📚 Common Mistakes in AS Maths Unit 2 (Jan 2020) – Key Error-prone Areas | AS数学Unit 2（2020年1月卷）易错点总结
The January 2020 AS Mathematics Unit 2 paper was designed to test core pure mathematical skills including differentiation, trigonometry, logarithms, sequences, and coordinate geometry. Many students lost marks not because they didn’t understand the concepts, but because they made small yet critical errors under exam pressure. This article summarises the most common mistakes seen in that paper, helping you avoid them in your own revision and future exams.
2020年1月的AS数学Unit 2试卷旨在考查纯数核心技能，涵盖微分、三角学、对数、数列与坐标几何等内容。许多学生丢分并非因为不理解概念，而是在考试压力下犯了细小但致命的错误。本文总结了该卷中最常出现的典型错误，帮助你在复习和应考时有效避坑。
1. Misapplying the Chain Rule in Differentiation | 微分中错误应用链式法则
When differentiating composite functions such as y = (3x² + 5)⁴, many candidates correctly identified the need for the chain rule, but either forgot to multiply by the derivative of the inner function or multiplied incorrectly.
对复合函数如 y = (3x² + 5)⁴ 求导时，许多考生能识别出需用链式法则，但要么忘了乘以内层函数的导数，要么乘错了内层导数。
The correct derivative is dy/dx = 4(3x² + 5)³ · (6x). A common mistake was writing 4(3x² + 5)³ only, omitting the factor 6x, or incorrectly differentiating 3x² + 5 as 3x instead of 6x.
正确导数应为 dy/dx = 4(3x² + 5)³ · (6x)。常见错误是只写了 4(3x² + 5)³，漏掉了因子 6x，或者将 3x² + 5 的导数错求为 3x 而不是 6x。
Always write out u = inner function and du/dx explicitly before applying the formula dy/dx = dy/du · du/dx. This step-by-step approach prevents rushing and losing the inner derivative.
务必在套用公式 dy/dx = dy/du · du/dx 之前，明确写出 u = 内层函数及 du/dx。这种分步方法能防止匆忙中遗漏内层导数。
2. Incorrectly Solving Trigonometric Equations | 解三角方程时丢失解
Trigonometric equations in the 0° to 360° range (or 0 to 2π radians) often cause students to stop after finding only the principal solution from the calculator. In the January 2020 paper, a question involving 2sin²x − sin x − 1 = 0 led many to omit one or more valid solutions.
在0°到360°（或0到2π弧度）区间内解三角方程时，学生常犯的错误是仅找到计算器给出的主值就止步不前。2020年1月试卷中有一道涉及 2sin²x − sin x − 1 = 0 的题，许多人因此遗漏了一个甚至多个有效解。
For example, the quadratic in sin x gives sin x = 1 or sin x = −½. For sin x = 1, the obvious solution is x = 90°, but the cosine or sine graph symmetry means other solutions exist. For sin x = −½, the calculator gives x = −30° (or 330°), but x = 210° is also valid. A complete solution requires drawing the CAST diagram or the sine graph to find all angles within the given interval.
比如，这个关于 sin x 的二次方程给出 sin x = 1 或 sin x = −½。对 sin x = 1，显然有 x = 90°，但若区间要求0°到360°，则不存在其他解；而对 sin x = −½，计算器给出 x = −30°（或330°），但 x = 210° 同样有效。完整求解需画出CAST图或正弦图像，找出给定区间内的所有角。
The error is often forgetting that sin(180° − θ) = sin θ and sin(180° + θ) = −sin θ, leading to incomplete answer sets and lost accuracy marks.
常见错误是忘记了 sin(180° − θ) = sin θ 和 sin(180° + θ) = −sin θ 等关系，导致解集不全，丢掉准确度分。
3. Errors in Logarithmic Manipulation | 对数运算常见错误
Logarithms appeared in the context of solving equations like 3²ˣ = 5. Many students took logs on both sides but incorrectly applied the power rule, writing 2x · log 3 = log 5 as 2x + log 3 = log 5 or misplacing parentheses.
对数出现在解诸如 3²ˣ = 5 的方程中。许多学生两边同时取对数后，错误地应用了幂法则，将 2x · log 3 = log 5 误写成 2x + log 3 = log 5，或是括号位置出错。
The correct working is: log(3²ˣ) = log 5 → 2x log 3 = log 5 → x = log 5 / (2 log 3). A typical slip is writing x = log 5 / 2 log 3, which a calculator might interpret as (log 5 / 2) × log 3 unless properly bracketed.
