(5/2)，得到完全不同的值。评分方案要求至少精确到小数点后两位，除非题目要求精确形式。
Laws of logs were misapplied frequently. log a + log b = log (a + b) was a serious error. The correct law is log a + log b = log (ab). Similarly, k log a = log aᵏ, not log (k a). Candidates needed to simplify expressions like 2 log x + log y to log(x²y) to solve equations.
对数运算法则频繁被误用。log a + log b = log (a + b) 是一个严重错误。正确的法则是 log a + log b = log (ab)。类似地，k log a = log aᵏ，而不是 log (k a)。考生需要把像 2 log x + log y 这样的表达式化简为 log(x²y) 以解方程。
Equations that required converting between index form and log form, such as log₃(2x − 1) = 2, were sometimes solved by writing 2x − 1 = 2³ = 8, but then candidates subtracted 1 from 8 instead of adding. Step-by-step checking prevents such slips.
需要在指数形式和对数形式之间转换的方程，如 log₃(2x − 1) = 2，有时候学生能正确写出 2x − 1 = 3² = 9，但在后续计算中却把 9 减 1 而不是加 1。逐步检查可以避免这类失误。
10. Sequences and Series | 数列与级数
Arithmetic sequences were a common topic. The formula for the nth term a + (n − 1)d was often used correctly, but some substituted n = 1 instead of n = 10 when asked for the 10th term, or they confused d and a. In one question, d = −3 but candidates used +3, leading to a wrong sequence.
等差数列是一个常见考点。通项公式 a + (n − 1)d 经常被正确使用，但有些题目要求第 10 项，有人却代入 n = 1，或者把 d 和 a 搞混。比如 d = −3 却用了 +3，导致数列错误。
The sum of an arithmetic series Sₙ = n/2 [2a + (n − 1)d] caused issues when n was large and students tried to add terms manually. Using the formula correctly required careful substitution of a and d. A typical error was computing (n − 1)d first without multiplying by n/2 at the correct stage, leading to an arithmetic slip.
等差数列求和公式 Sₙ = n/2 [2a + (n − 1)d] 在 n 较大而学生试图手动相加时出现问题。正确使用公式需要仔细代入 a 和 d。典型错误是先计算 (n − 1)d，却没有在正确的阶段乘以 n/2，导致计算错误。
Geometric sequences required the formula for the nth term arⁿ⁻¹. Mistakes included using arⁿ or writing the ratio r as a fraction incorrectly. For a sequence 3, 6, 12, …, r = 2, but for a descending sequence 100, 50, 25, …, r = ½ was sometimes mistakenly written as 2 or as 0.5 but used as 0.5 × 100 = 50, which is correct, but then the next term was often miscalculated.
等比数列需要用到通项公式 arⁿ⁻¹。错误包括使用 arⁿ，或把公比 r 写成分数时弄错。对于数列 3, 6, 12, …, r = 2，但对于递减数列 100, 50, 25, …, r = ½ 有时被错误地写成 2，或者虽然正确写成 0.5，但在求下一项的时候算错。
11. Integration with Boundary Conditions | 利用边界条件的积分
When given f'(x) and a point on the curve, candidates needed to integrate and then find the constant c. Many integrated correctly but then substituted the point wrongly, for instance swapping x and y or forgetting to include c in the equation before solving. A classic slip was writing y = 2x² + c, substituting (3, 20) as 20 = 2(3)² → 20 = 18, so c = 2, which is correct, but then not writing the final equation y = 2x² + 2.
当已知 f'(x) 和曲线上一点，考生需要积分然后求常数 c。很多人积分正确，但在代入点时出错，比如把 x 和 y 交换，或者解方程前忘记把 c 包括在内。一个经典失误是：写出 y = 2x² + c，代入 (3, 20) 得 20 = 2(3)² → 20 = 18，得出 c = 2（正确），但最后没写出方程 y = 2x² + 2。
For definite integrals used to find the area under a curve between two x-values, candidates sometimes forgot to multiply by the factor outside the integral. For example, area = 3 ∫₁² (x²) dx, they integrated x² to x³/3, applied limits, but omitted the factor 3. This led to an answer that was one-third of the required value.
用定积分求曲线与 x 轴在两点间的面积时，考生有时忘记乘以积分号外面的系数。例如，面积 = 3 ∫₁² (x²) dx，他们把 x² 积分为 x³/3，代入上下限，却漏掉了系数 3。这样得到的答案是所需值的三分之一。
12. Modelling with Quadratics | 二次函数建模
Real-life context problems such as projectile motion or maximum profit forced students to form a quadratic expression. Many misidentified the variables, setting the height equation as h = −t² + 5t + 6 but then solving for t when h = 0 gave t = 6 and t = −1. Discarding the negative solution was often forgotten, or they kept both and stated ‘t = −1 or 6’, which is not meaningful in context.
现实情境问题如抛体运动或最大利润问题，要求学生列出一个二次表达式。很多人识别变量出错，例如设高度方程为 h = −t² + 5t + 6，然后解 h = 0 得到 t = 6 和 t = −1。他们常常忘记舍去负值解，或者保留两个解写 ‘t = −1 或 6’，这在具体情境中毫无意义。
Maximising or minimising a quadratic by completing the square was another area where candidates struggled. Given y = −2x² + 12x − 5, they factorised out −2, obtaining y = −2(x² − 6x) − 5, then completed the square to y = −2[(x − 3)² − 9] − 5 = −2(x − 3)² + 18 − 5 = −2(x − 3)² + 13, but then some wrote the maximum point as (3, −13) forgetting the +18 − 5. Precision in arithmetic is key.
用配方法求二次函数的最大值或最小值是另一个考生感到棘手的领域。给定 y = −2x² + 12x − 5，他们提取 −2，得到 y = −2(x² − 6x) − 5，接着配方为 y = −2[(x − 3)² − 9] − 5 = −2(x − 3)² + 18 − 5 = −2(x − 3)² + 13，但有些人把最大值点写成 (3, −13)，忘了 +18 − 5。算术必须仔细。
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