The amphoteric hydroxide is:

Amphoteric hydroxide = reacts with both acids AND bases.

Step 1 — Check each hydroxide:

(1) $\mathrm{Be(OH)_2}$: Reacts with HCl (acid): $\mathrm{Be(OH)_2 + 2HCl \rightarrow BeCl_2 + 2H_2O}$ Also reacts with NaOH (base): $\mathrm{Be(OH)_2 + 2NaOH \rightarrow Na_2[Be(OH)_4]} \rightarrow$ Amphoteric ✓

(2) $\mathrm{Ca(OH)_2}$: Only reacts with acids $\rightarrow$ purely basic ✗

(3) $\mathrm{Mg(OH)_2}$: Only reacts with acids $\rightarrow$ purely basic ✗

(4) $\mathrm{Sr(OH)_2}$: Only reacts with acids $\rightarrow$ purely basic ✗

Note: Be is a small, highly polarising cation $\rightarrow$ its hydroxide shows amphoteric character (similar to Al, Zn).

Answer: Option (1) — $\mathrm{Be(OH)_2}$