📚 IGCSE Additional Mathematics 0606 Common Mistakes Summary | IGCSE 附加数学0606 易错点总结
This article summarises the most frequent errors students make in Cambridge IGCSE Additional Mathematics 0606, along with clear explanations and correct approaches. Mastering these pitfalls will boost your exam performance.
本文总结了学生在剑桥 IGCSE 附加数学 0606 中反复出现的易错点，并给出清晰的解释和正确的处理方法。吃透这些陷阱能有效提升解题准确度与考试成绩。
1. Functions and Inverse Functions | 函数与反函数
the domain must be the range of the original function, or they fail to check whether the original function is one-to-one over its natural domain.
易错点：用代数方法求出反函数表达式 f⁻¹(x) 后，未注明其定义域（即原函数的值域），或者忘记验证原函数是否一一对应。直接写出反函数却不限制 x 的取值范围会导致失分。
one-to-one. For example, f(x) = x² − 4x + 5 for x ≥ 2 gives range [1, ∞), so f⁻¹(x) = 2 + √x − 1 with domain x ≥ 1.
正确做法：先求出 f(x) 的值域，把它作为反函数的定义域；如果原函数不是一一对应的，应先限制定义域。例如 f(x) = x² − 4x + 5, x ≥ 2，值域为 [1, ∞)，则 f⁻¹(x) = 2 + √x − 1，定义域为 x ≥ 1。
Mistake: Confusing the order of composition. When asked to find gf(x) or fg(x), some students apply f first in both cases. Remember that gf(x) means do f first, then g.
易错点：混淆复合函数的顺序。gf(x) 表示先作用 f 再作用 g，许多同学误以为可以随意交换，有时直接算成 f(g(x))。
e notation carefully: gf(x) = g[f(x)] and fg(x) = f[g(x)]. Apply the inner function first.
正确做法：看清楚符号——gf(x) = g[f(x)]，先代入内层函数 f，再代入外层的 g。顺序不可颠倒。
2. Quadratic Functions and Discriminant | 二次函数与判别式
Mistake: Using the discriminant b² − 4ac carelessly when solving quadratic inequalities. Some students conclude that a quadratic is always positive simply because the discriminant is negative, but fail to check the sign of the leading coefficient a.
易错点：解二次不等式时，仅由判别式 b² − 4ac < 0 就断言函数值恒正，却忽略了二次项系数 a 的符号。若 a < 0 且 Δ < 0，函数值恒负而非恒正。
Correct: When Δ < 0, the quadratic does not change sign; the sign is the same as a. So ax² + bx + c > 0 for all x if and only if a > 0 and Δ < 0.
正确做法：Δ < 0 时，二次函数不会变号，符号完全取决于 a。因此 ax² + bx + c > 0 对一切 x 成立的充要条件是 a > 0 且 Δ < 0。
Mistake: Writing the completed square form (x + p)² + q incorrectly. A common slip is getting the sign of p wrong when factorising, or miscalculating q when the coefficient of x² is not 1.
易错点：配方法处理不当，尤其是二次项系数不为 1 时，提取系数后符号和常数项 q 容易算错。
+ 5. Complete the square inside: 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 8 + 5 = 2(x − 2)² − 3.
正确做法：对于 2x² − 8x + 5，先提取前两项的系数：2[x² − 4x] + 5。括号内配方得 2[(x − 2)² − 4] + 5，再展开化简为 2(x − 2)² − 3。
3. Equations and Inequalities | 方程与不等式
Mistake: Multiplying both sides of a rational inequality by the denominator without considering its sign. For example, solving (x − 1)/(x + 2) > 2 by multiplying by (x + 2)² is safe, but many multiply by (x + 2) directly and drop the inequality direction when x + 2 is negative.
易错点：解分式不等式时，直接两边乘以分母 (x + 2)，忽略了分母可能为负时不等号要反向，导致漏解或错解。
l values and test intervals. Alternatively, multiply by the square of the denominator, (x + 2)², which is non-negative.
