Step-by-step analysis:

1. Analyze Statement 1:

   • The statement is: "If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant."
   • The velocity of the center of mass, $\vec{v}_{CM}$, of a body remains constant if and only if its acceleration, $\vec{a}_{CM}$, is zero.
   • According to Newton's second law for translational motion of a system, the net external force $\vec{F}_{ext}$ is related to the acceleration of the center of mass by $\vec{F}_{ext} = M \vec{a}_{CM}$, where $M$ is the total mass.
   • Thus, for $\vec{v}_{CM}$ to be constant, the net external force $\vec{F}_{ext}$ must be zero.
   • Statement 1 claims that if the net external torque about the center of mass ($\ au_{CM}$) is zero, then $\vec{v}_{CM}$ is constant. This implies that if $\ au_{CM} = 0$, then $\vec{F}_{ext} = 0$.
   • Let's consider a counterexample. Suppose a non-zero net external force $\vec{F}_{ext}$ acts on the body, and its point of application is the center of mass itself.
   • The torque about the center of mass is given by $\vec{\tau}_{CM} = \vec{r}_{CM} \times \vec{F}_{ext}$, where $\vec{r}_{CM}$ is the position vector of the point of application of the force relative to the center of mass. In this case, $\vec{r}_{CM} = 0$.
   • Therefore, the torque about the center of mass is $\vec{\tau}_{CM} = 0 \times \vec{F}_{ext} = 0$. So the condition in the statement is met.
   • However, since $\vec{F}_{ext}$ is not zero, the center of mass will accelerate: $\vec{a}_{CM} = \frac{\vec{F}_{ext}}{M} \neq 0$. This means the velocity of the center of mass, $\vec{v}_{CM}$, will change.
   • Since we found a situation where the premise ($\ au_{CM} = 0$) is true but the conclusion ($"v_{CM}$ is constant") is false, Statement 1 is False.

2. Analyze Statement 2:

   • The statement is: "The linear momentum of an isolated system remains constant."
   • An 'isolated system' is defined in physics as a system on which the net external force is zero ($\"{F}_{ext} = 0$).
   • Newton's second law states that the net external force on a system is equal to the rate of change of its total linear momentum ($\vec{p}$): $\vec{F}_{ext} = \frac{d\vec{p}}{dt}$.
   • If the system is isolated, $\vec{F}_{ext} = 0$. Substituting this into the equation gives $\frac{d\vec{p}}{dt} = 0$.
   • This implies that the total linear momentum $\vec{p}$ of the system must be a constant vector. This is the well-known principle of conservation of linear momentum.
   • Therefore, Statement 2 is True.

3. Conclusion:

   • We have determined that Statement 1 is False and Statement 2 is True.
   • Looking at the options, this corresponds to option D.

Final Answer Selection:

• Statement-1 is False.
• Statement-2 is True.
• This matches option D.