See, if we have all the numbers in a sorted fashion in an array, we can use binary search which takes $O(logn)$ time.
So, the time complexities mentioned in Options A and B hold. The "but not" part makes these options wrong.
The question says that
n is represented by binary notation
Hence the size of each number is $O(logn)$.
When performing the binary search, all the $O(logn)$ bits of a number would be potentially checked to find a match. Hence, at the minimum this algorithm would take $O(logn*logn)$ time = $O(logn)^2$
Since this would be the minimum time, we can't get $O(loglogn)$, hence Option D is incorrect, too.