Answer :- B, C, D.
Option A : If f(n) = O(n^2) then f(n) = O(n).
False. Counter example, when f(n) = n^2.
Option B : If f(n) = O(3 $^{log_{10}n}$) then f(n) = O(n^2).
Here, Put $log_{10}n$ = k $\implies$ n = 10 $^{k}$.
then 3 $^{log_{10}n}$ becomes 3 $^{k}$ and n $^{2}$ becomes 10 $^{2k}$.
We know 3 $^{k}$ <= 10 $^{2k}$.
Therefore, 3 $^{log_{10}n}$ = O(n $^{2}$).
By transitivity, f(n) = O(n$^{2}$).
True.
Option C : log (n!) = O(nlogn).
True. This is standard result.
Option D : Growth of the sum 1 + ½ + 1/3 + ⋯ + 1/n can be described by θ(logn).
We know 1 + ½ + 1/3 + ⋯ + 1/n = log n + $\gamma$ = θ(logn).
(For more information, refer to the links given in following comments)
True.