The correct match between Item - I and Item - II is:
Item - I                                        Item - II
(a) High density polythene    (I) Peroxide catalyst
(b) Polyacrylonitrile             (II) Condensation at high temperature & pressure
(c) Novolac                          (III) Ziegler-Natta catalyst
(d) Nylon 6                          (IV) Acid or base catalyst

We start by recalling how each of the given polymers is manufactured and which catalyst or reaction condition is characteristically employed.

High density polythene, commonly abbreviated as HDPE, is obtained by polymerising ethene in the presence of the well-known Ziegler-Natta catalyst, a mixture of $$\text{TiCl}_4$$ and $$\text{(C}_2\text{H}_5)_3\text{Al}$$. So we can write the correspondence as

High density polythene}\; \longrightarrow \; \text{Ziegler-Natta catalyst (III)

Polyacrylonitrile, abbreviated as PAN, is produced by the free-radical polymerisation of acrylonitrile. A peroxide such as benzoyl peroxide or hydrogen peroxide supplies these free radicals. Therefore, the link is

Polyacrylonitrile}\; \longrightarrow \; \text{Peroxide catalyst (I)

Novolac is a phenol-formaldehyde resin formed by the condensation of phenol with formaldehyde when an acid or a base acts as the catalyst. Thus, we have

$$\text{Novolac} \; \longrightarrow \;$$ Acid or base catalyst (IV)

Nylon 6 is synthesised by the ring-opening polymerisation (a step-growth process) of $$\epsilon$$-caprolactam. This polymerisation is carried out at a high temperature of about $$533 \text{ K}$$ and a pressure of roughly $$10^4 \text{ Pa}$$, so it is categorised as condensation at high temperature and pressure. Hence,

Nylon 6}\; \longrightarrow \; \text{Condensation at high temperature \& pressure (II)

Collecting all four pairings, we obtain

$$\begin{aligned} (a)&: \text{HDPE} &\rightarrow&\; (III) \\ (b)&: \text{Polyacrylonitrile} &\rightarrow&\; (I) \\ (c)&: \text{Novolac} &\rightarrow&\; (IV) \\ (d)&: \text{Nylon 6} &\rightarrow&\; (II) \end{aligned}$$

Looking at the given options, this exact sequence corresponds to Option B.

Hence, the correct answer is Option B.