Statement 1 : ${\mathrm{BaCO}}_3$ is more soluble in  ${\mathrm{HNO}}_3$ than in plain water.

Statement 2 : Carbonate is a weak base and reacts with the ${\mathrm{H}}^{+}$ from the strong acid, causing the barium salt to dissociate.

Barium carbonate is more soluble in ${\mathrm{HNO}}_3$ than in water. Carbonate is a weak base and reacts with the ${\mathrm{H}}^{+}$ ion of  ${\mathrm{HNO}}_3$ causing the barium salt to dissociate.

${\mathrm{BaCO}}_3 + {\mathrm{HNO}}_3 \to \mathrm{Ba}{\left({\mathrm{NO}}_3\right)}_2 + {\mathrm{CO}}_2 + {\mathrm{H}}_2\mathrm{O}$