The question is:
I am happy .... you are back.
a) while
b) that
c) however
d) since
I think both b and d are correct
I asked some of my friends and they told to choose option b but I think option d works .. since related to reason
Both are technically correct, but I think answer b) sounds more natural -- "I'm happy that you are back". I've never heard someone say "I'm happy since you are back." But I have heard "I'm happy since you came back." But since that option isn't there, I would definitely go with answer b).
Yes. You are correct. Both work here - "since" indicating a reason, and "that - meaning: I am happy to have you here again" connecting the two clauses.
It could be a poorly designed question.
Further, I think "while" can work there but the second part of the sentence should be "you are here."