📚 Common Mistakes in AS Further Maths Unit 2 (Jan 2020 Paper) | AS进阶数学Unit 2（2020年1月卷）易错点总结
The January 2020 AS Further Mathematics Unit 2 paper tested a range of advanced pure topics, including complex numbers, matrix algebra, series, polar coordinates, and numerical methods. While the concepts are well defined, many candidates lost marks due to predictable errors in algebraic manipulation, formula application, and interpretation of geometric conditions. This article summarises the most frequent mistakes observed in the exam, providing clear explanations and correct approaches to help you avoid similar pitfalls in your revision.
2020年1月AS进阶数学Unit 2试卷考查了复数、矩阵代数、级数、极坐标和数值方法等一系列高等纯数内容。尽管概念定义清晰，许多考生因代数操作、公式应用和几何条件解读方面的常见错误而失分。本文汇总了该次考试中最频繁出现的错误，提供清晰的解释和正确方法，帮助你在复习时避开这些陷阱。
1. Misinterpreting Complex Conjugate Relationships | 误解共轭复数关系
When a cubic equation with real coefficients has a complex root a + bi, it is essential to recognise that its conjugate a – bi is also a root. A common error in the Jan 2020 paper occurred when students attempted to construct the quadratic factor from a given pair of complex conjugates. Instead of writing (z – (a + bi))(z – (a – bi)) = z² – 2az + (a² + b²), many candidates incorrectly expanded the product, omitting the minus sign before the conjugate or miscomputing the constant term.
当实系数三次方程有一个复数根 a + bi 时，必须认识到其共轭 a – bi 也是根。2020年1月试卷中的一个常见错误发生在学生试图从给定的一对共轭复数构造二次因式时。他们没有正确地写出 (z – (a + bi))(z – (a – bi)) = z² – 2az + (a² + b²)，许多考生在展开时漏掉了共轭前的负号，或错算了常数项。
Another frequent slip was failing to link the conjugate pair back to the original real coefficients. For instance, if a cubic z³ + pz² + qz + r = 0 has a real root c and complex pair a ± bi, then p = –(2a + c), but some incorrectly used p = 2a + c, forgetting the negative sum-of-roots relationship. Always check Vieta’s formulas carefully.
另一个常见疏漏是未能将共轭对与原实系数联系起来。例如，若三次方程 z³ + pz² + qz + r = 0 有一个实根 c 和一对复数根 a ± bi，则 p = –(2a + c)，但有些学生错误地使用了 p = 2a + c，忘记了根之和的负号关系。务必仔细验证韦达定理。
2. Errors in Applying De Moivre’s Theorem to Negative Powers | 棣莫弗定理负指数应用错误
De Moivre’s theorem states that (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for any integer n. The Jan 2020 paper included evaluation of negative powers, such as (cos θ + i sin θ)⁻². A typical mistake was to write cos(–2θ) + i sin(–2θ) = cos 2θ – i sin 2θ correctly, but then mistakenly claim the imaginary part is + sin 2θ. Another error was to incorrectly apply the reciprocal first, forgetting that 1/(cos θ + i sin θ) = cos θ – i sin θ equals cos(–θ) + i sin(–θ), which is consistent, but when combined with a power, the simplification demands care.
棣莫弗定理指出，对于任意整数 n，(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。2020年1月试卷涉及了负指数求值，如 (cos θ + i sin θ)⁻²。一个典型错误是正确写出 cos(–2θ) + i sin(–2θ) = cos 2θ – i sin 2θ，却又错误地声称虚部为 + sin 2θ。另一个错误是先错误地取倒数，忘记了 1/(cos θ + i sin θ) = cos θ – i sin θ 等于 cos(–θ) + i sin(–θ)，虽然一致，但在与幂次结合时化简容易出错。
The table below contrasts common incorrect manipulations with the correct application of De Moivre’s theorem for negative indices, based on exam responses.
下表中对比了基于试卷作答的常见错误操作与棣莫弗定理在负指数下的正确应用。
| Incorrect Attempt (错误尝试) | Correct Approach (正确方法) |
|---|---|
| (cos θ + i sin θ)⁻³ = cos 3θ – i sin 3θ (incorrect sign reasoning) | (cos θ + i sin θ)⁻³ = cos(–3θ) + i sin(–3θ) = cos 3θ – i sin 3θ (correct sign follows from sine oddness) |
| Writing (e^{iθ})⁻² = e^{–2iθ} but then simplifying cos 2θ + i sin 2θ | e^{–2iθ} = cos(–2θ) + i sin(–2θ) = cos 2θ – i sin 2θ |
| For (cos θ – i sin θ)⁴, assuming it equals cos 4θ – i sin 4θ directly | First rewrite as (cos(–θ) + i sin(–θ))⁴ = cos(–4θ) + i sin(–4θ) = cos 4θ – i sin 4θ; works only because power is integer. |
Always express the complex number in the standard form cos φ + i sin φ before applying the theorem, and use the even/odd properties of cosine and sine to set the final signs.
