AS Maths Unit 2 Mark Scheme Jan 20 Common Mistakes Summary | AS数学Unit 2 2020年1月评分方案易错点总结

📚 AS Maths Unit 2 Mark Scheme Jan 20 Common Mistakes Summary | AS数学Unit 2 2020年1月评分方案易错点总结

This article breaks down the most frequent errors seen in the January 2020 AQA AS Mathematics Unit 2 (7356/2) Paper, focusing on pure mathematics and mechanics. Each section is based on actual mark scheme notes and examiner reports, helping you avoid the same pitfalls and secure full marks in future exams.

本文深入分析2020年1月AQA AS数学Unit 2（试卷代号7356/2）考试中最常见的错误，涵盖纯数学和力学内容。每一部分都依据真实评分方案和考官报告编写，帮助你避开雷区，在后续考试中稳拿满分。

1. Incorrect Factorisation of Quadratics with a ≠ 1 | 二次项系数不为1时的错误因式分解

Many candidates lost marks when factorising expressions like 3x² − 10x + 8. The most common mistake was writing (3x − 2)(x − 4) as the final answer without checking the expansion. The correct factorisation is (3x − 4)(x − 2). Always multiply out mentally to verify: 3x × x = 3x², 3x × (−2) = −6x, (−4) × x = −4x, sum to −10x, and (−4) × (−2) = +8.

许多考生在分解如3x² − 10x + 8这样的二次式时丢分。最常见错误是直接写下(3x − 2)(x − 4)而不验证展开式。正确的因式分解为(3x − 4)(x − 2)。务必心算验证：3x × x = 3x²，3x × (−2) = −6x，(−4) × x = −4x，合并得−10x，常数项(−4) × (−2) = +8。

2. Misreading Domain Restrictions in Inverse Functions | 反函数定义域限制的误读

When finding the inverse of a function like f(x) = x² − 4x + 7, x ≥ 2, several students forgot to state the domain of f⁻¹(x) or gave it incorrectly as x ∈ ℝ. The correct domain of f⁻¹ is the range of f, which is [3, ∞) because the vertex of the quadratic is at x = 2, giving f(2) = 3. The mark scheme required stating f⁻¹(x) = 2 + √(x − 3) with domain x ≥ 3.

在求函数f(x) = x² − 4x + 7, x ≥ 2的反函数时，一些学生忘记标注f⁻¹(x)的定义域，或错误地写成x ∈ ℝ。反函数的正确定义域是原函数的值域，即[3, ∞)，因为二次函数顶点在x = 2处，f(2) = 3。评分方案要求写出f⁻¹(x) = 2 + √(x − 3)，且定义域为x ≥ 3。

3. Sign Errors When Expanding Brackets with Minus Signs | 去括号时符号错误

Dropping a negative sign when expanding expressions such as (x + 3) − 2(x − 1) was extremely common. Many wrote x + 3 − 2x − 2, missing that −2 × (−1) is +2. The correct simplification is x + 3 − 2x + 2 = −x + 5. Examiners noted that such slips lost the first method mark and often led to subsequent working being awarded zero.

在展开类似(x + 3) − 2(x − 1)的式子时，丢失负号极为常见。许多人写成x + 3 − 2x − 2，忽略了−2 × (−1) = +2。正确化简应为x + 3 − 2x + 2 = −x + 5。考官指出，这类失误会丢掉第一个方法分，并常导致后续过程得零分。

4. Radian vs Degree Mode in Trigonometric Equations | 三角函数方程中弧度与角度的混淆

The equation sin 2θ = 0.6 for 0 ≤ θ ≤ π required solving in radians. A significant minority had their calculator in degree mode, obtaining θ = 18.4° and 71.6°, instead of the correct radian measures 0.322 and 1.25 (to 3 s.f.). Always check the mode before starting a trigonometric question—rewriting the interval as 0 to 3.14 can help remind you.

