The reagent that gives a brilliant red precipitate with Nickel ions (\( \text{Ni}^{2+} \)) in a basic medium is dimethyl glyoxime (dmg). When \( \text{Ni}^{2+} \) reacts with dimethyl glyoxime, it forms a complex that produces a bright red precipitate. The reaction is as follows:
\(\text{Ni}^{2+} + \text{dmg} \rightarrow [\text{Ni}(\text{dmg})_2] \, \, (\text{Rosy red/Bright Red precipitate})\)
Thus, the correct answer is dimethyl glyoxime.
The Correct Answer is: dimethyl glyoxime