AS Mathematics Unit 1 (June 2019) Common Mistakes Summary | AS数学单元1（2019年6月）易错点总结

📚 AS Mathematics Unit 1 (June 2019) Common Mistakes Summary | AS数学单元1（2019年6月）易错点总结

The AS Mathematics Unit 1 examination in June 2019 covered core topics such as algebra, functions, coordinate geometry and calculus. Analysis of the mark scheme reveals several recurring mistakes that prevented candidates from securing full marks. This article highlights these common pitfalls and provides guidance on how to avoid them, enhancing exam technique.

2019年6月的AS数学单元1考试涵盖了代数、函数、坐标几何和微积分等核心主题。对评分方案的分析揭示了考生们反复出现的几类错误，导致他们未能拿到满分。本文重点总结这些常见易错点，并提供如何避免它们的指导，帮助提高应试技巧。

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1. Algebraic Manipulation and Quadratic Equations | 代数操作与二次方程

Many students correctly rearranged to x² = k, but then wrote x = k, forgetting the ± symbol. For instance, solving x² = 9, some gave only x = 3, omitting x = -3. The mark scheme often awards the final accuracy mark only when both solutions are clearly stated.

很多学生正确地化简到 x² = k，但之后只写 x = k，忘记了 ± 号。例如解 x² = 9 时，有人只给出 x = 3，遗漏了 x = -3。评分方案通常只有在明确写出两个解时才给最后的准确性分。

When factorising quadratics, sign errors in the factors led to incorrect solutions. A common slip was writing (x – 3)(x – 2) = 0 instead of the correct (x – 3)(x + 2) = 0, which then gave wrong root x = 2 instead of x = -2.

在因式分解二次式时，因式中的符号错误导致解不正确。一个常见的失误是将正确的 (x – 3)(x + 2) = 0 写成 (x – 3)(x – 2) = 0，从而得到错误的根 x = 2 而非 x = -2。

Completing the square also caused issues: candidates often mishandled the constant term when rewriting x² + bx + c. For example, x² + 6x + 5 should become (x + 3)² – 4, but many wrote (x + 3)² + 5, forgetting to subtract 9.

完成平方同样造成问题：考生在改写 x² + bx + c 时常常处理常数项出错。比如 x² + 6x + 5 应化为 (x + 3)² – 4，但许多人写成 (x + 3)² + 5，忘记减去 9。

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2. Differentiation Errors | 微分错误

In questions involving functions like (3x + 2)⁴, candidates often differentiated as 4(3x + 2)³, omitting the derivative of the inner function (3). The correct derivative is 12(3x + 2)³. This mistake stems from not applying the chain rule fully.

对于像 (3x + 2)⁴ 这样的函数，考生常将其导数写成 4(3x + 2)³，遗漏了内层函数的导数 (3)。正确导数是 12(3x + 2)³。这一错误源于未能完全应用链式法则。

When differentiating terms like 5/x², many incorrectly rewrote it as 5x⁻² and then differentiated to 5 × (-2)x⁻³ = -10x⁻³, but a sign error was common: some obtained 10x⁻³ or left it as 5x⁻³. Careless use of the power rule for negative exponents frequently cost marks.

微分 5/x² 时，许多人将其改写为 5x⁻² 然后微分，得到 5 × (-2)x⁻³ = -10x⁻³，但符号错误常见：有人得到 10x⁻³ 或仍保留 5x⁻³。对负指数幂规则的不细致运用常常导致失分。

For exponential functions like e²ˣ, the derivative is 2e²ˣ, yet some wrote just e²ˣ. Remembering to multiply by the derivative of the exponent is crucial. Similarly, with ln(5x), the derivative is 1/x, not 1/(5x).

对于 e²ˣ 这样的指数函数，导数是 2e²ˣ，但有些人只写 e²ˣ。记住要乘上指数的导数是关键。类似地，ln(5x) 的导数是 1/x，而不是 1/(5x)。

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3. Integration Mistakes | 积分错误

In indefinite integrals, the ‘+ C’ was frequently omitted. The mark scheme explicitly requires the constant of integration for full marks. This seemingly small oversight cost candidates the final accuracy mark in many questions.

在不定积分中，“+ C”经常被遗漏。评分方案明确要求必须写出积分常数才能拿到满分。这一看似微小的疏忽在许多题目中使考生丢掉了最后的准确性分。

When finding the area under a curve, candidates sometimes evaluated a definite integral that gave a negative value, but failed to recognise that area is always positive. For functions that cross the x-axis, splitting the integral into sections and taking absolute values was often forgotten.

