On what does the position of the center of mass of a rigid body depend?

For a body,the center of volume is defined as $\frac{\int \vec{r} \, dV}{\int dV}$ over the complete body,where $dV$ is a small volume element of the body and $\vec{r}$ is the position vector of that small volume from the origin. Which of the following statements is correct?

Obtain an expression for the position vector of the centre of mass of a system of $n$ particles in two dimensions.

Two objects of masses $200 \text{ g}$ and $500 \text{ g}$ possess velocities $10 \hat{i} \text{ m/s}$ and $(3 \hat{i} + 5 \hat{j}) \text{ m/s}$ respectively. The velocity of their centre of mass in $\text{m/s}$ is:

$A$ thin rod of length $L$ is placed along the $X$-axis such that one of its ends is at $x = 0$ and the other is at $x = L$. Its linear mass density (mass/length) varies with $x$ as $k(x/L)^n$,where $n$ is any positive number or zero. If the graph of the center of mass $x_{cm}$ versus $n$ is plotted for the rod,which of the following graphs is correct?

Write the meaning of homogeneous bodies.