Choose the grammatically correct sentence:

The correct answer is Option (1) → Ankit and Fatima always check each other’s answers after a mock test.

“Each other” for two people:

“Each other” is used when two people do the same action to the other. “Each other’s” shows that each one checks the answers of the other. Ankit and Fatima are two people, so “each other’s” is right. The sentence has no other error.

Why the Other Options are Incorrect?

Option 2: The shopkeeper which sold us the cooler has shut his shop. — Incorrect. The given sentence says “The shopkeeper which”. This is wrong: “which” is used for things and animals, not for people. A shopkeeper is a person, so use “who”.
Corrected: The shopkeeper who sold us the cooler has shut his shop.

Option 3: The cracked phone on the table is her’s, not mine. — Incorrect. The given sentence says “her’s”. This is wrong: possessive pronouns such as hers, yours and theirs never take an apostrophe.
Corrected: The cracked phone on the table is hers, not mine.

Option 4: Grandpa has lost his reading glasses and cannot find it anywhere. — Incorrect. The given sentence says “cannot find it”. This is wrong: “glasses” is plural, so the pronoun that stands for it must be plural too.
Corrected: Grandpa has lost his reading glasses and cannot find them anywhere.