If the sum of the squares of the reciprocals of the roots α and β of the equation 3 x 2 + λ x - 1 = 0 is 15 , then 6 α 3 + β 3 2 is equal to

Step-by-step solution

Given 1 α 2 + 1 β 2 = 15 For equation 3 x 2 + λ x - 1 = 0 α + β = - λ 3 , α β = - 1 3 ⇒ α 2 β 2 = 1 9 then α 2 + β 2 = α + β 2 - 2 α β = λ 2 9 - 2 - 1 3 = λ 2 9 + 2 3 ⇒ α 2 + β 2 = λ 2 + 6 9 Now 1 α 2 + 1 β 2 = 15 ⇒ α 2 + β 2 α 2 β 2 = 15 ⇒ λ 2 + 6 9 × 1 α 2 β 2 = 15 ⇒ λ 2 + 6 9 × 1 1 9 = 15 ⇒ λ 2 + 6 = 15 &#8658