Let ( , ) be roots of equation (x^2-70 x+ =0 ), where ( 2 , 3 ). If ( ) assumes the minimum possible value, then ( ( -1 + -1 )( +35) | - | ) is equal to :

Step-by-step solution

Given, x 2 - 70 x + λ = 0 Sum of the roots will be α + β = 70 Product of roots will be α β = λ ⇒ α ( 70 - α ) = λ ⇒ d λ d α = 70 - 2 α Now, λ is increasing for all α ≤ 35 And 2 and 3 does not divide λ so taking α = 5 we get, β = 70 - α = 65 As for α = 1 , β = 69 & for α = 4 , β = 66 which is not possible as they are multiple of 2 & 3 Hence, α = 5 , β = 65 , λ = 325 Now solving, α - 1 + β - 1 λ + 35 | α - β | = 5 - 1 + 65 - 1 325 + 35 | 5 - 65 | = 2 + 8 · 360 60 = 60