正确做法：移项通分，化为一个分式与 0 比较：(x − 1)/(x + 2) − 2 > 0 ⇒ (−x − 5)/(x + 2) > 0，再通过数轴穿根法或符号表求解。或者两边乘以分母的平方 (x + 2)²，这样可避免讨论符号。
Mistake: Forgetting to check extraneous solutions when squaring both sides of an equation, especially with square roots. For √x + 3 t the resulting roots must be tested in the original equation because squaring introduces false solutions.
易错点：解无理方程 √x + 3 = x − 5 时，直接两边平方，求得两根后未代回原方程检验，导致保留增根。
t non-negative) are valid. For the above, x = 11 works, but x = 2 does not because the right side is negative.
正确做法：求出解后必须代入原方程检验，舍去使被开方数无意义或等式不成立的增根。上述方程中 x = 11 成立，x = 2 不成立（右边为负）。
4. Indices and Surds | 指数与根式
Mistake: Assuming √a² + b² = a + b, or √a² = a without considering the absolute value. In Additional Mathematics, √x² = |x| is crucial when simplifying expressions with unknown signs.
易错点：错误地认为 √a² + b² = a + b，或化简时直接写 √x² = x 而忽略绝对值。当 x 为负数时这样会出错。
Correct: √x² = |x|, and √x⁴ en x is known to be non-negative can we drop the absolute value. Always check the context.
正确做法：√x² = |x|，而 √x⁴ = x²（因为 x² 非负）。只有当题目明确 x ≥ 0 时，才可直接写成 x。
s of indices, e.g. (a + b)ⁿ mistakenly expanded as aⁿ + bⁿ, or writing aᵐ × bⁿ as (ab)ᵐ⁺ⁿ.
易错点：错误使用指数运算法则，如将 (a + b)ⁿ 展开为 aⁿ + bⁿ，或者把 aᵐ × bⁿ 写成 (ab)ᵐ⁺ⁿ。指数法则只适用于同底数幂或同指数幂的乘除。
ᵐ = aᵐ bᵐ, aᵐ × aⁿ = aᵐ⁺ⁿ, and (aᵐ)ⁿ = aᵐⁿ. Distinguish clearly between products and sums.
正确做法：乘方的积 (ab)ᵐ = aᵐ bᵐ，同底数幂相乘 aᵐ × aⁿ = aᵐ⁺ⁿ，幂的乘方 (aᵐ)ⁿ = aᵐⁿ。加法不可以拆分成指数。
5. Logarithms | 对数
g a / log b = log(a/b). These are very common. The correct law is log a + log b = log(ab).
易错点：滥用错误对数恒等式，如 log(a + b) = log a + log b，或 log a / log b = log(a/b)。正确法则为 log a + log b = log(ab) 和对数的换底公式。
base: logₐ b = log b / log a. None of these allow splitting addition inside the argument.
正确做法：对数运算法则：log(ab) = log a + log b，log(a/b) = log a − log b；换底公式 logₐ b = log b / log a。对数内部是加或减时不可拆分。
g x − 3 > 0 and x + 1 > 0 can lead to negative solutions that appear to work algebraically but are invalid.
易错点：解对数方程时未先写定义域（真数 > 0）。解出 x 后可能包含使真数 ≤ 0 的根，看似代数成立实则无意义。
any root that does not satisfy the original domain. Only x = 5 is acceptable in this case.
正确做法：先确定定义域 x > 3。根据对数性质合并得 log₂[(x − 3)(x + 1)] = 3，解得两个值，但必须舍去不满足 x > 3 的根。本例仅 x = 5 有效。
6. Coordinate Geometry | 坐标几何
original segment instead of the negative reciprocal. The gradient must satisfy m₁ × m₂ = −1.
易错点：求线段的垂直平分线方程时，虽然正确计算了中点坐标，却忘了使用垂线斜率（负倒数），仍用原线段斜率。垂直条件为斜率乘积等于 −1。
Correct: If the segment has slope m, the perpendicular bisector has slope −1/m (provided m ≠ 0). Then use the midpoint with point-slope form.
