Which of the following half reaction equations represent the reaction at the cathode

In an electrochemical cell, the cathode is the electrode where reduction occurs. Reduction is the gain of electrons by a species. So, the half-reaction equation that involves the gain of electrons is the reaction at the cathode. Option B represents the reaction at the cathode because it involves the gain of two electrons, which is a reduction half-reaction. The species B\(^{2+}\) is being reduced to B\(_{(s)}\), which means that B\(^{2+}\) is gaining two electrons to become a neutral atom of B. Option A is not the reaction at the cathode because it involves the oxidation of A\(_{(s)}\) to A\(^{3+}\). This is an oxidation half-reaction, and it occurs at the anode. Option C is not the reaction at the cathode because it involves the reduction of A\(^{3+}\) to A\(_{(s)}\). This is a reduction half-reaction, but it occurs at the anode in an electrolytic cell. Option D is not the reaction at the cathode because it involves the oxidation of B\(_{(s)}\) to B\(^{2+}\). This is an oxidation half-reaction, and it occurs at the anode. In summary, option B represents the reaction at the cathode because it involves the gain of two electrons, which is a reduction half-reaction.