务必先将复数表达为标准形式 cos φ + i sin φ，再应用定理，并利用余弦和正弦的奇偶性确定最终符号。
3. Drawing and Interpreting Loci on Argand Diagrams | 阿尔冈图轨迹的绘制与解读
Locus questions in the Jan 2020 paper asked candidates to sketch sets defined by conditions like |z – 2i| = |z + 3| and arg(z – 1) = π/4. A fundamental error was confusing the perpendicular bisector with a circle. For |z – a| = |z – b|, the locus is the perpendicular bisector of the segment joining points a and b, yet many students incorrectly sketched a circle centred at the midpoint.
2020年1月试卷中的轨迹题要求考生画出满足条件如 |z – 2i| = |z + 3| 和 arg(z – 1) = π/4 的图形。一个根本性错误是将垂直平分线误认为圆。对于 |z – a| = |z – b|，轨迹是连接点 a 和 b 的线段的垂直平分线，但许多学生错误地画成以中点为圆心的圆。
Similarly, when interpreting arg(z – 1 – i) = π/4, some candidates drew the half-line from the point (1,1) but with the wrong direction or without excluding the point itself. The half-line must start from the point, making an angle of π/4 with the positive real axis, and the starting point is an open circle. Common mistakes included drawing a full line or placing the ray in the second quadrant. Additionally, for a circle locus |z + 2 – i| = 3, forgetting to identify the centre correctly as (–2,1) instead of (2,–1) was a frequent sign error.
同样，在解释 arg(z – 1 – i) = π/4 时，一些考生画出了从点 (1,1) 出发的射线，但方向错误或未排除该点本身。射线必须从该点出发，与正实轴成 π/4 角，且起点为空心圆。常见错误包括画成一条完整的直线或将射线画在第二象限。此外，对于圆轨迹 |z + 2 – i| = 3，忘记正确识别圆心为 (–2,1) 而非 (2,–1) 也是频发的符号错误。
To avoid these errors, describe the transformation in words: “the distance from z to 2i equals the distance from z to –3” leads to the perpendicular bisector, while “the argument of (z – (1+i)) is π/4” points to a ray. Always label the starting point and direction clearly.
为避免这些错误，请用语言描述变换：“z 到 2i 的距离等于 z 到 –3 的距离”可导出垂直平分线，而“(z – (1+i)) 的辐角为 π/4”则指向一条射线。务必清晰标注起点和方向。
4. Matrix Transformations Involving Reflections | 涉及反射的矩阵变换
The paper tested matrix representation of linear transformations, particularly reflection in the line y = x and reflection in the line y = (tan θ)x. A surprising number of students confused the matrix for reflection in y = x with that for rotation by 90°. The correct matrix is [[0,1],[1,0]], but some wrote [[0,–1],[1,0]] or forgot that reflection is self-inverse and its determinant is –1. Another common slip occurred when a question required finding the image of a point after a transformation: they multiplied the matrix by the column vector in the wrong order or used row vector instead.
试卷考查了线性变换的矩阵表示，特别是关于直线 y = x 和 y = (tan θ)x 的反射。令人惊讶的是，许多学生将关于 y = x 的反射矩阵与旋转 90° 的矩阵混淆。正确矩阵是 [[0,1],[1,0]]，但有人写成 [[0,–1],[1,0]] 或忘记了反射是自逆的且行列式为 –1。另一个常见失误发生在需要求变换后点的像时：他们以错误的顺序用矩阵乘以列向量，或用行向量代替。
For reflection in the line through the origin making an angle θ with the x-axis, the matrix is [[cos 2θ, sin 2θ],[sin 2θ, –cos 2θ]]. In the Jan 2020 paper, candidates often omitted the minus sign in the bottom-right entry, writing cos 2θ instead of –cos 2θ, or swapped sin 2θ and –cos 2θ. Remember that the determinant must be –1, which provides a quick check: cos² 2θ + sin² 2θ = 1 should have a minus sign somewhere. Also, when asked to find the reflection of a specific point, always verify that the line connecting the original point and its image is perpendicular to the mirror line.
