Step 1: Identify \(X\), \(Y\) and \(Z\).
• Used in black ink \(\rightarrow\) Carbon black (Lamp black)
• Used in water filters \(\rightarrow\) Activated charcoal
• Used as reducing agent in metallurgy \(\rightarrow\) Coke
Therefore,
\[
X=\text{Carbon black}
\]
\[
Y=\text{Activated charcoal}
\]
\[
Z=\text{Coke}
\]
Step 2: Check each statement.
Option (A)
Heating wood in absence of air produces
\[
\text{Wood charcoal},
\]
not carbon black.
Hence,
\[
\boxed{\text{False}}
\]
Option (B)
Activated charcoal is not produced by burning hydrocarbons in limited air.
Burning hydrocarbons in limited air produces carbon black (lamp black).
\[
\boxed{\text{False}}
\]
Option (C)
Coke is obtained by destructive distillation of coal.
That is,
\[
\text{Heating coal at high temperature in absence of air}.
\]
\[
\boxed{\text{True}}
\]
Option (D)
The three allotropes are obtained by different processes.
\[
\boxed{\text{False}}
\]
Final Answer:
\[
\boxed{\text{Z is produced by heating coal at high temperatures in absence of air}}
\]
\[
\boxed{\text{Answer = (C)}}
\]