We need to check from the options for which of the given set of numbers mean is the same as the median.

We know that mean comes equal to the median for numbers which for an arithmetic sequence i.e. they have same gap between any two consecutive terms. For example a series like \(a, a +\mathrm{d}, \mathrm{a}+2 \mathrm{~d}, \mathrm{a}+3 \mathrm{~d} \&\)$ so on is an arithmetic series, where d is the common difference between any two consecutive terms.

So, we need to check from the options that which of the series forms an arithmetic sequence.

I. $\(\mathrm{x}, \mathrm{x}+1, \mathrm{x}+2, \mathrm{x}+3, \mathrm{x}+4-\)$ which is an arithmetic sequence having a common difference of 1 , so it will have mean $\(=\)$ median $\(=x+2\)$.

II. $\(x, 2 x, 3 x, 4 x, 5 x-\)$ which is an arithmetic sequence having a common difference of $\(x(2 x-x=3 \mathrm{x}-2 \mathrm{x}=\mathrm{x}\)$ ), so it will have mean $\(=\)$ median $\(=3 \mathrm{x}\)$.

III. $\(\frac{1}{x}, \frac{2}{x}, \frac{3}{x}, \frac{4}{x}, \frac{5}{x}\)$ - which is an arithmetic sequence having a common difference of $\(\frac{1}{x}\left(\frac{2}{x}-\frac{1}{x}=\frac{3}{x}-\frac{2}{x}=\frac{1}{x}\right)\)$, so it will have mean $\(=\)$ median $\(=\frac{3}{x}\)$.

Hence all three options I, II \& III are correct, so the answer is (E).