The coloration of transition metal ions arises from d-d transitions within the metal's d-orbitals, influenced by the oxidation state and surrounding ligands.
- Option (1) Ti\(^{4+}\), V\(^{3+}\): Ti\(^{4+}\) is colorless due to a lack of d-electrons; V\(^{3+}\) is colored due to d-d transitions. These ions do not exhibit the same color.
- Option (2) Cr\(^{2+}\), Cu\(^{2+}\): Cr\(^{2+}\) and Cu\(^{2+}\) ions possess similar electronic configurations and display characteristic colors owing to their d-d transitions, resulting in identical coloration.
- Option (3) Cr\(^{3+}\), Ni\(^{2+}\): Cr\(^{3+}\) is typically green, whereas Ni\(^{2+}\) exhibits a different characteristic color (commonly blue-green). These ions do not share the same color.
- Option (4) Mn\(^{3+}\), Fe\(^{2+}\): Mn\(^{3+}\) is usually purple, and Fe\(^{2+}\) is pale green. These ions are not the same color.
Therefore, the correct selection is Cr\(^{2+}\), Cu\(^{2+}\) (option 2).