External Torque and Center of Mass Velocity Constancy

To evaluate the given statements, we analyze the translational and rotational dynamics of a body:

1. Analysis of STATEMENT-1:
The motion of the center of mass of a system/body of mass $M$ is governed by Newton's second law for translational motion:
$\vec{F}_{\text{ext}} = M\vec{a}_{\text{cm}} = M\frac{d\vec{v}_{\text{cm}}}{dt}$
where $\vec{F}_{\text{ext}}$ is the net external force acting on the body, and $\vec{v}_{\text{cm}}$ is the velocity of its center of mass.

The net external torque about the center of mass is given by:
$\vec{\tau}_{\text{cm}} = \sum_{i} (\vec{r}_i - \vec{r}_{\text{cm}}) \times \vec{F}_{i}^{\text{ext}}$

If an external force acts such that its line of action passes directly through the center of mass:

• The torque about the center of mass is zero ($\vec{\tau}_{\text{cm}} = \vec{0}$).
• However, the net external force is non-zero ($\vec{F}_{\text{ext}} \neq \vec{0}$).

Consequently, the acceleration of the center of mass is non-zero:
$\vec{a}_{\text{cm}} = \frac{\vec{F}_{\text{ext}}}{M} \neq \vec{0}$
Thus, the velocity of the center of mass $\vec{v}_{\text{cm}}$ changes with time and does not remain constant.

Therefore, STATEMENT-1 is False.

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2. Analysis of STATEMENT-2:
An isolated system is defined as a system on which no external forces act, meaning:
$\vec{F}_{\text{ext}} = \vec{0}$

By the law of conservation of linear momentum:
$\vec{F}_{\text{ext}} = \frac{d\vec{P}}{dt} = \vec{0} \implies \vec{P} = \text{constant}$

Thus, the total linear momentum of an isolated system remains conserved (constant).

Therefore, STATEMENT-2 is True.

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Conclusion:

• Statement-1 is False.
• Statement-2 is True.

This corresponds to Option (D).