I, as orthocresol♦ questioned in the comments section, don't know where these terms came from. Thus, any attempt to rationalize these terms would be a speculation. For clarity, it could be argued that each of the structures of these alkyl radicals are drawn in such a way that each of them is attached to a functional group such as hydroxyl or halide or etc. For sake of an argument, let's assume the functional group is hydroxyl, thus the compound from the first radical would be 2-pentanol (or pent-2-ol). Thus, let's try to give a meaningful rationalization to the terminology in the question:
- Why it is called 'active sec pentyl' (case 1): If you carefully look at the alkyl radical in case 1, you would realize it is a secondary radical so that it is a sec-pentyl group. Thus, the compound is optically active alcohol (in our case), pent-2-ol, which can exists in both (R)- and (S)-forms. Thus, I assume 'active' part comes to indicate optically active compounds.
- Why it is called 'sec pentyl' (case 2): If you carefully look at the alkyl radical in case 2, you would realize it is also a secondary radical so that it is also a sec-pentyl group. Thus, the relevant compound in our case is pent-3-ol, which can exists in only one form, because it is not optically active. Thus, I assume 'active' part is dropped from 'active sec pentyl' to indicate the radical forms only the non-optically active compounds, yet it is a 'sec pentyl'.
The compound related to 'case 3' is 3-methylbutan-1-ol, which is optically active so that it has been given the terminology, 'active pentyl'.