Step 1: Approach
Check for ties between corners and for unboundedness.
Step 2: Region
Bounded triangle with vertices $(0,0),(6,0),(0,2)$ after removing the redundant constraint $x+3y\le9$.
Step 3: Parallel check
The objective line $2x+5y=$ const has slope $-\dfrac25$, and the edge $x+3y=6$ has slope $-\dfrac13$. They are not parallel, so two corners cannot tie.
Step 4: Values
$z$ is 12 at $(6,0)$, larger than 10 at $(0,2)$. The maximum is unique, option (A).
Final Answer:
The maximum value 12 occurs at only one corner, (6, 0), so the solution is unique, option (A).
\[ \boxed{\text{Unique solution}} \]