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KVPY2017MathematicsComplex Number

Let z₁, z₂, , z₇ be the vertices of a regular heptagon that is inscribed in the unit circle with centre at the origin in the complex plane. Let w = _ 1 i < j 7 z _ i z _ j , then | w | is equal to

Options

  1. A0
  2. B1
  3. C2
  4. D3

Correct answer

A. 0

Step-by-step solution

given expression can be written as (z₁+z₂+ +z₇ )²- _ i=1 ⁷ z_ i ² 2 = _ 1 i < j -7 z_ i z_ j Sum of all seven root of unity =0 (By property) z₁²+z₂²+z₃²+ .+z₇²=0 (By property)

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