Common Mistakes in AS Maths Unit 1 (June 2022 Mark Scheme) | AS数学单元1（2022年6月评分方案）易错点总结

📚 Common Mistakes in AS Maths Unit 1 (June 2022 Mark Scheme) | AS数学单元1（2022年6月评分方案）易错点总结

Understanding where marks are lost in the June 2022 AS Pure Mathematics Unit 1 paper is crucial for targeted revision. This article analyses the most frequent errors candidates made, as highlighted in the official mark scheme, and explains how to avoid them in future assessments.

了解2022年6月AS纯数学单元1试卷中易失分的地方对于针对性复习至关重要。本文分析了官方评分方案中指出的考生最常犯的错误，并解释了如何在今后的考试中避免这些错误。

1. Misapplying the Distributive Law in Algebraic Expansion | 代数展开中错误使用分配律

A common error arose when expanding expressions such as (x+3)(x-2)(x+1). Many candidates correctly multiplied the first two brackets but then failed to distribute every term of the resulting quadratic across the third linear factor. This led to missing terms or sign errors in the final cubic expansion. The mark scheme requires fully expanded and simplified polynomials, so any omission or mis-sign costs accuracy marks.

在展开如(x+3)(x-2)(x+1)这样的表达式时，常见错误是考生正确相乘了前两个括号，但随后未能将所得二次式的每一项都与第三个一次因式相乘。这导致最终的三次展开式中漏项或符号错误。评分方案要求完全展开并化简多项式，因此任何遗漏或符号错误都会导致准确度分数丢失。

2. Errors in Solving Quadratic Equations by Factorisation | 因式分解解二次方程的错误

When solving quadratic equations like 2x² – 5x – 3 = 0, a significant number of students wrote the factors as (2x-1)(x+3) instead of the correct (2x+1)(x-3). This mix-up happens when the signs of the constant term and coefficient of x are not carefully cross-checked. The mark scheme typically awards method marks for setting up the factor pairs, but final answer marks are only given for the correct roots x = -½ and x = 3.

在解像2x² – 5x – 3 = 0这样的二次方程时，相当多的学生将因式误写为(2x-1)(x+3)，而正确答案是(2x+1)(x-3)。这种混淆发生在常数项和x项系数的符号未被仔细交叉核对时。评分方案通常会为设置因式对的过程给出方法分，但最终答案分只给正确的根x = -½ 和 x = 3。

3. Mishandling the Leading Coefficient in Completing the Square | 配方法中首项系数的处理不当

For a quadratic like 3x² + 12x + 5, candidates often wrote the completed square form as 3(x+2)² – 7 or 3(x+2)² – 4. The correct expression is 3(x+2)² – 7, but many forgot to multiply the constant adjustment by the leading coefficient. The mark scheme insists on working inside the bracket first: 3[x² + 4x] + 5 → 3[(x+2)² – 4] + 5 → 3(x+2)² – 12 + 5 = 3(x+2)² – 7. Not doing so cost the final accuracy mark.

对于像3x² + 12x + 5这样的二次式，考生常将配方式写为3(x+2)² – 7或3(x+2)² – 4。正确的表达式是3(x+2)² – 7，但许多人忘记把括号内的常数调整项乘以首项系数。评分方案要求在括号内先操作：3[x² + 4x] + 5 → 3[(x+2)² – 4] + 5 → 3(x+2)² – 12 + 5 = 3(x+2)² – 7。未这样做会导致最终准确度分数丢失。

4. Discriminant Confusion and Inequality Direction | 判别式混淆与不等式方向

A typical question asked to find the set of values of k for which the equation x² + kx + 9 = 0 has no real roots. Candidates correctly set Δ < 0, but some wrote k² - 36 < 0 and then incorrectly solved it as k < -6 or k > 6, instead of -6 < k < 6. The mark scheme penalises this logical slip because the inequality sign must flip when dealing with a negative range in a quadratic inequality.

一个典型题目要求找出使方程x² + kx + 9 = 0没有实数根的k值集合。考生正确设出Δ < 0，但有人写出k² - 36 < 0后错误地解为k < -6 或 k > 6，而不是-6 < k < 6。评分方案会对此逻辑失误扣分，因为处理二次不等式的负值区间时，不等号方向必须正确翻转。

5. Substitution Errors in Simultaneous Equations | 联立方程中的代入错误

When solving a pair of equations, such as y = 2x – 1 and x² + y² = 10, many candidates substituted correctly but then made arithmetic mistakes when expanding (2x-1)². The error (2x-1)² = 4x² – 1 was surprisingly frequent, missing the middle term -4x. The mark scheme grants method marks for substitution and forming a quadratic, but the final solutions for x and y must be accurate; often both pairs of coordinates are required.

在解一组方程时，比如y = 2x – 1和x² + y² = 10，许多考生代入正确，但在展开(2x-1)²时出现算术错误。错误(2x-1)² = 4x² – 1出人意料地频繁，漏掉了中间项-4x。评分方案会为代入和构建二次方程给出方法分，但x和y的最终解必须准确；通常需要写出两对坐标。

6. Gradient of Perpendicular Lines and Negative Reciprocal | 垂直线的斜率与负倒数

In coordinate geometry, the relationship m₁m₂ = -1 for perpendicular lines is well known, but under exam pressure many candidates incorrectly used the rule m₂ = 1/m₁ or m₂ = -m₁. For a line with gradient 3/4, the perpendicular gradient is -4/3. Marks were lost when students wrote 4/3 or -3/4 instead. The mark scheme expects the negative reciprocal and will not award the method mark if the perpendicular gradient is not correctly identified.

