Identify the sentences which are grammatically correct.A. Either she…

The correct answer is Option (2) → B and C only

Correlative conjunctions:

• They work in fixed pairs: either ... or, neither ... nor, both ... and, not only ... but also.
• The right partner must be used, and each part stands just before one of the two words or groups it joins.
• When "not only" begins a sentence, the helping verb comes before the subject: "Not only did he study".
  Example: You can either stay here or come with us.

A. Either she sings and dances. — Incorrect.

• "Either" pairs with "or", not "and", and it offers a choice: "She either sings or dances."
• Correct form: She either sings or dances.

B. Both the teacher and the students agreed. — Correct.

• The pair both ... and joins the two subjects correctly.

C. Not only did he study, but he also played. — Correct.

• The pair not only ... but also is complete, and the fronted "not only" is followed by the inverted order "did he study".

D. Neither he is honest not loyal. — Incorrect.

• "Neither" pairs with "nor", not "not", and the pair should stand before the two adjectives: "He is neither honest nor loyal."
• Correct form: He is neither honest nor loyal.