In which one of the following transition metal complexes, does the metal exhibit zero oxidation state?

\([Ni(CO)_4]\)

The correct answer is option 1.\([Ni(CO)_4]\).

The transition metal complex in which the metal exhibits a zero oxidation state is: \([Ni(CO)_4]\)

\([Ni(CO)_4]\) (Nickel tetracarbonyl): In this complex, nickel is in the zero oxidation state. The carbon monoxide \((CO)\) ligands are neutral, meaning they do not contribute to the oxidation state of the nickel. Therefore, the oxidation state of nickel is 0.

\([Fe(H_2O)_6]X_3: Here, iron is typically in the +3 oxidation state because the X counterion would be necessary to balance the charge. Each water molecule \((H_2O)\) is neutral, so the charge of the complex is determined by the iron ion and the X counterions.

\([Co(NH_3)_6]Cl_3\): In this complex, cobalt is in the +3 oxidation state. Each ammonia \((NH_3)\) ligand is neutral, and the three chloride ions (Cl^-) balance the +3 charge of the cobalt ion.

\([Fe(H_2O)_6]SO_4\): In this complex, iron is typically in the +2 oxidation state. Each water molecule \((H_2O)\) is neutral, and the sulfate ion \((SO_4^{2-})\) balances the +2 charge of the iron ion.

Therefore, \([Ni(CO)_4]\) is the correct answer, as nickel in this complex is in the zero oxidation state.