If G and L are the greatest and least values of the expression`(2x^(2)-3x+2)/(2x^(2)+3x+2), x epsilonR` respectively. If `L^(2)ltlamdaltG^(2), lamda epsilon N` the sum of all values of `lamda` is

Options

A. 1035
B. 1081
C. 1225
D. 1176

Answer

D

Explanation

Let `y=(2x^(2)-3x+2)/(2x^(2)+3x+2)`
`implies2x^(2)y+3xy+2y=2x^(2)-3x+2`
`implies2(y-1)x^(2)+3(y+1)x+2(y-1)=0`
As `x epsilonR`
`:.Dge0`
`=9(y+1)^(2)-4.2(y-1).2(y-1)ge0`
`implies9(y+1)^(2)-16(y-1)^(2)ge0`
`implies(3y+3)^(2)-(4y-4)^(2)ge0`
`implies(7y-1)(7-y)ge0`
`implies(7y-1)(y-7)le0`
`:.1/7leyle7`
`:.G=7` and `L=1/7`
`:.GL=1`
NOw `(G^(100)+L^(100))/2ge(GL)^(100)implies(G^(100)+L^(100))/2ge1`
`impliesG^(100)+L^(100)ge2`
We have `L^(2)lt lamdalt G^(2)`
`(1/7)^(2)ltlamdalt 7^(2)`
`implies1/49lt lamdalt 49`
`implies lamda=1,2,3,..48` as `lamda epsilon N`
`:.` Sum of all values of `lamda=1+2+3+.............+48=(48xx49)/2=1176`