Concept:
When converting an interrogative sentence from direct speech to indirect speech, we apply several structural grammar rules:
• Reporting Verb Change: The reporting verb "said to" shifts into an inquiring action verb like "asked", "enquired", or "interrogated" because the direct statement is a question.
• Connector Application: For interrogatives that start with an auxiliary verb (like "Are", "Is", "Do", "Have"), we remove the quotation marks and introduce the conjunctions if or whether.
• Tense Shifting Rule (Backshifting): If the reporting clause is in the past tense ("said"), the present continuous tense ("Are you going") in the reported clause shifts back to the past continuous tense ("was/were going").
• Pronoun Adaptations: The second-person pronoun "you" changes based on the object of the reporting verb ("him"), becoming "he". The possessive determiner "your" changes to "his".
• Sentence Structure: The interrogative sentence structure inversion changes back into an assertive sentence format (subject preceding verb).
Step 1: Step-by-Step implementation of transformations.
Let us track the transformation of each segment explicitly:
\[
\text{Direct: } \underbrace{\text{The gatekeeper said to him,}}_{\text{Reporting Clause}} \quad \text{``}\underbrace{\text{Are you going to your House?}}_{\text{Reported Interrogative Clause}}\text{''}
\]
• Change "said to him" into "
asked him".
• Drop quotation marks and insert the connector "
if".
• Transform the question format "Are you going..." to an assertive path. Under backshifting, "you" matches "him" $\rightarrow$ "
he".
• Change present continuous "are going" to past continuous for singular subject "he" $\rightarrow$ "
was going".
• Change "your House" to "
his house".
Step 2: Evaluating the options based on these steps.
Let us verify the given options:
• Option (A): Uses "is going". This fails because it leaves the tense in the present, violating the backshifting rule.
• Option (B): Grammatically flawless. "The gatekeeper asked him if he was going to his house." It applies all transformation rules correctly.
• Option (C): Changes the reporting verb to present tense "asks", which is incorrect as the original sentence used the past tense "said".
• Option (D): Transforms "are going" into the simple past "goes/went" (expressed poorly as "goes"), failing to maintain the continuous aspect of the action.
Hence, Option (B) is the correct indirect speech conversion.