Which of the following polymer is a polyamide?

A polyamide is a polymer that contains repeating amide functional groups (-CO-NH-) in its main chain. These groups are formed through condensation reactions, typically between diamines and dicarboxylic acids, or from amino acids.

Now, let's examine each option:

Option A: Terylene
Terylene is a synthetic polymer known as polyester. It is formed by the condensation reaction between ethylene glycol (a diol) and terephthalic acid (a dicarboxylic acid). The repeating unit contains ester linkages (-CO-O-), not amide groups. Therefore, Terylene is not a polyamide.

Option B: Nylon
Nylon is a synthetic polymer. For example, Nylon-6,6 is made from hexamethylenediamine (a diamine) and adipic acid (a dicarboxylic acid). The reaction is:
$$H_{2}N-(CH_{2})6-NH_{2} + HOOC-(CH_{2})4-COOH \rightarrow [-NH-(CH_{2})6-NH-CO-(CH_{2})4-CO-]_n + H_{2}O$$
The repeating unit contains the amide group (-CO-NH-). Hence, Nylon is a polyamide.

Option C: Rubber
Natural rubber is a polymer of isoprene (2-methyl-1,3-butadiene). Its repeating unit is:
$$[-CH_{2}-C(CH_{3})=CH-CH_{2}-]_n$$
It contains only carbon-carbon double bonds and single bonds, with no amide groups. Synthetic rubbers, like styrene-butadiene rubber (SBR), also lack amide groups. Thus, rubber is not a polyamide.

Option D: Vulcanised rubber
Vulcanised rubber is natural rubber treated with sulfur to form cross-links. The base polymer remains polyisoprene, which has no amide groups. The vulcanization process adds sulfur bridges but does not introduce amide linkages. Therefore, vulcanised rubber is not a polyamide.

Only Nylon contains the characteristic amide groups in its polymer chain.

Hence, the correct answer is Option B.