To identify the reagent producing a brilliant red precipitate with Nickel ions in a basic medium, each option is evaluated:
When dimethyl glyoxime is added to a solution containing Nickel ions (\(Ni^{2+}\)) under basic conditions (typically with ammonia or another base), a red precipitate of the nickel(II) dimethylglyoxime complex is generated:
\(Ni^{2+} + 2(C_4H_8N_2O_2) + 2OH^- \rightarrow Ni(C_4H_7N_2O_2)_2 \, (red \, precipitate) + 2H_2O\)
Therefore, dimethyl glyoxime is the correct reagent as it forms a red precipitate with Nickel ions in a basic medium.
In summary, dimethyl glyoxime is the reagent that yields a brilliant red precipitate with Nickel ions in a basic medium.
Kjeldahl's method cannot be used for the estimation of nitrogen in which compound? 
In the group analysis of cations, Ba$^{2+}$ & Ca$^{2+}$ are precipitated respectively as