JEE Main202431 Jan 2024Morning ShiftMathematicsQuadratic EquationActual
For 0 < c < b < a , let ( a + b – 2 c ) x 2 + ( b + c – 2 a ) x + ( c + a – 2 b ) = 0 and α ≠ 1 be one of its root. Then, among the two statements (I) If α ∈ - 1 , 0 , then b cannot be the geometric mean of a and c . (II) If α ∈ 0 , 1 , then b may be the geometric mean of a and c .
Options
- ABoth (I) and (II) are true
- BNeither (I) nor (II) is true
- COnly (II) is true
- DOnly (I) is true
Correct answer
A. Both (I) and (II) are true
Step-by-step solution
Given: f ( x ) = ( a + b – 2 c ) x 2 + ( b + c – 2 a ) x + ( c + a – 2 b ) ⇒ f ( 1 ) = a + b – 2 c + b + c – 2 a + c + a – 2 b = 0 ⇒ f ( 1 ) = 0 Using product of roots, ⇒ α · 1 = c + a - 2 b a + b - 2 c ⇒ α = c + a - 2 b a + b - 2 c If, - 1 < α < 0 Statement-I: ⇒ - 1 < c + a - 2 b a + b - 2 c < 0 ⇒ b + c < 2 a and b > a + c 2 So, b cannot be G.M. between a and c . as A . M ≥ G . M So, statement I is correct. Statement-II: ⇒ 0 < α < 1 ⇒ 0 < c + a - 2 b a + b - 2 c < 1 ⇒ c + a - 2 b a + b - 2 c > 0 & c + a - 2 b a +