Which of the following is the strongest base in the aqueous medium? 

(CH₃)₂NH

The correct answer is option 3. \((CH_3)_2NH\).

In aqueous medium, the strongest base among the given options can be determined based on the availability of the lone pair of electrons on the nitrogen atom for protonation. Stronger bases generally have more available electron pairs and less steric hindrance around the nitrogen.

Comparison of Bases:

1. \(CH_3NH_2\) (Methylamine):

Methylamine is a primary amine with a lone pair of electrons on the nitrogen that is relatively available for protonation. It is a strong base compared to many other amines.

2. \(NCCH_2NH_2\) (Aminomethylcyanide):

This molecule has a cyano group \((−CN)\) attached to the carbon next to the amine group. The cyano group is electron-withdrawing, which decreases the availability of the lone pair on the amine nitrogen for protonation, making it a weaker base.

3. \((CH_3)_2NH\) (Dimethylamine):
   - Dimethylamine is a secondary amine with two methyl groups. The electron-donating effect of the methyl groups increases the availability of the lone pair of electrons on the nitrogen, making it a stronger base compared to methylamine.

4. \(C_6H_5NHCH_3\) (N-Methylphenylamine):

In this compound, the amine group is attached to a phenyl ring. The phenyl ring exerts an electron-withdrawing effect through resonance, which reduces the availability of the lone pair on the nitrogen, making it a weaker base compared to dimethylamine.

Conclusion:

Among the options, dimethylamine \((CH_3)_2NH\) is the strongest base in aqueous medium due to the electron-donating effects of the two methyl groups that increase the availability of the lone pair of electrons on the nitrogen for protonation.

Thus, the strongest base in the aqueous medium is: \((CH_3)_2NH\) (Dimethylamine)