Reading a sentence to see whether it "sounds wrong" is unreliable under time pressure, because the sentences are written to sound plausible. A checklist is faster and more accurate.
Run the same eight checks on every sentence: subject-verb agreement, tense consistency, pronoun agreement, preposition, article, comparison form, parallel structure, and misplaced modifier. The error is nearly always one of those.
Try each one before opening the answer. A solution you have read is not a solution you can produce under time pressure, which is the only kind that counts.
Q1 Find the error: "Each of the students have submitted their assignment on time."
"have" should be "has".
The subject is "Each", which is singular, not "the students". "Of the students" is a prepositional phrase and cannot supply the subject. The correct form is "Each of the students has submitted…". This is the single most common error type in the topic.
Q2 Find the error: "He is one of the best player in the team."
"player" should be "players".
"One of the" is always followed by a plural noun, because you are selecting one from a group. "One of the best players in the team" is correct. The verb stays singular - "He is" - because the subject is "He".
Q3 Find the error: "The teacher asked the students to read the chapter, write a summary and that they should discuss it."
The third item breaks the parallel structure.
The list is "to read", "write" and "that they should discuss". The first two are infinitives sharing the "to"; the third is a clause. It should be "…read the chapter, write a summary and discuss it."