Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) If and are the distinct roots of the equation x^2 + x + 1 = 0 , then the quadratic equation with roots 1 ( +1)²⁰²⁶ and 1 ( +1)²⁰²⁶ is (1) x^2 + x + 1 = 0 (Q) If and are the distinct roots of the equation x^2 + x + 1 = 0 , then the quadratic equation with roots 1 ( +1)²⁰²⁷ and 1 ( +1)²⁰²⁷ is (2) x^2 - x + 1 = 0

Step-by-step solution

For (P): Since and are roots of x^2 + x + 1 = 0 , we have ^2 + + 1 = 0 + 1 = - ^2 and ^3 = 1 . 1 ( +1)²⁰²⁶ = 1 (- ^2)²⁰²⁶ = 1 ⁴⁰⁵² Since 4052 = 3 1350 + 2 , ⁴⁰⁵² = ( ^3)¹³⁵⁰ ^2 = ^2 . Thus, 1 ^2 = ^3 ^2 = . Similarly, 1 ( +1)²⁰²⁶ = . The quadratic equation with roots and is the original equation x^2 + x + 1 = 0 . So, (P) (1). For (Q): 1 ( +1)²⁰²⁷ = 1 (- ^2)²⁰²⁷ = - 1 ⁴⁰⁵⁴ Since 4054 = 3 1351 + 1 , ⁴⁰⁵⁴ = . Thus, - 1 = - . Similarly, the other root is - . Sum of roots = - - = -( + ) = -(-1) = 1 . Product of roots =