Step 1: Understanding the Concept
In the H\(_2\)-O\(_2\) fuel cell, porous carbon electrodes carrying a catalyst dip into a hot aqueous solution of an alkali.
Step 2: Detailed Explanation
The electrolyte is aqueous KOH. At the anode, hydrogen is oxidised: \(2\text{H}_2 + 4\text{OH}^- \to 4\text{H}_2\text{O} + 4e^-\).
At the cathode, oxygen is reduced: \(\text{O}_2 + 2\text{H}_2\text{O} + 4e^- \to 4\text{OH}^-\).
The OH\(^-\) ions carry current through the solution, so a hydroxide solution is needed.
Final Answer:
The carbon rods dip in hot aqueous KOH, option (B).
\[ \boxed{\text{KOH (B)}} \]