正确做法：先求线段中点，再计算垂线斜率 −1/m（m ≠ 0），然后利用点斜式写出方程。
equation must be x = k. Some students try to use slope-intercept form and create nonsense.
易错点：用斜截式表示竖直线。竖直线斜率不存在，方程应为 x = 常数。强行写 y = mx + c 会导致错误。
ne parallel to the y-axis, simply write x = constant. For a horizontal line, y = constant.
正确做法：平行于 y 轴的直线方程为 x = 常数；平行于 x 轴的方程为 y = 常数。
7. Trigonometry | 三角学
= 30° and 150°. Students often stop after finding the principal value from the calculator.
易错点：解三角方程时漏解。在 0° 到 360° 内，sin x = 0.5 有两个解：30° 和 150°，但许多同学仅用计算器求出一个主值就结束。
range. For sin θ = k, the second solution is 180° − θ (in degrees) or π − θ (in radians).
正确做法：利用 ASTC 图或三角图像找出给定区间内的全部解。正弦方程的第二解为 180° − θ（度）或 π − θ（弧度）。
Mistake: Confusing radian and degree modes on the calculator, especially when solving equations involving calculus and trigonometry within the same problem. The derivative of sin x is cos x only when x is in radians.
易错点：混淆弧度制与角度制。在微积分与三角混合的题目中，微分公式 d/dx (sin x) = cos x 仅在 x 使用弧度时成立。用角度制所得结果完全不同。
Correct: In Additional Mathematics, calculus with trigonometric functions always assumes arguments in radians. Set calculator to radian mode and convert any given degrees to radians before differentiating or integrating.
正确做法：附加数学中，涉及三角函数的微积分都默认使用弧度。解题前将计算器设为弧度模式，并确保所有角都转为弧度。
8. Differentiation | 微分
Mistake: Forgetting the chain rule when differentiating composite functions, e.g. wrongly writing the derivative of (3x + 2)⁵ as 5(3x + 2)⁴ without multiplying by the derivative of the inner function (3).
易错点：复合函数求导时忘记链式法则。比如对 (3x + 2)⁵ 求导，只得到 5(3x + 2)⁴，漏乘内层函数的导数 3。
Correct: d/dx [f(g(x))] = f'(g(x)) · g'(x). So for y = (3x + 2)⁵, dy/dx = 5(3x + 2)⁴ × 3 = 15(3x + 2)⁴.
正确做法：链式法则：dy/dx = 外函数的导数（代入内函数）乘以内函数的导数。上例结果为 15(3x + 2)⁴。
Mistake: Misusing the quotient rule when differentiating a rational function where the denominator is a simple power of x. Some students apply the full quotient rule instead of rewriting with negative exponents, which often leads to algebraic errors.
易错点：对可化为幂函数的简单分式仍坚持使用复杂的商法则。如对 y = 3/x²，可写成 y = 3x⁻² 直接求导，用商法则反而容易算错。
6x⁻³. Use the quotient rule only when the numerator and denominator are both non-constant expressions.
正确做法：分母是单项式时，化为负指数再微分：d/dx (3x⁻²) = −6x⁻³。仅当分子分母均为非常量多项式时才用商法则。
9. Integration | 积分
Mistake: Omitting the constant of integration '+ C' in indefinite integrals. Even when the question asks for the general form, a missing + C can cost marks.
易错点：不定积分遗漏积分常数 + C。凡是求不定积分（原函数），都必须加上“+ C”，否则在打分严格的试卷中会被扣分。
ite ∫ f(x) dx = F(x) + C, unless the problem provides initial conditions that determine C.
正确做法：不定积分答案一律写为 F(x) + C，除非题目给出了具体条件可求出 C 值。
Mistake: Integrating 1/(ax + b) incorrectly. Some students write ∫ 1/(ax + b) dx = ln|ax + b| + C, forgetting to divide by a. The correct result is (1/a) ln|ax + b| + C.
