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Complex Numbers And Quadratic Equations

Question
CBSEENMA11015618

If z2 + z + 1 = 0, where z is a complex number, then the value ofopen parentheses straight z plus 1 over straight z close parentheses squared space plus open parentheses straight z squared space plus 1 over straight z squared close parentheses squared space plus open parentheses straight z cubed space plus 1 over straight z cubed close parentheses squared space plus..... open parentheses straight z to the power of 6 plus 1 over straight z to the power of 6 close parentheses squared space is

  • 18

  • 54

  • 6

Solution

E.

12

z2 + z + 1 = 0 ⇒ z = ω or ω2
straight z space plus 1 over straight z to the power of 4 space equals space minus space 1 comma space straight z to the power of 5 space plus 1 over straight z to the power of 5 space equals space minus space 1 space and space straight z to the power of 6 space plus 1 over straight z to the power of 6 space equals space 2

∴ The given sum = 1 + 1 + 4 + 1 + 1 + 4 = 12