STATEMENT-1: If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant. STATEMENT-2: The linear momentum of an isolated system remains constant.

Step 1: Analyze Statement-1.

Consider a force $\vec{F}$ applied directly at the center of mass of a body of mass $m$.

The torque about the center of mass is zero.

$$\vec{\tau}_{\text{COM}} = 0$$

The acceleration of the center of mass is non-zero.

$$\vec{a}_{\text{CM}} = \frac{\vec{F}}{m} \neq 0$$

Since the acceleration is non-zero, the velocity $\vec{v}_{\text{CM}}$ is not constant, which makes Statement-1 false.

Step 2: Analyze Statement-2.

An isolated system has no net external force acting on it.

$$\vec{F}_{\text{ext}} = 0$$

The relation between net external force and linear momentum is given by Newton's second law.

$$\vec{F}_{\text{ext}} = \frac{d\vec{P}}{dt}$$

Integrating this relation shows that the total linear momentum is conserved.

$$\vec{P} = \text{constant}$$

Thus, Statement-2 is true.

Step 3: Compare the statements.

Statement-1 is false and Statement-2 is true.

Answer: Statement-1 is False, Statement-2 is True.