对于关于通过原点且与 x 轴夹角为 θ 的直线的反射，矩阵为 [[cos 2θ, sin 2θ],[sin 2θ, –cos 2θ]]。在2020年1月试卷中，考生常常遗漏右下角元素中的负号，错写成 cos 2θ 而不是 –cos 2θ，或交换了 sin 2θ 和 –cos 2θ 的位置。请记住行列式必须为 –1，这可以快速检验：cos² 2θ + sin² 2θ 本身应有一处带负号。此外，当要求求特定点的反射像时，务必验证原点和其像的连线垂直于镜线。
5. Maclaurin Series Expansion Errors | 麦克劳林级数展开错误
Series questions required expansions of functions like ln(1 + x) and e^(x) sin x. A pervasive mistake was truncating the series too early without checking the requirement of the question. For example, when asked to expand up to the term in x³, some students omitted the x³ term in ln(1 + x) = x – ½x² + ⅓x³ – … , and instead stopped at x². Others misapplied the standard series for sin x or cos x, forgetting the alternating signs or factorial denominators.
级数题要求展开诸如 ln(1 + x) 和 e^(x) sin x 的函数。一个普遍的错误是过早截断级数而没有检查题目要求。例如，当要求展开至 x³ 项时，一些学生遗漏了 ln(1 + x) = x – ½x² + ⅓x³ – … 中的 x³ 项，而在 x² 处就停止了。还有人错误应用了 sin x 或 cos x 的标准级数，忘记了交错符号或阶乘分母。
When multiplying two series, as in e^(x) sin x, the errors multiplied. Candidates often computed e^(x) = 1 + x + x²/2 + x³/6 + … and sin x = x – x³/6 + … but then incorrectly multiplied, missing cross-terms like x·(–x³/6) or (x²/2)·x, leading to a wrong coefficient for x³. The correct product up to x³ is (1 + x + x²/2 + x³/6)(x – x³/6) = x + x² + (1/2 – 1/6)x³ + higher, giving x + x² + (1/3)x³. A systematic grid method helps avoid these algebraic slips.
当将两个级数相乘时，如 e^(x) sin x，错误成倍增加。考生通常计算 e^(x) = 1 + x + x²/2 + x³/6 + … 和 sin x = x – x³/6 + …，但在相乘时出错，遗漏了如 x·(–x³/6) 或 (x²/2)·x 的交叉项，导致 x³ 的系数错误。正确的乘积至 x³ 项为 (1 + x + x²/2 + x³/6)(x – x³/6) = x + x² + (1/2 – 1/6)x³ + 高阶，得到 x + x² + (1/3)x³。系统性的网格法可帮助避免这些代数疏漏。
Also, when a composite function such as ln(1 + sin x) appeared, some tried to differentiate repeatedly using unfamiliar chain rules and made errors in evaluating derivatives at 0. Using known series and substituting correctly is often safer.
此外，当出现复合函数如 ln(1 + sin x) 时，一些考生试图使用不熟悉的链式法则反复求导，并在求导数值时出错。使用已知级数并正确代入往往更稳妥。
6. Selecting the Wrong Particular Integral for Second-Order ODEs | 二阶常微分方程特解形式选错
The Jan 2020 paper featured solving a second-order linear differential equation with constant coefficients. A classic pitfall was choosing the particular integral (PI) for the right-hand side. For an RHS of the form e^(kx), the standard PI is λ e^(kx), but if k coincides with a root of the auxiliary equation, the PI must be multiplied by x (or x²). Many candidates forgot to check the auxiliary equation roots and proceeded with the standard trial function, resulting in an inconsistent system or an incorrect general solution.
2020年1月试卷包含了求解二阶常系数线性微分方程的内容。一个典型陷阱是针对右侧函数选择特积分 (PI) 的形式。对于形如 e^(kx) 的右边，标准特解为 λ e^(kx)，但如果 k 与辅助方程的根重合，则特解必须乘以 x（或 x²）。许多考生忘记检查辅助方程的根，直接套用标准试验函数，导致矛盾方程组或错误的通解。
Similarly, when the RHS was a polynomial, like 2x² + 3, some used a PI of the form ax² + bx + c, which is correct, but missed that if 0 is a root of the auxiliary equation, they must multiply by x. Another common oversight was failing to differentiate the trial PI correctly when substituting into the ODE, especially when the PI includes terms like x e^(kx) requiring product rule. Practise writing the full expression for y_p, y_p’ and y_p” before substitution to minimise algebraic mistakes.
同样，当右边是多项式，如 2x² + 3，一些人采用形如 ax² + bx
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