📚 AS Level Maths Unit 2 (Jan 19) Common Mistake Summary | AS 数学第2单元2019年1月试卷易错点总结
The January 2019 AS Pure Mathematics Unit 2 paper tested core topics such as logarithms, trigonometry, calculus, sequences, and binomial expansion. Many students lost marks due to avoidable errors in algebraic manipulation, domain checking, and formula application. This article highlights the most common pitfalls and shows you how to avoid them.
2019年1月AS纯数第2单元试卷涵盖对数、三角、微积分、数列和二项展开等核心内容。许多学生因代数操作疏忽、未检查定义域或公式误用而失分。本文梳理高频易错点并给出避错策略。
1. Logarithmic Domain Errors | 忽略对数定义域
When solving equations like ln(3x – 2) = ln(x + 4), it is essential to check that arguments remain positive. Many candidates solved 3x – 2 = x + 4 to get x = 3, but failed to verify that both 3(3) – 2 = 7 > 0 and 3 + 4 = 7 > 0, so it is valid. However, in other cases, such as ln(x – 5) + ln(x + 2) = ln(10), after simplifying to (x – 5)(x + 2) = 10, candidates obtained x = 3, x = –4, but only x = 3 is valid because x – 5 must be > 0 → x > 5. Forgetting this check cost marks. Always write domain conditions before solving.
解方程如 ln(3x–2)=ln(x+4) 时，必须确保真数大于零。许多学生由 3x–2 = x+4 得 x=3，但未验证 3(3)–2=7>0 与 3+4=7>0，所幸这个解有效。然而在如 ln(x–5)+ln(x+2)=ln(10) 的题目中，化简得 (x–5)(x+2)=10，解得 x=3 与 x=–4，但只有 x=3 有效，因 x–5>0 要求 x>5。忽略此检查就会失分。解题前务必写出定义域条件。
2. Trapezium Rule Calculation Slips | 梯形法则计算失误
In the trapezium rule, using incorrect strip width h or misreading values from the table are common. Given h = (b – a)/n, some students forget to divide correctly or use the wrong number of strips. Also, the formula requires y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁). A typical mistake is to add all y-values and multiply by h, missing the factor of 2 for the intermediate ordinates. Always write the formula and substitute carefully. Double-check the table values and the number of intervals.
梯形法则中，常错误计算带宽 h 或读错表格值。h = (b–a)/n，有学生忘记除以 n 或误用段数。此外，公式要求 y₀+yₙ+2(y₁+y₂+…+yₙ₋₁)。常见错误是将所有 y 值相加再乘以 h，漏掉了中间值的加倍。解题时须写出公式并细心代入。反复核对表格数据和区间数量。
3. Trigonometric Equations with Multiple Angles | 多倍角三角方程求解疏忽
For equations like sin(2θ) = 0.5 for 0° ≤ θ ≤ 360°, some candidates solve 2θ = 30°, 150°, and then stop, only getting θ = 15°, 75°. They forget to add the periodicity: 2θ = 30° + 360°k and 150° + 360°k, so also 2θ = 390°, 510° giving θ = 195°, 255°. Without adding full cycles, solutions within the given θ-range are missed. Always adjust the range for the multiple angle first: 0° ≤ 2θ ≤ 720°, find all solutions for 2θ, then divide.
解方程如 sin(2θ)=0.5，θ∈[0°,360°]，有些学生解 2θ=30°,150°，然后仅得 θ=15°,75°。他们忘记了加周期：2θ = 30°+360°k 及 150°+360°k，从而还有 2θ=390°,510°，得 θ=195°,255°。未加完整周期便漏掉了给定区间内的解。应先调整倍角范围：0°≤2θ≤720°，求出 2θ 的所有解后再除以系数。
4. Differentiating Exponential and Logarithmic Functions | 指数与对数函数求导错误
When differentiating eᵏˣ, the derivative is keᵏˣ. A frequent mistake is forgetting the coefficient k. For example, d/dx(e³ˣ) = 3e³ˣ, not e³ˣ. For ln(kx), using the chain rule gives 1/x, but candidates sometimes incorrectly write 1/(kx) × k = 1/x, which is correct, but they might forget to simplify. Also, d/dx(ln(ax + b)) = a/(ax + b). In the Jan 19 paper, a question required differentiating y = 2e²ˣ – ln(3x + 1); many failed to combine correctly, writing 4e²ˣ – 1/(3x+1) instead of 4e²ˣ – 3/(3x+1). Be rigorous with the chain rule.
求导 eᵏˣ 时，结果为 keᵏˣ。常犯错误是漏掉系数 k。如 d/dx(e³ˣ) = 3e³ˣ，而非 e³ˣ。对 ln(kx)，使用链式法则得 1/x，但有的学生错误写成 1/(kx) × k =1/x 是对的，但他们忘记化简。对 d/dx(ln(ax+b)) = a/(ax+b)。在 Jan 19 试卷中，要求对 y=2e²ˣ–ln(3x+1) 求导，许多人错误地写成 4e²ˣ–1/(3x+1) 而非 4e²ˣ–3/(3x+1)。务必严谨使用链式法则。
5. Forgetting the Constant of Integration | 遗漏积分常数
Indefinite integrals must include the constant of integration, + C. In the Jan 19 paper, after finding an indefinite integral, a follow-up part asked for the function f(x) given a point on the curve. Candidates who omitted + C could not find the specific constant. Worse, in differential equations, missing + C leads to an incomplete solution. Always write “+ C” immediately after integrating.
