The determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|
(A) a, b, c are in AP
(B) a, b, c are in GP
(C) a is a root of the equation ax<sup>2</sup> + bx + c = 0
(D) (x - a) is a factor of ax<sup>2</sup> + 2bx + c
Step-by-Step Solution
Key Concept: The determinant is equal to -(ax^2+2bx+c)(ac-b^2). For this to be zero, either ac-b^2=0 (which means a,b,c are in GP) or ax^2+2bx+c=0.
Let the determinant be D. Performing R3 -> R3 - aR1 - bR2, we get the determinant as -(ax^2+2bx+c)(ac-b^2). Thus, D=0 if ac-b^2=0 (i.e., a,b,c are in GP) or ax^2+2bx+c=0. If ax^2+2bx+c=0, then (x-a) is a factor if a is a root, i.e., aa^2+2ba+c=0.
Correct Answer: B,D