I want to know if any students passed the maths exam this year. Which of the following sentences should I use?
In the sentence "I want to know if any students passed the math exam this year," the "if"-clause is an embedded question. "If" could be changed to "whether": "I want to know whether any students passed the math exam this year."
Now that you know that the "if"-clause is an embedded question, which of the three root questions you've given do you think it resembles the most? Isn't it the root question with "any students"? That is the phrase you've used in the embedded question.
Suppose you asked me, "Did any students pass the math exam this year?," and I needed to inquire of somebody else whether that was so. Would I say, "Subhajit wants to know if any student / any of the students passed the math exam this year." Why wouldn't I use the phrase you used?
One more question, does the first sentence mean the same as second sentence?
The second sentence ("Did any student pass the math exam this year?") would likely be used in a context in which there were just a handful of students, or in which the math exam was so hard that the vast majority students were expected to fail, or in which the person asking the question wanted the answerer to name the exam-passer(s).
Subhajit, I happened to notice your post above late last night, before you edited it. In the original version of your post, you claimed that it is believed by teachers in India that "any" cannot be followed by a plural noun. Let me assure you that it can.
"Any" is normally
followed by a plural noun when the noun it is followed by is a count noun. Obviously, with noncount nouns, we can only use the singular: "Is there any water left in the jug?" Maybe the superstition you speak of is caused by pronouns like "anyone."
(a) Was there anyone in the park?
(b) Were there any people in the park?
(c) (?) Was there any person in the park?
Incidentally, in the U.S., nobody says "maths." We say "math."