Step 1: Understand carbylamine test.
Carbylamine test is a test used to identify primary amines. In this test, a primary amine reacts with chloroform and alcoholic potassium hydroxide to form an isocyanide.
\[
RNH_2 + CHCl_3 + 3KOH \rightarrow RNC + 3KCl + 3H_2O
\]
Step 2: Check the given options.
\(R_2NH\) is a secondary amine. Secondary amines do not give carbylamine test.
\(RNH_2\) is a primary amine. Primary amines give carbylamine test.
\(RNC\) is an isocyanide product, not the amine that gives the test.
\(RCN\) is a nitrile and does not give carbylamine test.
Step 3: Final conclusion.
Only primary amines give carbylamine test. Therefore, the correct answer is:
\[
\boxed{RNH_2}
\]