Step 1: Evaluating Each Statement
- Statement I: Oxidation of PbS by \( H_2O_2 \)
- Lead sulfide (\( PbS \)) reacts with hydrogen peroxide to form lead sulfate (\( PbSO_4 \)):
\[
PbS + 4 H_2O_2 \rightarrow PbSO_4 + 4 H_2O
\]
- This is a well-known oxidation reaction.
- Statement I is correct.
- Statement II: Industrial Use in Sodium Perborate
- Hydrogen peroxide is widely used in the industrial production of sodium perborate (\( NaBO_3 \)), which is used in detergents.
- Statement II is correct.
- Statement III: Solubility in Water
- Hydrogen peroxide (\( H_2O_2 \)) is highly soluble in water due to its hydrogen bonding capability.
- Statement III is incorrect.
Step 2: Conclusion
Only Statements I and II are correct.
Thus, the correct answer is \( \mathbf{(1)} \) I, II only.