Author's younger sister Laxmi had given him the autograph book for ___________________.
It is mentioned in the passage that ‘Laxmi had given me an autograph book to get the autographs of visiting players’. "To get the autographs" is an infinitive which means: "In order to get the autographs". So option C is the right answer.
Option A is incorrect because "writing down scores" is not mentioned as the purpose for which Lakshmi gave the book.
Option B: The narrator mentions that he carefully put down MCC touring team in 1933-34 on one of the pages. However, the purpose of putting down the names (or listing the names) was to get the autograph of each of the players against their names. Hence, listing down the team members is not the purpose of giving the autograph book. Hence, this option is incorrect.
Option D: It is mentioned in the passage that ‘Laxmi had given me an autograph book to get the autographs of visiting players’. Though Gandhiji's autograph was taken in the same book, it was not given particularly for getting Gandhiji's autograph. Hence, this option is incorrect.