Adela's last name – "Quested" – suggests that her quest is one important way of understanding the passage in the title, A Passage to India. Following the quest structure, Adela arrives in India, and feels the mysterious appeal of the force and life of the country.
Adela attempts to see the "real" India, but the real India eludes her. Even when she does meet Indians, at the Bridge Party, for example, they are in such artificial and formal situations that she doesn't feel as though she can have a genuine conversation with them.
Adela finally seems to have met her goal when she meets Mrs. Moore's charming Aziz at Fielding's tea party. On their excursion to the Marabar Caves, however, the thrill of hanging out with Aziz quickly sours when she believes he has attacked her.
Adela's ordeal only continues in the days leading up to the trial as she veers between her hatred of Aziz and her suspicion that Aziz is innocent. On the stand, she realizes her mistake and withdraws her charge. Far from solving her dilemma, this act opens up a whole new can of worms as she now has to deal with her ejection from Anglo-Indian society.
For a quest, this seems like a rather humble goal. Other mythical quests involve the Holy Grail, the Fountain of Youth, Shalimar … and all Adela gets is the realization that she's not all that good at making friends? The anticlimactic end of Adela's quest fits in with the novel's general tendency to emphasize the fundamental meaninglessness of existence. This is not as depressing as it sounds. Freed from the attitude that life is meaningful and ought to be examined like a book, Adela no longer studies life, but actually lives.