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If z 1 , z 2 and z 3 are three complex numbers such that z 1 = z 2 = z 3 = 1 z 1 + 1 z 2 + 1 z 3 = 1 , then z 1 + z 2 + z 3 is

Correct answer

1

Step-by-step solution

z 1 = z 2 = z 3 = 1 (given) Now z 1 = 1 ⇒ z 1 2 = 1 ⇒ z 1 z 1 ¯ = 1 ⇒ 1 z 1 = z 1 ¯ Similarly, 1 z 2 = z 2 ¯   &   1 z 3 = z 3 ¯ Now 1 z 1 + 1 z 2 + 1 z 3 = 1 ⇒ z 1 + z 2 + z 3 ¯ = 1 ( ∵ z 1 ¯ + z 2 ¯ = z 1 + z 2 ¯ ) and | z 1 + z 2 + z 3 ¯ | = 1     ( ∵ | z 1 ¯ | = | z | ) ⇒ z 1 + z 2 + z 3 = 1

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