题目要求解sin 2θ = 0.6，区间0 ≤ θ ≤ π，必须使用弧度制。不少人计算器仍处于角度模式，求得θ = 18.4°和71.6°，而不是正确的弧度值0.322和1.25（保留三位有效数字）。在处理三角题目前务必检查计算器模式——将区间重写为0到3.14有助提醒自己。

5. Misapplying the Chain Rule in Differentiation | 链式求导法则的错误应用

When differentiating y = (2x³ − 5)⁴, some candidates forgot to multiply by the derivative of the inner function, writing dy/dx = 4(2x³ − 5)³. The complete derivative is dy/dx = 4(2x³ − 5)³ × 6x² = 24x²(2x³ − 5)³. Another error was misusing the power: differentiating the outside as 4(…)³ is correct, but then students often multiplied by the derivative of 2x³ − 5 incorrectly as just 2 instead of 6x².

对y = (2x³ − 5)⁴求导时，部分考生忘记乘以内部函数的导数，写成dy/dx = 4(2x³ − 5)³。完整的导数是dy/dx = 4(2x³ − 5)³ × 6x² = 24x²(2x³ − 5)³。另一个错误是幂次使用不当：对外层求导得4(…)³是正确的，但随后学生常将内部函数2x³ − 5的导数错乘为2，而不是6x²。

6. Area Under a Curve: Forgetting to Subtract the Baseline | 曲线下方面积：忘记减去基线

In a question requiring the area enclosed between a curve y = 4 − x² and the line y = 2 − x, many integrated only the curve from x = −2 to x = 2, ignoring that the area is the integral of the difference of the two functions. The correct setup was ∫₋₂² [(4 − x²) − (2 − x)] dx. Failing to subtract the line function led to an overestimated area and zero marks for accuracy.

在求曲线y = 4 − x²与直线y = 2 − x所围面积的问题中，许多人仅从x = −2到x = 2对曲线积分，忽略了面积应为两函数之差的积分。正确设置是∫₋₂² [(4 − x²) − (2 − x)] dx。未减去直线函数导致面积被高估，准确性得分为零。

7. Mechanics: Confusing Displacement and Distance in Kinematics | 力学：运动学中位移与路程的混淆

A particle moving along a straight line with velocity v = 2t − 5 (m/s) for 0 ≤ t ≤ 6 required the distance travelled. Many simply integrated velocity to get displacement from t = 0 to t = 6, obtaining −3 m, and then took absolute value as distance. The correct method was to identify the turning point at t = 2.5, compute the integral in two stages (from 0 to 2.5 and 2.5 to 6) and sum the absolute values to get the total distance of 14.5 m.

一质点沿直线运动，速度v = 2t − 5 (m/s)，时间区间0 ≤ t ≤ 6，求路程。许多考生直接对速度积分得到t=0到t=6的位移为−3 m，然后取绝对值作为路程。正确做法应是识别出转折点t = 2.5，分两段积分（0到2.5和2.5到6），并将绝对值相加，得到总路程14.5 m。

8. Resolving Forces: Incorrect Use of Sine and Cosine | 力的分解：正弦与余弦的错误使用

When resolving a 12 N force inclined at 30° to the horizontal, a number of students mixed up the components, writing the horizontal component as 12 sin 30° and the vertical as 12 cos 30°. The horizontal component is adjacent to the angle, so it is 12 cos 30°, while the vertical is 12 sin 30°. Drawing a clear right triangle and labelling the components before starting calculations can prevent this slip.

在将一个与水平方向成30°的12 N力进行分解时，不少学生搞混了分量，将水平分量写成12 sin 30°，垂直分量写成12 cos 30°。水平分量紧邻角度，因此是12 cos 30°，而垂直分量为12 sin 30°。在开始计算前，画一个清晰的直角三角形并标出分量，可以避免这种失误。

9. Algebraic Manipulation in Proof Questions | 证明题中的代数操作失误

A proof item asked to show that the sum of the squares of any two consecutive even numbers is always two more than a multiple of 8. Candidates correctly expressed the numbers as 2n and 2n + 2, but then incorrectly expanded (2n)² + (2n + 2)² as 4n² + 4n² + 4, missing the middle term. The correct expansion is 4n² + (4n² + 8n + 4) = 8n² + 8n + 4 = 8(n² + n) + 4, which is two more than a multiple of 8, since 8(n² + n) + 4 = 8(n² + n) + 2 + 2. The conclusion needs to be clearly linked to the expression.

