Question:

Let the number 2, b, c be an A.P. and A = [ 1 1 1; 2 b c; a b^2 c^2 ]. If det (A) ∈ [2, 16], then c lies in the interval:

[3,2+23/4

[2,3)

(2+23/4,4

[4,6]

Correct option is D. [4,6]
put b = 2 + c/2 in determinant of A
|A| = c³ - 6c² + 12c - 8 ∈ [2, 16]
⇒ (c - 2)³ ∈ [8, 64]
⇒ c ∈ [4, 6]