"P" is an optically active compound with molecular formula $$C_6H_{12}O$$. When "P" is treated with 2,4-dinitrophenylhydrazine, it gives a positive test. However, in presence of Tollens reagent, "P" gives a negative test. Predict the structure of "P".

To determine the structure of compound P, we need to interpret each of the chemical tests given in the question.

The compound gives a positive test with 2,4-dinitrophenylhydrazine (2,4-DNP).

A positive 2,4-DNP test indicates the presence of a carbonyl group. Therefore, compound P must be either an aldehyde or a ketone.

The compound gives a negative Tollens' test.

Tollens' reagent oxidizes aldehydes but does not react with ketones. Since the test is negative, compound P must be a ketone.

This immediately eliminates Option (C), which is an aldehyde.

The compound is also optically active.

For a molecule to be optically active, it must be chiral. In this case, it should contain at least one carbon atom attached to four different groups.

Now let us examine the remaining ketone options.

Option (A): Hexan-2-one

$$CH_3-C(=O)-CH_2-CH_2-CH_2-CH_3$$

None of the carbon atoms are bonded to four different groups. Therefore, the molecule is achiral and optically inactive.

Option (D): 4-methylpentan-2-one

$$CH_3-C(=O)-CH_2-CH(CH_3)-CH_3$$

The branched carbon is attached to one hydrogen atom, one side chain, and two identical $$CH_3$$ groups. Since two substituents are identical, it is not a chiral center.

Hence, this compound is also achiral.

Option (B): 3-methylpentan-2-one

$$CH_3-C(=O)-CH(CH_3)-CH_2-CH_3$$

The carbon at position 3 is attached to:

1. A hydrogen atom.
2. A methyl group $$(-CH_3)$$.
3. An ethyl group $$(-CH_2CH_3)$$.
4. An acetyl group $$(-C(=O)CH_3)$$.

Since all four groups are different, this carbon is a chiral center.

Therefore, this compound is optically active.

Hence, Option (B) is the only compound that satisfies all the given conditions. It is a ketone, gives a positive 2,4-DNP test, gives a negative Tollens' test, and contains a chiral center.

Therefore, the correct answer is Option (B).