Step 1: Recall the carbylamine reaction.
The carbylamine reaction is a test for primary amines (aliphatic or aromatic). In this reaction, a primary amine reacts with chloroform (CHCl$_3$) and alcoholic KOH to form isocyanides (carbylamines), which have a very unpleasant odor.
General reaction:
\[
R–NH_2 + CHCl_3 + 3KOH \;\; \xrightarrow{\Delta} \;\; R–NC + 3KCl + 3H_2O
\]
Step 2: Check each option.
- (A) CH$_3$–CH$_2$–NH$_2$: This is a primary aliphatic amine. It gives the carbylamine test.
- (B) CH$_3$–NH$_2$: This is also a primary aliphatic amine, so it gives the carbylamine test.
- (C) CH$_3$–NH–CH$_3$: This is a secondary amine. Secondary (and tertiary) amines do not give the carbylamine reaction.
- (D) C$_6$H$_5$–NH$_2$: Aniline is a primary aromatic amine. It also gives the carbylamine test.
Step 3: Final Answer.
Thus, the amine which does not give the carbylamine reaction is the secondary amine, CH$_3$–NH–CH$_3$.
\[
\boxed{CH_3–NH–CH_3 \; \text{(secondary amine)}}
\]
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