Understanding the Concept:
This question tests a series of fundamental functional group transformations:
Dehydrohalogenation ($\text{E}2$ elimination) using a strong base like alcoholic $\text{KOH}$.
Hydrohalogenation (electrophilic addition) across an alkene using $\text{HCl}$.
Nucleophilic substitution ($\text{S}_{\text{N}}2$) using cyanide ion, followed by complete acid hydrolysis of the nitrile group ($-\text{CN} \rightarrow -\text{COOH}$).
Step 1: Track transformations from starting material [C] to intermediate [X].
Starting with cyclohexyl bromide [C], treating it with hot alcoholic $\text{KOH}$ induces an elimination reaction that removes $\text{HBr}$, forming cyclohexene [X]:
\[
\text{Cyclohexyl bromide} + \text{alc. KOH} \rightarrow \text{Cyclohexene [X]}
\]
Step 2: Track the addition from [X] to [Y].
Reacting cyclohexene [X] with $\text{HCl}$ results in electrophilic addition across the symmetric double bond, converting it back to chlorocyclohexane [Y]:
\[
\text{Cyclohexene [X]} + \text{HCl} \rightarrow \text{Chlorocyclohexane [Y]}
\]
Step 3: Determine the conversion from [Y] through to final product [Z].
Treating chlorocyclohexane [Y] with alcoholic $\text{KCN}$ introduces a nitrile group via nucleophilic substitution, producing cyanocyclohexane. Subsequent treatment with dilute $\text{HCl}$ undergoes complete hydrolysis, transforming the nitrile function into a carboxylic acid group. This yields cyclohexanecarboxylic acid [Z] ($\text{C}_6\text{H}_{11}\text{COOH}$).