Step 1: Identify the group reagent.
Ba$^{2+}$ and Ca$^{2+}$ belong to alkaline earth metals and are analysed using group reagent:
\[
(NH_4)_2CO_3 \text{ in presence of } NH_4OH \text{ and } NH_4Cl
\] Step 2: Formation of precipitates.
Both Ba$^{2+}$ and Ca$^{2+}$ form insoluble carbonates:
\[
Ba^{2+} + CO_3^{2-} \rightarrow BaCO_3 \ (\text{white ppt})
\]
\[
Ca^{2+} + CO_3^{2-} \rightarrow CaCO_3 \ (\text{white ppt})
\] Step 3: Conclusion.
Thus, both ions give carbonate test.