Examine Assertion (A)

Assertion: Barium carbonate is insoluble in water and is highly stable.

• In Group 2, the carbonates such as $\text{MgCO}_3, \text{CaCO}_3, \text{SrCO}_3, \text{BaCO}_3$ are generally insoluble in water.
• Also, the thermal stability of alkaline earth metal carbonates increases down the group.
• Since $\text{Ba}^{2+}$ is a large cation, $\text{BaCO}_3$ is among the more thermally stable carbonates.

Therefore, Assertion (A) is true.

Examine Reason (R)

Reason: The thermal stability of the carbonates increases with increasing cationic size.

• For Group 2 carbonates, as cation size increases from $\text{Mg}^{2+}$ to $\text{Ba}^{2+}$, the polarizing power of the cation decreases.
• Smaller cations polarize the $\text{CO}_3^{2-}$ ion more strongly, destabilizing it.
• Larger cations such as $\text{Ba}^{2+}$ polarize less, so the carbonate ion remains more stable.
• Hence, thermal stability increases with increasing cation size.

Therefore, Reason (R) is true.

Check whether (R) explains (A)

• The statement that $\text{BaCO}_3$ is highly stable is directly explained by the fact that carbonate thermal stability increases with increasing cationic size.
• Since $\text{Ba}^{2+}$ is large, $\text{BaCO}_3$ is more stable.
• In standard assertion-reason questions, this is accepted as the explanation for the stability part of the assertion.

Hence, (R) is the correct explanation of (A).

Final choice

The correct option is: $\boxed{\text{A}}$

Both (A) and (R) are true and (R) is the true explanation of (A).