Complex Set

IIT JEE 2015 Papers › Algebra questions › Complex Set

Let S = (z_1,z_2,...,z_{2010}) We construct for each z_i \in S the set S_i = (z_iz_1,z_i,z_2,...,z_iz_{2010})

If it is true that S_i = S for 1 \le i \le 2010 then prove that

(1) |z_i| =1

(2) z \in S \Rightarrow \overline{z} \in S

#Mathematics #Algebra

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2 Answers

kaymant ·2010-12-14 12:10:52

Since S_i = S, so the product of the elements in the two sets must be the same. That is
z₁ z₂ ... z₂₀₁₀ = z_i ²⁰¹⁰ z₁ z₂ ... z₂₀₁₀
which means that z_i must be one of the 2010-th roots of unity. As such |z_i| =1 for each 1 â‰¤ i â‰¤ 2010.
This incidentally also means that S is the set of the 2010-th roots of unity and the (2) one follows.