求曲线下方面积时，考生有时算出的定积分出现负值，却未能认识到面积总是正的。对于穿过 x 轴的函数，常常忘记将积分分段并取绝对值。

Another frequent slip was misapplying the power rule for integration: integrating xⁿ to xⁿ⁺¹/(n+1) but mishandling the new exponent. For example, ∫ x⁻² dx should be -x⁻¹ + C, yet some wrote -x⁻³/3 + C or similar.

另一个常见失误是错误应用积分的幂规则：将 xⁿ 积分为 xⁿ⁺¹/(n+1)，但处理新指数时出错。例如 ∫ x⁻² dx 应为 -x⁻¹ + C，有人却写成 -x⁻³/3 + C 等。

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4. Coordinate Geometry: Lines and Circles | 坐标几何：直线与圆

To find a line perpendicular to a given line, the product of gradients should be -1. Many candidates simply used the same gradient or forgot to flip and change the sign. For example, if a line has gradient 2/3, the perpendicular gradient is -3/2, but some gave 3/2 or -2/3.

求与给定直线垂直的直线时，斜率乘积应为 -1。许多考生直接用了相同的斜率，或者忘记取负倒数。例如，一条直线斜率为 2/3，垂直线的斜率应为 -3/2，但有人给出 3/2 或 -2/3。

The midpoint formula ((x₁+x₂)/2, (y₁+y₂)/2) was sometimes confused with the distance formula √((x₂-x₁)² + (y₂-y₁)²). This led to lost marks in circle geometry problems where the centre and radius were needed, as candidates calculated the wrong distance or midpoint.

中点公式 ((x₁+x₂)/2, (y₁+y₂)/2) 有时与距离公式 √((x₂-x₁)² + (y₂-y₁)²) 相混淆。在需要求圆心和半径的圆几何题中，这导致考生算出错误的距离或中点，从而失分。

When finding the equation of a line given two points, errors in slope calculation were frequent: subtracting y-coordinates in the wrong order produced a sign error, which then affected the entire equation.

已知两点求直线方程时，斜率计算错误频发：y坐标相减的顺序错误导致符号错误，进而影响整个方程。

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5. Functions and Their Inverses | 函数及其反函数

After rearranging y = f(x) to make x the subject, some candidates forgot to swap x and y to write f⁻¹(x). For instance, from y = 2x + 3, they correctly found x = (y-3)/2, but then left the answer as f⁻¹(y) = (y-3)/2 or f⁻¹(x) = (x-3)/2 but did not actually swap? Actually the correct swap produces f⁻¹(x) = (x-3)/2. The error was often leaving the expression in terms of y and calling it f⁻¹(y).

在将 y = f(x) 变形为以 x 为主体后，一些考生忘记交换 x 和 y 以写出 f⁻¹(x)。例如从 y = 2x + 3 正确得到 x = (y-3)/2，但将答案留在关于 y 的形式并标为 f⁻¹(y)，而未最终替换变量。

When finding the range or domain, candidates often did not consider restrictions like denominators not being zero or square roots requiring non-negative arguments. In composite functions, they sometimes used values that were undefined, leading to incorrect domains.

在求值域或定义域时，考生常未考虑分母不能为零或平方根下非负等限制。在复合函数中，他们有时使用了未定义的值，导致定义域错误。

Misunderstanding the notation f⁻¹(x) as 1/f(x) was another classic error, though less common, it appeared when candidates hastily simplified expressions.

将 f⁻¹(x) 误解为 1/f(x) 是另一个经典错误，虽然不常见，但当考生匆忙简化表达式时会出现。

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6. Graph Sketching and Transformations | 草图绘制与图像变换

In curve sketching, marks were lost for not indicating intercepts, turning points or asymptotes on the axes. Even if the shape was roughly correct, the mark scheme often required labelled coordinates of key points to award full marks.

在曲线草图中，未能标出截距、转折点或渐近线导致失分。即使形状大体正确，评分方案往往要求关键点的坐标标注才能给满分。

When applying multiple transformations such as y = af(bx + c) + d, candidates performed the transformations in the wrong order. The correct sequence is to apply stretches/compressions first, then translations. For example, transforming f(x) to 2f(3x – 1) should involve a horizontal compression by 1/3, then a translation right by 1/3, then a vertical stretch by 2; many reversed these steps.

进行 y = af(bx + c) + d 等多重变换时，考生常弄错变换顺序。正确顺序是先进行拉伸压缩，再平移。例如将 f(x) 变换为 2f(3x – 1)，应是先水平压缩至 1/3，再向右平移 1/3，最后垂直拉伸2倍；许多人颠倒了这些步骤。

Sketching reciprocal or logarithmic graphs, candidates frequently missed the asymptotes or drew them crossing axes incorrectly. A

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