正确的推导应为：log(3²ˣ) = log 5 → 2x log 3 = log 5 → x = log 5 / (2 log 3)。常见笔误是写成 x = log 5 / 2 log 3，若不加括号，计算器可能会理解为 (log 5 / 2) × log 3。
Also, when simplifying logₐ b + logₐ c = logₐ(bc), many incorrectly extended this to logₐ(b + c), which has no simplification. Recognising that logₐ(b + c) ≠ logₐ b + logₐ c prevents critical sign and value errors.
此外，在化简 logₐ b + logₐ c = logₐ(bc) 时，许多人错误地将其推广到 logₐ(b + c)，认为它也可拆分，然而 logₐ(b + c) 没有任何简化形式。认清 logₐ(b + c) ≠ logₐ b + logₐ c 能避免严重的符号与数值错误。
4. Forgetting the Constant of Integration | 忘记积分常数C
Indefinite integration questions consistently catch students out, and the Jan 2020 paper was no exception. After integrating a function like f ‘(x) = 4x³ − 6x + 1, many gave the answer as x⁴ − 3x² + x, omitting the crucial ‘+ C’.
不定积分问题始终是学生的失分重灾区，2020年1月卷也不例外。对 f ‘(x) = 4x³ − 6x + 1 积分后，许多学生给出的答案是 x⁴ − 3x² + x，漏掉了至关重要的 ‘+ C’。
Even when given a point to find the particular solution, candidates often forgot to include C initially, then substituted the coordinate into an expression without the constant, ruining the entire part of the question.
即使题目给出某点以求特解，考生也常忘记先写 ‘+ C’，然后将坐标代入一个没有常数的表达式，导致整个大题部分全错。
Train yourself to write ‘+ C’ automatically after every indefinite integral, and only then use additional information to determine C. Examiners always reserve at least one mark for the constant.
请训练自己在每次完成不定积分后，惯性地写上 ‘+ C’，然后再利用额外信息求出 C。考官总会为常数的保留至少一分。
5. Mistakes in Coordinate Geometry: Distance and Midpoint | 坐标几何中的距离与中点错误
A straightforward question involving the distance between two points (x₁, y₁) and (x₂, y₂) became a common source of error when students forgot to square the differences or mixed up the midpoint formula with the gradient formula.
一道涉及两点 (x₁, y₁) 与 (x₂, y₂) 间距的直接题，因学生忘记将差值平方，或将中点公式与斜率公式混淆，成了常见错误源。
The distance formula is √[(x₂ − x₁)² + (y₂ − y₁)²]. A typical blunder was to compute √[(x₂ − x₁) + (y₂ − y₁)], omitting the squares. Similarly, the midpoint was sometimes given as (x₂ − x₁)/2 instead of (x₁ + x₂)/2.
距离公式为 √[(x₂ − x₁)² + (y₂ − y₁)²]。典型的硬伤是计算成了 √[(x₂ − x₁) + (y₂ − y₁)]，漏了平方。同样，中点有时被写成 (x₂ − x₁)/2 而非 (x₁ + x₂)/2。
When using the circle equation (x − a)² + (y − b)² = r², students occasionally wrote the centre as (−a, −b) instead of (a, b), revealing confusion with the sign inside the brackets.
在使用圆方程 (x − a)² + (y − b)² = r² 时，学生偶尔将圆心写成 (−a, −b) 而非 (a, b)，暴露出对括号内符号的理解混乱。
6. Mishandling Algebraic Fractions | 错误处理代数分式
Adding and subtracting algebraic fractions, such as (3/(x+1)) + (2/(x−2)), required finding a common denominator. Many candidates attempted to cross-multiply incorrectly, resulting in expressions like (3(x−2) + 2(x+1)) / ((x+1)+(x−2)), which is wholly wrong.
代数分式的加减，如 (3/(x+1)) + (2/(x−2))，需要通分。许多考生尝试交叉相乘，但方法错误，得出类似 (3(x−2) + 2(x+1)) / ((x+1)+(x−2)) 的式子，这完全错误。
The correct common denominator is the product (x+1)(x−2), giving the numerator 3(x−2) + 2(x+1). Errors also occurred when expanding brackets, particularly with negative signs.
