Vacillate: Hesitate:

1. Understanding the Relationship:

- "Vacillate" and "hesitate" both relate to a lack of decisiveness or firmness in making decisions or taking action.

- "Vacillate" suggests an oscillation or wavering between different options, while "hesitate" implies a temporary pause or reluctance before making a decision or taking action.

2. Rule of Analogy:

- In an analogy question, you're looking for a pair of words that have a similar relationship to the given pair.

- The relationship between "vacillate: hesitate" is one of similarity in meaning, particularly in terms of indecision or uncertainty.

3. Analysis of Options:

- Option ক pairs "persevere" with "waver." However, "persevere" means to continue in a course of action despite difficulty or delay, which contrasts with the indecision implied by "waver" or "hesitate."

- Options খ and গ offer pairs of words with contrasting meanings that do not parallel the relationship between "vacillate" and "hesitate."

- Option ঘ pairs "irresolute" with "indecisive," which accurately mirrors the relationship between "vacillate" and "hesitate." Both "irresolute" and "indecisive" convey a lack of firmness or determination in decision-making, akin to vacillation and hesitation.

4. Conclusion:

- Applying the rule of analogy, we identify option ঘ, "irresolute: indecisive," as the correct pair of words. They share a similar relationship of indecision or uncertainty, akin to the relationship between "vacillate" and "hesitate."