图 2 给出了图像

的示意图，其中 $a$ 是正常数。

该图像

• 与 $y$ 轴交于点 $P$

• 与 $x$ 轴交于点 $Q$ 和 $R$

• 在点 $S$ 处取得最小值

(a) 用 $a$ 的最简形式求下列点的坐标：

(i) 点 $P$

(ii) 点 $Q$ 和点 $R$

(iii) 点 $S$

(b) 用 $a$ 的最简形式求满足

的 $x$ 值。

这是绝对值函数。与 $y$ 轴交点令 $x=0$；与 $x$ 轴交点令 $y=0$，也就是 $|3x-5a|=2a$；最低点出现在绝对值内部等于 $0$ 的位置。

The graph has equation

For (i), find the $y$-intercept.

For the $y$-intercept, set $x=0$:

Since $a>0$,

So

Therefore

For (ii), find the $x$-intercepts.

For the $x$-intercepts, set $y=0$:

So

Therefore

From

we get

From

we get

Therefore the two $x$-intercepts are

For (iii), find the minimum point.

The minimum value of $|3x-5a|$ is $0$, and this occurs when

So

At this value of $x$,

Therefore

解绝对值方程时要按符号分情况。官方评分路线等价于分别取两边绝对值内部同为非负、同为负的情况，并解出两个有效的 $x$ 值。

We need to solve

First take the case where both expressions inside the absolute values are non-negative. Then

and

So

Hence

Therefore

so

This value satisfies $3x-5a\ge0$ and $x-2a\ge0$, so it is valid.

Now take the case where both expressions inside the absolute values are negative. Then

and

So

Simplify:

Thus

Therefore

so

This value satisfies $3x-5a<0$ and $x-2a<0$, so it is valid.

The mixed-sign cases do not give additional valid solutions: if $3x-5a\ge0$ and $x-2a<0$, the equation gives $x=\frac94a$, which does not satisfy $x<2a$; the other mixed-sign condition $3x-5a<0$ and $x-2a\ge0$ is impossible because it would require $x<\frac53a$ and $x\ge2a$ at the same time.

Hence the required values are