不定积分必须包含积分常数 +C。Jan 19 试卷中，在求出不定积分后，后续问及已知曲线过某点求函数 f(x)，漏写 +C 的学生无法求出具体的常数值。更糟的是在微分方程中，缺少 +C 导致解不完整。积分后务必立即写上 “+C”。
6. Binomial Expansion Validity and Accuracy | 二项展开式的有效范围与近似精度
When expanding (1 + x)ⁿ with rational n, the expansion is valid only for |x| < 1 (or appropriate interval). In an approximation question, candidates substitute x = 0.1 into the expansion to estimate a value, but sometimes ignore that the remainder term should be estimated. In Jan 19, the question asked to expand (1+3x)¹/³ and use it to approximate ∛1.03. A common error was to misidentify x: here 3x = 0.03 → x = 0.01. Some students used x = 0.03 directly, leading to an incorrect approximation. Also, the expansion is valid for |3x| < 1 → |x| < 1/3; x = 0.01 is fine. Always check validity before using the expansion for approximation.
对有理数 n 展开 (1+x)ⁿ 时，展开式仅在 |x|<1（或相应区间）内有效。在近似题中，学生代入 x=0.1 估算，但有时忽视需要估计余项。Jan 19 试卷要求展开 (1+3x)¹/³ 并用以近似 ∛1.03。常见错误是错误识别 x：此处 3x=0.03→x=0.01。有学生直接用 x=0.03，导致近似错误。此外，展开有效需 |3x|<1→|x|<1/3，x=0.01 满足。务必在用展开式近似前验证有效性。
7. Arithmetic and Geometric Series Formula Confusion | 等差与等比数列公式混淆
In questions on savings or growth, applying the wrong formula is frequent. An arithmetic series uses a + (n–1)d for the nth term and Sₙ = n/2 [2a + (n–1)d]. A geometric series uses arⁿ⁻¹ and Sₙ = a(1 – rⁿ)/(1 – r) for |r| < 1. In Jan 19, a context with compound interest required geometric series, but some used arithmetic. Also, solving inequalities like Sₙ > 5000 with geometric series demands careful log manipulation: take logs correctly and remember to reverse the inequality if the log base is < 1 (natural log does not reverse). Misapplication leads to lost marks.
在储蓄或增长类问题中，用错公式很常见。等差数列第 n 项为 a+(n–1)d，前 n 项和 Sₙ=n/2[2a+(n–1)d]；等比数列用 arⁿ⁻¹ 和 Sₙ=a(1–rⁿ)/(1–r)（当 |r|<1）。Jan 19 中复利背景需等比数列，却有人用等差。此外，解不等式如等比数列和 Sₙ>5000 时，取对数要小心：正确取对，且若对数底小于1才需反转不等号（自然对数无需反转）。误用会导致失分。
8. Integration Using Partial Fractions | 部分分式积分失误
A question required integrating a rational function like (3x+5)/(x² + 3x + 2). The first step is to factor the denominator into (x+1)(x+2) and express as A/(x+1) + B/(x+2). Common errors: algebraic mistakes finding A and B, or forgetting that the integral of 1/(x+a) is ln|x+a|, and missing the absolute value or modulus. Also, some candidates integrated term by term but forgot the constant. After integrating, ensure you write ln|linear factor|. If a definite integral, evaluate carefully with limits.
需要积分有理函数如 (3x+5)/(x²+3x+2) 时，首先分解分母为 (x+1)(x+2)，并写成部分分式 A/(x+1)+B/(x+2)。常见错误：求 A、B 时计算错误，或者积分 1/(x+a) 得 ln|x+a| 而漏掉绝对值符号。还有学生在逐项积分后遗漏常数。若为定积分，代入上下限需仔细。积分后务必写上 ln|线性因子|。
9. Implicit Differentiation Oversights | 隐函数求导疏忽点
When differentiating an equation like x² + 2xy + y³ = 10 with respect to x, apply the product rule for terms like xy, and multiply by dy/dx for y terms. Many lost marks by differentiating 2xy as 2y + 2x(dy/dx) but then forgetting to include dy/dx when differentiating y³, writing 3y² instead of 3y²(dy/dx). Also, after finding dy/dx, candidates must simplify and sometimes substitute a point to find the gradient of a tangent. Check that the point satisfies the original equation.
对方程如 x²+2xy+y³=10 关于 x 求导时，对项 xy 使用乘法法则，对含 y 项乘以 dy/dx。许多人将 2xy 求导得 2y+2x(dy/dx)，但在求导 y³ 时忘记乘 dy/dx，仅写 3y² 而非 3y²(dy/dx)。此外，求出 dy/dx 后，需要代入点坐标求切线斜率，务必检查该点是否满足原方程。
10. Graphical Interpretation and Inequalities | 图形解读与不等式
A question involved sketching f(x) = |2x – 3|, or solving an inequality like |2x – 3| > x. Students often treat the modulus by squaring both sides without considering cases,
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