Let n be a positive integer and a complex number with unit modulus is a solution of the equation Z n + Z + 1 = 0 , then the value of n can be

Step-by-step solution

Let, Z 1 satisfies Z n + 1 = - Z ⇒ Z 1 also satisfies Z n + 1 = - Z ⇒ Z 1 also satisfies Z n + 1 = 1 ⇒ Z 1 n + 1 = 1 … 1 Now, Z 1 = 1 ⇒ Z 1 n = 1 … 2 From 1 & 2 , we get, Z 1 n must be the point of intersection of Z = 1 & Z n + 1 = 1 ⇒ Z 1 n = ω or ω 2 where, ω is non-real cube root of unity ⇒ Z 1 can be ω or ω 2 ⇒ n is of the form of 3 k + 2