Concept:
English grammar uses specific prepositions with means of transportation.
When we refer to a mode of transport in a general sense and there is no article before the vehicle noun, we use the preposition “by”.
The structure is:
\[
\text{by + vehicle}
\]
Examples:
• by train
• by bus
• by car
• by plane
• by ship
Step 1: Analyzing the structure of the sentence.
The sentence is:
\[
\text{He frequently travels _ _ _ _ train.}
\]
Notice that the word “train” appears without any article such as “a” or “the.”
Therefore, the standard transport expression should be used.
Step 2: Examining each option carefully.
• on
“On” generally requires an article or determiner.
Example:
\[
\text{on the train}
\]
Hence, incorrect.
• in
“In” also requires a determiner.
Example:
\[
\text{in a train}
\]
Hence, incorrect.
• by
This is the correct preposition used with a bare vehicle noun.
Hence, correct.
• across
This indicates movement from one side to another.
It cannot indicate a means of transportation.
Hence, incorrect.
Step 3: Completing the sentence.
The sentence becomes:
\[
\text{He frequently travels by train.}
\]
Therefore,
\[
\boxed{\text{by}}
\]
is the correct answer.