Let z k = cos ⁡ 2 k π 10 + i sin ⁡ 2 k π 10 ; k = 1 , 2 , … , 9 List – I List – II (A) For each z k there exists a z j s u c h z k . z j = 1 (P) True (B) There exists a k ϵ 1 , 2 , … , 9 such that z 1 . z = z k has no solution z in the set of complex numbers (Q) False (C) 1 - z 1 1 - z 2 … 1 - z 9 10 equals (R) 1 (D) 1 - ∑ k = 1 9 cos ⁡ 2 k π

Step-by-step solution

P z k i s 10 t h root of unity ⇒ z k - will also be 10 t h root of unity. Take z j as z k - Q z 1 ≠ 0 t a k e z = z k z 1 , we can always find z. R z 10 - 1 = z - 1 z - z 1 … z - z 9 ⇒ z - z 1 z - z 2 … z - z 9 = 1 + z + z 2 + … + z 9 ∀ z ϵ complex number. Put z = 1 1 - z 1 1 - z 2 … 1 - z 9 = 10 S 1 + z 1 + z 2 + … + z 9 = 0 ⇒ R e 1 + R e z 1 + … + R e z 9 = 0 ⇒ R e z 1 + R e z 2 + … + R e z 9 = - 1 ⇒ 1 - ∑ k = 1 9 cos ⁡ 2 k π 10 = 2