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Let z 1 = 1 ,   z 2 = 2 ,   z 3 = 3 and z 1 + z 2 + z 3 = 3 + 5 i , then the value of Re z 1 z 2 - + z 2 z 3 - + z 3 z 1 - is equal to (Where z 1 ,   z 2 and z 3 are complex numbers)

Options

  1. A1
  2. B- 1
  3. C- 1 2
  4. D0

Correct answer

D. 0

Step-by-step solution

z 1 + z 2 + z 3 2 = 3 2 + 5 = 14 ⇒ z 1 + z 2 + z 3 z 1 - + z 2 - + z 3 - = 14 ⇒ z 1 2 + z 2 2 + z 3 2 + z 1 z 2 - + z 2 z 3 - + z 3 z 1 - + z 1 - z 2 + z 2 - z 3 + z 3 - z 1 = 14 ⇒ 1 + 4 + 9 + 2   Re z 1 z 2 - + z 2 z 3 - + z 3 z 1 - = 14 ⇒ 2 Re z 1 z 2 - + z 2 z 3 - + z 3 z 1 - = 0 ⇒ Re z 1 z 2 - + z 2 z 3 - + z 3 z 1 - = 0

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