在坐标几何中，垂直线的关系m₁m₂ = -1广为人知，但在考试压力下，许多考生错误地使用了m₂ = 1/m₁或m₂ = -m₁。对于斜率为3/4的直线，垂直线的斜率应为-4/3。当学生写成4/3或-3/4时会丢分。评分方案期望负倒数，如果垂直线斜率未被正确识别，将不给方法分。

7. Careless Mistakes in Basic Differentiation | 基础微分中的粗心错误

Differentiating functions like f(x) = 4/x² – 3√x caused frequent errors. Candidates forgot to rewrite terms as 4x⁻² – 3x½ before differentiating. Common slips included differentiating 4x⁻² as -4x⁻¹ (missing the power reduction) or keeping the square root as x½ together with the wrong coefficient. The mark scheme requires the derivative in its simplest form, e.g., f'(x) = -8x⁻³ – (3/2)x⁻½. Losing a negative sign or misplacing a fraction led to answer marks being withheld.

对像f(x) = 4/x² – 3√x这样的函数求导时经常出错。考生忘记在求导前将各项重写为4x⁻² – 3x½。常见的失误包括将4x⁻²微分为-4x⁻¹（漏掉了幂次下降），或者保留平方根符号但系数错误。评分方案要求以最简形式给出导数，例如f'(x) = -8x⁻³ – (3/2)x⁻½。丢失负号或分数位置错误会导致答案分不能被授予。

8. Omitting the Constant of Integration | 遗漏积分常数

In problems that required finding an equation of a curve from its derivative, e.g., dy/dx = 6x² – 2x and a point on the curve, candidates often integrated to get y = 2x³ – x² but then stopped. The mark scheme explicitly requires the ‘+ C’ to be introduced and then evaluated using the given point. Without ‘+ C’, no further marks could be earned even if the rest was correct, because the constant of integration is essential for the complete solution.

在需要从导数求曲线方程的问题中，例如dy/dx = 6x² – 2x以及曲线上一点，考生经常积分得到y = 2x³ – x²后就停住了。评分方案明确要求先引入’+ C’，然后利用给定点求值。没有’+ C’，即使其余部分正确，也无法得到后续分数，因为积分常数对于完整解至关重要。

9. Area Under a Curve: Misuse of Limits or Sign | 曲线下方面积：上下限误用或符号错误

Finding the area between a curve and the x‑axis, especially when the curve crosses the axis, caused difficulties. Some candidates integrated between the given x‑limits without checking whether the region was above or below the axis. For a part below the axis, the definite integral is negative, so the physical area must be taken as the absolute value or the integral of the negative function. The mark scheme penalised answers that gave a negative area without justification or that failed to split the integral at the root.

求曲线与x轴之间的面积，尤其是当曲线穿过轴线时，让很多考生感到困难。有些人直接在给定的x上下限之间积分，没有检查区域是在轴上方还是下方。对于轴下方的部分，定积分为负，因此实际面积必须取绝对值或对负函数积分。评分方案会惩罚那些没有合理解释就给出负面积，或未在根处分割积分的答案。

10. Exponential and Logarithm Equation Errors | 指数与对数方程错误

When solving 2e³ˣ⁺¹ = 10, weaker candidates sometimes took the natural logarithm incorrectly, writing ln(2e³ˣ⁺¹) = ln10 as 3x+1 + ln2 = ln10 instead of ln2 + 3x+1 = ln10. The rules for logs must be applied to the product as a whole: ln(ab) = lna + lnb. More seriously, some applied ln before isolating the exponential term, leading to entangled expressions. The mark scheme demands a clear step‑by‑step approach: divide by 2, then take ln.

在解2e³ˣ⁺¹ = 10时，基础较弱的考生有时错误地取自然对数，将ln(2e³ˣ⁺¹) = ln10写为3x+1 + ln2 = ln10，而不是ln2 + 3x+1 = ln10。对数法则必须应用于整个乘积：ln(ab) = lna + lnb。更严重的是，有些人在分离指数项之前就取ln，导致表达式纠缠不清。评分方案要求清晰的逐步操作：先除以2，再取ln。

11. Trigonometric Equations: Missing Solutions | 三角方程：漏解

A question on sinθ = 0.4 for 0° ≤ θ ≤ 360° often saw students giving only θ = 23.6° and moving on. They forgot the second solution in the range, θ = 180° – 23.6° = 156.4°, and sometimes omitted the fact that sin is positive in both the first and second quadrants. The mark scheme expects all solutions within the given interval, and answers that omit one or more valid solutions lose at least one mark per missing value.

一道关于sinθ = 0.4，0° ≤ θ ≤ 360°的题目中，许多学生只给出θ = 23.6°就结束了。他们忘记了在该区间内的第二个解θ = 180° – 23.6° = 156.4°，有时还忽略了正弦在第一和第二象限均为正的事实。评分方案期望在给定范围内给出所有解，遗漏一个或多个有效解的答案每缺失一个值至少扣一分。

12. Graph Transformations: Order of Operations | 图形变换：操作顺序

When describing the transformation of y = f(x) to y = -f(2x), candidates often reversed the order or described the stretch incorrectly. The correct sequence is a stretch parallel to the x‑axis by scale factor ½, followed by a reflection in the x‑axis. Writing ‘stretch by factor 2’ instead of ½ for the x‑direction was a typical mistake. The mark scheme is strict about the precise wording: both the type of transformation and its numerical parameter must be exact.

在描述从y = f(x)到y = -f(2x)的变换时，考生常常颠倒顺序或错误描述伸缩。正确顺序是平行于x轴、比例因子为½的伸缩，然后关于x轴的反射。典型错误是将x方向的伸缩写成“因子2”而不是½。评分方案对精确措辞要求严格：变换类型及其数值参数都必须准确无误。

Published by TutorHao | AS Pure Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）