易错点：对 1/(ax + b) 积分时漏除 a。∫ 1/(ax + b) dx = (1/a) ln|ax + b| + C，而不是直接 ln|ax + b| + C。
: if the numerator is the derivative of the denominator, i.e. a, then ∫ a/(ax + b) dx = ln|ax + b| + C. When the numerator is 1, adjust by a factor of 1/a.
正确做法：运用反链式法则，当分子恰好是分母的导数 a 时，积分得 ln|ax + b| + C。若分子为 1，则需乘以系数 1/a。
Mistake: Finding the area between a curve and the x-axis but forgetting to split the interval where the curve goes below the axis. The definite integral yields a signed area; negative regions contribute negative values unless absolute values are taken.
易错点：计算曲线与 x 轴围成的实际面积时，未把 x 轴下方的部分取绝对值。定积分直接相加会因面积正负抵消得到错误结果。
Correct: Find where the curve crosses the x-axis. Integrate |f(x)| f(x) dx on each subinterval and take the absolute value of negative parts before summing.
正确做法：先求曲线与 x 轴的交点，分区段积分。对于 f(x) < 0 的区间，取其积分的绝对值或直接用 |f(x)| 积分，再将各段面积相加。
10. Vectors | 向量
Mistake: Treating position vectors and free vectors as the same thing. A position vector is anchored at the origin, whereas a free vector only describes a displacement. When finding the vector from point A to point B, students sometimes mistakenly use the position vector of B alone.
易错点：混淆位置向量与自由向量。点 A 到点 B 的向量应为 AB = OB − OA，即终点位置向量减起点位置向量，有的同学直接写 OB。
and b are the position vectors of A and B. This gives the displacement vector from A to B.
正确做法：AB = OB − OA = b − a，用终点位置向量减去起点位置向量，得到从 A 到 B 的位移向量。
negative scalar. The vectors (2, −4) and (−1, 2) are parallel because (−1, 2) = −½(2, −4).
易错点：判断向量是否平行时，只考虑正数倍，忽略负数倍。例如 (2, −4) 与 (−1, 2) 平行，因为 (−1, 2) = −½(2, −4)，但部分学生认为不平行。
parallel if u = k v for some non-zero scalar k. The scalar k can be positive or negative.
正确做法：两非零向量平行 ⇔ 存在非零实数 k，使 u = k v。比例常数 k 可为负数。
11. Permutations and Combinations | 排列与组合
Mistake: Confusing permutations and combinations. For a scenario where order does not matter (e.g., choosing a committee of 5 from 20 people), many students use permutations ²⁰P₅ instead of the combination ²⁰C₅, giving a much larger answer.
易错点：混淆排列与组合。选择委员会这类不考虑顺序的问题，学生常误用排列公式 ²⁰P₅ 而不是组合 ²⁰C₅，导致答案扩大很多倍。
Correct: Use combinations (nCr) when order is not important, and permutations (nPr) when the arrangement or sequence matters. Ask: 'Does the order change the outcome?'
正确做法：顺序不重要时用组合 nCr，顺序重要时用排列 nPr。判断的关键是问自己：“交换顺序是否会改变结果？”
Mistake: Mishandling permutations with identical items. For example, the number of ways to arrange the letters of 'MISSISSIPPI' requires dividing by the factorials of repeated letters: 11! / (4! 4! 2!). Some students either forget to divide, or divide incorrectly.
易错点：处理含重复元素的排列时忘记除以相同元素的阶乘。如排列单词 'MISSISSIPPI' 中的字母，须除以 4!（四个 S）和 4!（四个 I）以及 2!（两个 P）。
Correct: For n objects where there are p of one type, q of another, etc., the number of distinct arrangements is n! / (p! q! ...). Count repetitions carefully.
正确做法：n 个对象中有 p 个相同的一类，q 个相同的另一类……则不同排列数为 n! / (p! q! …)。务必仔细统计各字母出现次数。
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