Amongst the following, the strongest base in aqueous medium is 

(CH₃)₂NH

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

To determine the strongest base among the given compounds in an aqueous medium, we need to consider the basicity of each compound. Basicity is influenced by the availability of the lone pair of electrons on the nitrogen atom for protonation. The more available these lone pairs are, the stronger the base.

Let us analyze each compound:

1. \(CH_3NH_2\) (Methylamine):

Methylamine is a primary amine. The lone pair on the nitrogen is relatively available for protonation. It is a strong base due to the +I (inductive) effect of the methyl group, which donates electron density to the nitrogen.

2. \(N≡CCH_2NH_2\) (Aminomethanenitrile):

This compound contains a nitrile group \((-CN)\) adjacent to the amine group. The electron-withdrawing effect of the nitrile group reduces the electron density on the nitrogen, making the lone pair less available for protonation. This decreases the basicity.

3. \((CH_3)_2NH\) (Dimethylamine): Dimethylamine is a secondary amine with two methyl groups. These methyl groups have a strong +I effect, increasing the electron density on the nitrogen and making the lone pair more available for protonation. This makes dimethylamine a very strong base.

4. \(C_6H_5NHCH_3\) (N-Methylaniline):
 In N-methylaniline, the phenyl group \((C_6H_5^-)\) is electron-withdrawing through both resonance and inductive effects, reducing the electron density on the nitrogen. This decreases the availability of the lone pair for protonation, making it a weaker base compared to aliphatic amines.

Based on the electron-donating or electron-withdrawing effects of the substituents attached to the nitrogen, the strongest base in an aqueous medium is: \((CH_3)_2NH\) (Dimethylamine).

Dimethylamine is the strongest base among the given compounds due to the strong electron-donating effect of the two methyl groups, which increase the electron density on the nitrogen, making the lone pair highly available for protonation.