Which part of the sentence shows an improper usage of the articles 'a…

The correct answer is Option (1) → (A)

"A few" and "the few":

• A few means some, a small number, spoken of in general.
• The few means the small number that there were, all of them. "The" is needed when the noun is made definite, for example by a clause that follows it.
  Example: The few friends he had stood by him in difficult times.

(A) A few books — Incorrect.

• The books are particular ones, those "that the students had", and they were "all" in poor condition, so it should read "The few books".
• Correct form: The few books that the students had were all in a pathetic condition.

(B) that the students had were all — Correct.

• "The students" refers to particular students.

(C) in a pathetic condition — Correct.

• "A" is used before a singular countable noun mentioned for the first time.

Contrast: "a few" means some, in general. "The few" means all of the small number that there were.