Positive carbylamine test is shown by :

p-Methylbenzylamine

The correct answer is option 4. p-Methylbenzylamine.

Let us delve into the carbylamine test and the chemical principles behind it.

The carbylamine test is a chemical test used to differentiate primary (1°), secondary (2°), and tertiary (3°) amines based on their reactivity with chloroform (CHCl₃) and alcoholic potassium hydroxide (KOH dissolved in alcohol), leading to the formation of foul-smelling isocyanides, also known as carbylamines.

Here's a step-by-step explanation of the test:

1. Reaction Setup: The amine (RNH₂) is mixed with chloroform and alcoholic potassium hydroxide solution. The chloroform serves as the solvent, and alcoholic potassium hydroxide acts as the reagent. The mixture is then heated.

\[ \text{RNH}_2 + \text{CHCl}_3 + \text{KOH (alcoholic)} \]

2. Reaction: Under the conditions of heating, the primary amine reacts with chloroform and alcoholic potassium hydroxide to form an isocyanide compound, also known as carbylamine. The reaction proceeds via nucleophilic substitution, with the nitrogen atom of the amine attacking the chloroform molecule.

\[ \text{RNH}_2 + \text{CHCl}_3 + \text{KOH (alcoholic)} \rightarrow \text{RNC} + 3\text{H}_2\text{O} + \text{KCl} \]

Where RNH₂ represents the primary amine and RNC represents the isocyanide formed.

3. Product Formation: The isocyanide compound (carbylamine) produced in the reaction is characterized by its foul, pungent odor. This odor is distinctive and can be used as an indicator of a positive carbylamine test.

4. Interpretation: If the amine is a primary amine, the reaction will proceed, and a foul-smelling isocyanide compound will be produced, indicating a positive carbylamine test. If the amine is a secondary or tertiary amine, it will not react with chloroform and alcoholic potassium hydroxide to form the isocyanide compound, and therefore, no foul odor will be produced, indicating a negative carbylamine test.

Now, let's apply this understanding to the options provided:

1. N-Methylaniline: This is a secondary amine, as it has two alkyl groups attached to the nitrogen atom. Therefore, it will not give a positive carbylamine test.

2. N,N-Dimethylaniline: Similarly, this is also a secondary amine and will not give a positive carbylamine test.

3. Triethylamine: This is a tertiary amine, as it has three alkyl groups attached to the nitrogen atom. Tertiary amines do not react in the carbylamine test, so it will not give a positive result.

4. p-Methylbenzylamine: This is a primary amine, as it has a single hydrogen atom attached to the nitrogen atom. Primary amines react in the carbylamine test, so it will give a positive result, with the formation of the foul-smelling isocyanide compound.

Therefore, option (4) p-Methylbenzylamine is the correct answer for showing a positive carbylamine test.