1.He delayed doing/to do his homework.

2.He suggested going/to go on a picnic.
3.He quit smoking/to smoke.
4.He avoided meeting/to meet the lady.
5.He proposed going abroad/to go abroad.

(Which one is correct? Do we have to use only gerunds in the sentences above?)

1. doing

2. going

3. smoking

4. meeting

5. going abroad

Yes, gerunds -- they are serving as direct objects in these sentences.

Yes. Only the gerunds fit in those sentences.

In the sentences provided, both gerunds and infinitives can be used, depending on the intended meaning.

1. "He delayed doing his homework" - In this sentence, the gerund "doing" is correct because it is used after the verb "delayed" as the direct object of the action "doing his homework."

Alternatively, we can use the infinitive "to do" like this: "He delayed to do his homework." Here, "to do" functions as the infinitive form of the verb "do," and is used after the verb "delayed" to express purpose.

2. "He suggested going on a picnic" - In this sentence, the gerund "going" is correct because it is used after the verb "suggested" as the direct object of the action "going on a picnic."

As for the infinitive, we can use it like this: "He suggested to go on a picnic." Here, "to go" functions as the infinitive form of the verb "go," and is used after the verb "suggested" to express purpose.

3. "He quit smoking" - In this sentence, the gerund "smoking" is correct as it is used as the direct object of the verb "quit."

Using the infinitive, we would say: "He quit to smoke." Here, "to smoke" functions as the infinitive form of the verb "smoke" and is used after the verb "quit" to express purpose, but this construction is less common.

4. "He avoided meeting the lady" - In this sentence, the gerund "meeting" is correct as it is used after the verb "avoided" as the direct object of the action "meeting the lady."

Instead of the gerund, we could use the infinitive: "He avoided to meet the lady." Here, "to meet" functions as the infinitive form of the verb "meet" and is used after the verb "avoided" to express purpose, though this construction may sound less natural to some.

5. "He proposed going abroad" - In this sentence, the gerund "going" is correct as it is used after the verb "proposed" as the direct object of the action "going abroad."

However, we would not use the infinitive in this case. Saying "He proposed to go abroad" might lead to a different interpretation, implying that the proposal itself was to go abroad, not just the action of going abroad.

Therefore, in the given sentences, both gerunds and infinitives can be used. The choice between them depends on the intended meaning and what sounds more natural in each context.