在 $\bigg(\dfrac{x}{a}+\dfrac{a}{x^2}\bigg)^7$ 的展开式中，已知

求常数 $a$ 的可能值。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

每选一次 $a/x^2$ 来替代 $x/a$，总指数便减少 $3$。先找对应的选取次数，再计算系数之比。

If $a/x^2$ is selected $j$ times, the term is

For $x^4$ we have $j=1$ and coefficient $7/a^5$. For $x$ we have $j=2$ and coefficient $21/a^3$. Since $a\ne0$, their ratio is $1/(3a^2)=3$, giving $a=\pm1/3$.