The correct answer is: B. He asked his teacher whether he needed to read this chapter.
The sentence “Need I read this chapter?” is a polite way of asking whether it is necessary to read the chapter. The speaker is not sure whether they are required to read the chapter, and they are asking the teacher for clarification.
Option A is incorrect because the speaker is not asking whether there is a need for the teacher to read the chapter. They are asking whether they need to read the chapter.
Option C is incorrect because the speaker is not asking whether it is necessary for the teacher to read the chapter. They are asking whether they need to read the chapter.
Option D is incorrect because the speaker is not asking whether they have to read the chapter. They are asking whether they need to read the chapter.