The question asks us to identify which of the given sentences are grammatically correct. This involves understanding the correct placement of adverbs like "hardly" and "barely" when used with an auxiliary verb like "could".

Adverb Placement with Auxiliary Verbs

When adverbs of degree or frequency such as "hardly," "barely," "scarcely," "never," "rarely," etc., are used with an auxiliary verb (like "could," "will," "can," "have," "be"), they typically follow the auxiliary verb and precede the main verb. These adverbs convey a negative or restrictive meaning, indicating something is almost not happening or is difficult.

• Rule: Auxiliary Verb + Adverb + Main Verb
• Example: "He could hardly breathe."
• Example: "She can barely see."

Analysis of Each Sentence

Let's examine each sentence provided in the question to determine its grammatical correctness:

Sentence i: "After his surgery, Raja hardly could walk."

• In this sentence, the adverb "hardly" is placed before the auxiliary verb "could".
• According to standard English grammar rules, adverbs like "hardly" and "barely" usually come after the auxiliary verb when "could" is acting as an auxiliary verb.
• Therefore, this sentence is grammatically incorrect.

Sentence ii: "After his surgery, Raja could barely walk."

• Here, the adverb "barely" is correctly placed after the auxiliary verb "could" and before the main verb "walk".
• This placement follows the standard rule for adverbs of degree, making the sentence grammatically correct.
• Therefore, this sentence is grammatically correct.

Sentence iii: "After his surgery, Raja barely could walk."

• Similar to sentence i), the adverb "barely" is placed before the auxiliary verb "could".
• This placement is generally considered unnatural and grammatically incorrect in standard English.
• Therefore, this sentence is grammatically incorrect.

Sentence iv: "After his surgery, Raja could hardly walk."

• In this sentence, the adverb "hardly" is correctly placed after the auxiliary verb "could" and before the main verb "walk".
• This adheres to the grammatical rule for adverb placement, making the sentence grammatically correct.
• Therefore, this sentence is grammatically correct.

Correct Sentences Summary

Based on the analysis of adverb placement rules with auxiliary verbs, the following sentences are found to be grammatically correct:

• Sentence ii: "After his surgery, Raja could barely walk."
• Sentence iv: "After his surgery, Raja could hardly walk."

These two sentences demonstrate the proper positioning of the adverbs "barely" and "hardly" after the auxiliary verb "could," which is the standard and accepted grammar rule for such constructions. The sentences that place these adverbs before the auxiliary verb are considered incorrect in this context.