正确的公分母应为乘积 (x+1)(x−2)，分子为 3(x−2) + 2(x+1)。展开括号时也频频出错，尤其是涉及负号时。
Another pitfall was cancelling terms incorrectly before establishing a common denominator, such as cancelling the ‘x’s in (3x/(x+1)) with the denominator, which violates fundamental algebraic rules.
另一大陷阱是在未通分前约项，例如把 (3x/(x+1)) 中的 x 与分母约掉，这违反了基本的代数运算法则。
7. Misinterpreting Iterative Methods | 迭代法理解偏差
When a recurrence formula like xₙ₊₁ = ½(xₙ + 3/xₙ) was given, students often miscalculated the first few iterations by substituting incorrectly or not using enough decimal places, causing the subsequent values to drift and lose marks for accuracy.
当给出如 xₙ₊₁ = ½(xₙ + 3/xₙ) 的递推公式时，学生常因代入错误或未保留足够小数位而算错前几次迭代，导致后续值偏差，失去精度分。
A frequently seen error was writing x₂ = ½(x₁ + 3/x₁) but using x₀ instead of x₁, or confusing the index numbering. Remember that x₁ is substituted to find x₂, and so on.
一个常见错误是写对了式子 x₂ = ½(x₁ + 3/x₁)，却代入了 x₀ 而非 x₁，或是搞混了下标编号。应牢记由 x₁ 代入求得 x₂，依次类推。
Always record calculations to at least 5 decimal places, even if the final answer requires rounding. Premature rounding was a significant reason for losing the final accuracy mark in this trial paper.
计算过程中务必至少保留5位小数，即使最终答案要求四舍五入。提前舍入是在这份模拟卷中丢失最终准确度分的重要原因。
8. Problems with Domain and Range in Functions | 函数定义域与值域问题
Questions involving the inverse function f⁻¹(x) required stating the domain of the inverse, which equals the range of the original function. Many students simply gave the domain of the original function, failing to appreciate the swap.
涉及反函数 f⁻¹(x) 的题目要求写出反函数的定义域，它等于原函数的值域。许多学生直接照搬原函数的定义域，未能领悟这种互换关系。
For a function defined as f(x) = x² + 1 for x ≥ 0, the range is [1, ∞), so the inverse function f⁻¹(x) = √(x − 1) has domain [1, ∞). A common wrong answer was to state the domain of f⁻¹ as [0, ∞).
例如，对于 f(x) = x² + 1 且 x ≥ 0，值域为 [1, ∞)，因此反函数 f⁻¹(x) = √(x − 1) 的定义域为 [1, ∞)。常见的错误是将反函数定义域写成 [0, ∞)。
Furthermore, when sketching graphs of functions and their inverses, some candidates reflected across the line x = 0 (y-axis) instead of y = x, yielding a graph that was not the correct inverse.
此外，在画函数及其反函数的图像时，部分考生以直线 x = 0（y轴）为对称轴反射，而非以 y = x 反射，结果画出的并非正确的反函数图像。
9. Sign Errors in the Binomial Expansion | 二项式展开中的符号错误
The expansion of (1 + 2x)⁻² using the binomial theorem for negative powers was a classic slip-up area. Students often forgot that the term in the expansion contains minus signs from the negative exponent and from the negative coefficient of x when expanding (1 − 3x)⁻¹ etc.
利用负指数二项式定理展开 (1 + 2x)⁻² 是经典的易错区。学生常忘记展开式中既含有来自负指数的负号，又有来自 x 的负系数带来的负号，例如展开 (1 − 3x)⁻¹ 时。
The general expansion (1 + ax)ⁿ = 1 + nax + [n(n−1)/2!](ax)² + … works for |ax| < 1. Substituting negative n correctly requires careful bracket usage. A common error: expanding (1 − 2x)⁻³, the second term is (−3)(−2x) = +6x, but many wrote −6x due to a sign slip.
一般展开式 (1 + ax)ⁿ = 1 + nax + [n(n−1)/2!](ax)² + … 对 |ax| < 1 成立。准确代入负指数 n 要求小心使用括号。常见错误：展开 (1 − 2x)⁻³ 时，第二项应为 (−3)(−2x) = +6x，但不少人因符号疏忽写成了 −6x。
Remember to check that the expansion is valid for the given x; stating the validity range |ax| < 1 earned a mark that many candidates threw away by not writing it down.
