Option B: increases by 3
If each observation $x_i$ is increased by 3, the new mean $= \dfrac{\Sigma(x_i+3)}{n} = \dfrac{\Sigma x_i + 3n}{n} = \bar{x} + 3$.
The mean is the sum of all observations divided by $n$. Adding 3 to every observation adds $3n$ to the total sum; dividing by $n$ gives an increase of exactly 3 in the mean. Options A, C, D are wrong because the mean does change, it changes by 3 (not 6 or 3n). This is a standard property of mean tested frequently in MCQs.