Choose the grammatically correct sentence:

The correct answer is Option (4) → After retiring, Mr Menon devoted himself to teaching the children of his village.

“To” as a preposition + -ing form:

• In some phrases, “to” is a preposition and not part of a to-verb. Object to, confess to, be addicted to and devote oneself to are such phrases. A verb after them takes the -ing form.
• This sentence has “devoted himself to teaching”, which is right.

Why the Other Options are Incorrect?

Option 1: The residents of our society objected to pay the extra charge for parking. — Incorrect. The given sentence says “objected to pay”. In “object to”, “to” is a preposition, so the verb after it must take the -ing form.
Corrected: The residents of our society objected to paying the extra charge for parking.

Option 2: At last, the boy confessed to break the window of the staff room. — Incorrect. The given sentence says “confessed to break”. In “confess to”, “to” is a preposition, so the -ing form “breaking” is needed.
Corrected: At last, the boy confessed to breaking the window of the staff room.

Option 3: My cousin has become addicted to play video games late at night. — Incorrect. The given sentence says “addicted to play”. In “addicted to”, “to” is a preposition, so the -ing form “playing” is needed.
Corrected: My cousin has become addicted to playing video games late at night.