记住，需确认展开式对所给 x 有效；写出有效范围 |ax| < 1 就能拿下一分，而这一分却被许多考生因为没写而白白丢掉。
10. Misreading the Question in Sequences and Series | 数列与级数审题失误
An arithmetic sequence problem asked for the sum of the first 20 terms, but several students found the 20th term instead, or used the wrong formula (geometric sum formula for an arithmetic series).
一道等差数列题要求计算前20项之和，但不少学生却求了第20项，或者用了错误的公式（如用等比数列求和公式处理等差数列）。
The sum of an arithmetic series is Sₙ = n/2 [2a + (n−1)d]. A common misinterpretation was to plug n = 20 but then compute the nth term a + (n−1)d as the answer, confusing term and sum. Another typical error was using n/2 (a + l) with an incorrect last term l.
等差数列求和公式为 Sₙ = n/2 [2a + (n−1)d]。常见的误解是代入 n=20 后，却计算了第 n 项 a + (n−1)d 作为答案，混淆了单项与总和。另一个典型错误是用了 n/2 (a + l) 但末项 l 却求错了。
For geometric series, students sometimes misidentified the first term a or the common ratio r when the expression given was not in standard form, such as Σ 2·3ᵏ⁻¹ from k=0.
对于等比数列，当求和符号给出的形式不是标准形时，比如 Σ 2·3ᵏ⁻¹ 从 k=0 开始，学生有时会错判首项 a 或公比 r。
11. Using Degrees instead of Radians | 弧度与角度混用
In calculus and trigonometric equations, the mode of your calculator can make or break your answer. A notable error in the Jan 2020 paper was applying radian formulas while the calculator remained in degree mode, leading to nonsensical gradient values in trig differentiation.
在微积分和三角方程中，计算器的模式可能决定答案的对错。2020年1月卷中的一个显著错误是，在使用弧度公式时计算器却停留在角度模式，导致在三角微分中得出荒谬的梯度值。
For example, differentiating y = sin(2x) gives dy/dx = 2cos(2x). If you evaluate at x = 0.5, you must treat 0.5 as radians, not degrees. A calculator in degree mode would interpret 0.5° instead of 0.5 rad, producing a completely different numeric answer.
例如，对 y = sin(2x) 求导得 dy/dx = 2cos(2x)。若在 x = 0.5 处求值，必须将 0.5 视为弧度，而非角度。处于角度模式的计算器会将其理解为 0.5°，而非 0.5 弧度，得出完全不同的数值答案。
Always double-check the radian mode (RAD) on your calculator, especially when the question involves π or has no degree symbol. Most AS pure topics assume radian measure unless specified otherwise.
务必反复确认计算器处于弧度模式（RAD），尤其是当题目中出现 π 或无度数符号时。AS纯数大多数专题默认使用弧度制，除非另有说明。
12. Improper Use of the Quadratic Formula | 二次公式使用不当
When solving quadratic equations arising in various contexts, the quadratic formula x = [−b ± √(b² − 4ac)] / (2a) was sometimes applied incorrectly. A frequent mistake was computing the discriminant b² − 4ac as b² − 4ac but then dividing by 2a before applying the square root, or misreading the sign of b.
在各种情景下解一元二次方程时，公式 x = [−b ± √(b² − 4ac)] / (2a) 有时被错误套用。常见错误是算出了判别式 b² − 4ac，却先除以 2a 再开方，或者错读了 b 的符号。
For example, to solve 2x² − 3x − 5 = 0, a=2, b=−3, c=−5. Some students wrote x = [−3 ± √(9 − 40)] / 4, missing the double negative in the −b term: it should be −(−3) = 3.
例如，解 2x² − 3x − 5 = 0，其中 a=2，b=−3，c=−5。一些学生写成 x = [−3 ± √(9 − 40)] / 4，遗漏了 −b 中的双重负号：正确应为 −(−3) = 3。
Also, when the discriminant is negative, concluding there are “no solutions” without considering complex numbers is acceptable at AS level, but writing “no real solutions” is more precise and demonstrates understanding.
此外，当判别式为负时，在AS阶段下结论说“无解”虽可接受，但更严谨的说法是“无实数解”，这才能体现理解深度。
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