I. Rate of growth of $(n+k)^m$ is same as that of $(n^m)$ as $k$ and $m$ are constants. (If either $k$ or $m$ is a variable then the equality does not hold), i.e., for sufficiently large values of $n,$
$$(n+k)^m \leq a n^m \text{ and}$$
$$n^m \leq b (n+k)^m$$
where $a$ and $b$ are positive constants. Here, $a$ can be $k^m$and $b$ can be $1.$
So, TRUE.
II. $2^{n+1} = 2\times (2^n) = \Theta\left(2^n\right)$ as $2$ is a constant here.
As $2^{n+1}$ is both upper and lower bounded by $2^n$ we can say $2^{n+1} = O\left(2^n\right).$ $(\Theta$ implies both $O$ as well as $\Omega)$
So, TRUE.
III. $2^{2n+1}$ has same rate of growth as $2^{2n}$.
$2^{2n} = {2^n}^2 = 2^n \times 2^n$
$2^n$ is upper bounded by $(2^n)^2$, not the other way round as $2^{2n}$ is increasing by a factor of $2^n$ which is not a constant.
So, FALSE.
Correct Answer: $A$