Which among the following salts forms basic solution when dissolved in water?

Determining salt hydrolysis behavior requires analyzing the acid-base character of constituent ions.

Ammonium nitrate ($\mathrm{NH}_4\mathrm{NO}_3$) derives from weak base $\mathrm{NH}_4\mathrm{OH}$ and strong acid $\mathrm{HNO}_3$. Its cation hydrolyzes: $\mathrm{NH}_4^+ + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{NH}_4\mathrm{OH} + \mathrm{H}^+$, yielding acidic solution.

Sodium carbonate ($\mathrm{Na}_2\mathrm{CO}_3$) comes from strong base $\mathrm{NaOH}$ and weak acid $\mathrm{H}_2\mathrm{CO}_3$. Its anion hydrolyzes: $\mathrm{CO}_3^{2-} + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{HCO}_3^- + \mathrm{OH}^-$, producing basic solution.

Sodium nitrate ($\mathrm{NaNO}_3$) forms from strong acid and strong base; neither ion hydrolyzes, resulting in neutral solution.

Copper(II) sulfate ($\mathrm{CuSO}_4$) originates from weak base $\mathrm{Cu}(\mathrm{OH})_2$ and strong acid. The hydrated copper ion acts as acid: $[\mathrm{Cu}(\mathrm{H}_2\mathrm{O})_6]^{2+} + \mathrm{H}_2\mathrm{O} \rightleftharpoons [\mathrm{Cu}(\mathrm{H}_2\mathrm{O})_5\mathrm{OH}]^+ + \mathrm{H}_3\mathrm{O}^+$, giving acidic solution.

Only sodium carbonate yields a basic aqueous solution.