Let z be a complex number satisfying z 3 + 2 z 2 + 4 z ¯ - 8 = 0 , where z ¯ denotes the complex conjugate of z . Let the imaginary part of z be nonzero. Match each entry in List-I to the correct entries in List-II. List-I List-II P z 2 is equal to 1 12 Q z - z ¯ 2 is equal to 2 4 R z 2 + z + z ¯ 2 is equal to 3 8 S z + 1 2 is equal to 4 10 5 7 The correct option is
B. P → 2   Q → 1   R → 3   S → 5
Given, z 3 + 2 z 2 + 4 z ¯ - 8 = 0   . . . . . . . 1 Now on taking conjugate both side we get, z 3 + 2 z ¯ 2 + 4 z - 8 = 0   . . . . . . . . 2 Note   z ¯ = z Now subtracting both equations we get, 2 z 2 - z ¯ 2 + 4 z ¯ - z = 0 ⇒ z - z ¯ 2 z + z ¯ - 4 = 0 ∵   z = z ¯ (not possible) or 4 x = 4 ⇒ x = 1 as   z + z ¯ = 2 x So, z = 1 + λ i ⇒ z = 1 + λ 2   &   z ¯ = 1 - λ i Now putting the value of z ,   z   &   z ¯ in given equation we get, 1 + λ 2 3 / 2 + 2 1 - λ 2 + 2 λ i + 4 1 - λ i - 8 = 0 ⇒ 1 + λ 2 3 / 2 + 2 1 - λ 2 = 4 ⇒ 1 + λ 2 3 / 2 = 2 1 + λ 2 ⇒ 1 + λ 2 1 + λ 2 - 2 = 0 ⇒ λ 2 = 3 Now solving, P z 2 = 1 + λ 2 = 1 + 3 = 4 Q z - z ¯ 2 = 1 + λ i - 1 - λ i 2 = 2 λ i 2 = 4 λ 2 = 12 R z 2 + z + z ¯ 2 = 4 + 1 + λ i + 1 - λ i 2 = 4 + 4 = 8 S z + 1 2 = 1 + λ i + 1 2 = 4 + λ 2 = 4 + 3 = 7 ∴   P → 2 ,   Q → 1 ,   R → 3 ,   S → 5
Related: Mathematics — Complex Number · All PYQ Banks