Let a be the sum of all coefficients in the expansion of ( 1 – 2 x + 2 x 2 ) 2023 ( 3 - 4 x 2 + 2 x 3 ) 2024 and b = lim x → 0 ∫ 0 x log 1 + t t 2024 + 1 d t x 2 . If the equations c x 2 + d x + e = 0 and 2 b x 2 + a x + 4 = 0 have a common root, where c , d , e ∈ R , then d : c : e equals

Step-by-step solution

Given: a is the sum of all coefficients in 1 - 2 x + 2 x 2 2023 3 - 4 x 2 + 2 x 3 2024 ⇒ a = 1 - 2 × 1 + 2 × 1 2023 3 - 4 × 1 + 2 × 1 2024 ⇒ a = 1 . . . i Now, b = lim x → 0 ∫ 0 x log 1 + t t 2024 + 1 d t x 2 Using L-Hospital's rule and Newton Leibnitz Theorem, we get ⇒ b = lim x → 0 log 1 + x x 2024 + 1 2 x ⇒ b = lim x → 0 1 2 x 2024 + 1 ⇒ b = 1 2 . . . i i Also, 2 b x 2 + a x + 4 = 0 ⇒ x 2 + x + 4 = 0 , which gives complex conjugates as roots. Let α and α be those roots. Then, c x 2 + d x + e = 0 will also have α