一道证明题要求证明任意两个连续偶数的平方和总是比8的倍数多2。考生正确地将两数设为2n和2n + 2，但在展开(2n)² + (2n + 2)²时，错误地写成4n² + 4n² + 4，遗漏了中间项。正确展开为4n² + (4n² + 8n + 4) = 8n² + 8n + 4 = 8(n² + n) + 4，这比8的倍数多2，因为8(n² + n) + 4 = 8(n² + n) + 2 + 2。并且需要将结论清晰地与表达式联系起来。

10. Misinterpreting the Meaning of a Gradient in Context | 在具体情境中误解斜率的含义

A modelling question gave the temperature T of a cooling liquid as T = 80e⁻⁰·⁰⁵ᵗ, and asked for the rate of change at t = 10. Several students correctly found dT/dt = −4e⁻⁰·⁰⁵ᵗ and substituted t = 10 to get −2.43 (3 s.f.), but then failed to include units (°C per minute) or interpreted a negative rate as an increase. The mark scheme emphasised stating that the temperature is decreasing at a rate of 2.43 °C per minute.

一道建模题给出了冷却液体的温度T = 80e⁻⁰·⁰⁵ᵗ，要求求t = 10时的变化率。不少学生正确地求得dT/dt = −4e⁻⁰·⁰⁵ᵗ并代入t=10得到−2.43（三位有效数字），但随后遗漏了单位（°C每分钟），或将负变化率解读为上升。评分方案强调要说明温度正以每分钟2.43 °C的速率下降。

11. Integration: Missing the Constant or Incorrect Limits Substitution | 积分：遗漏常数或定积分代入错误

For an indefinite integral like ∫ (3x² + 2/x) dx, forgetting the + C was penalised in a “fully correct” bracket. For a definite integral, a common slip was failing to change the variable when using substitution, or when evaluating ∫ 2x(3x² + 1)⁴ dx from 0 to 1, missing that the term 2x is the derivative of 3x² + 1 up to a constant. Using the substitution u = 3x² + 1, du/dx = 6x, led to (1/3) ∫ u⁴ du; missing the 1/3 factor lost the accuracy mark.

对于如∫ (3x² + 2/x) dx的不定积分，忘记加C会在”完整正确”的评分标准中被扣分。对于定积分，常见的失误是在使用换元法时忘记改变积分变量，或者在计算∫₀¹ 2x(3x² + 1)⁴ dx时，忽略了2x恰好是3x² + 1导数的常数倍。令u = 3x² + 1，du/dx = 6x，得到(1/3) ∫ u⁴ du；遗漏系数1/3会导致准确性失分。

12. Mechanics: Failing to Convert Units When Applying SUVAT | 力学：应用SUVAT公式时未转换单位

A projectile question gave initial speed in km/h and time in seconds. Many candidates substituted directly into s = ut + ½ at² without converting 54 km/h to 15 m/s. The mark scheme awarded no marks for the substitution if the units were inconsistent. Always check that all values are in SI units (metres, seconds, m/s, m/s²) before starting calculations.

一个抛体问题给出了以km/h为单位的初速度和以秒为单位的时间。许多考生在未将54 km/h转换为15 m/s的情况下直接代入s = ut + ½ at²。如果单位不一致，评分方案对代入步骤不给予任何分数。在开始计算前，务必检查所有数值是否采用国际单位制（米、秒、m/s、m/s²）。

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

更多咨询请联系16621398022（同微信）