ω3n + ω3n+1 + ω3n+2 का मान क्या है जहां ω एकत्व का घन मूल है?

  1. -1
  2. \(\frac{{ - {\rm{\;}}1{\rm{\;}} + {\rm{\;\;i}}\sqrt 3 }}{2}\)
  3. 1
  4. 0

Answer (Detailed Solution Below)

Option 4 : 0
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धारणा:

एकत्व के घन मूल 1, ω और ω2 हैं

यहाँ, ω = \(\frac{{ - {\rm{\;}}1{\rm{\;}} + {\rm{\;\;i}}\sqrt 3 }}{2}\) और ω2\(\frac{{ - {\rm{\;}}1{\rm{\;}} - {\rm{\;\;i}}\sqrt 3 }}{2}\)

एकता के घन मूलों का गुण

  • ω3 = 1
  • 1 + ω + ω2 = 0
  • ω = 1 / ω 2 और ω2 = 1 / ω
  • ω3n = 1

 

गणना:

हमें ω3n + ω3n+1 + ω3n+2 का मूल्य खोजना होगा

⇒ ω3n + ω3n+1 + ω3n+2 

= ω3n (1 + ω + ω2)           (∵ 1 + ω + ω2 = 0)

= 1 × 0 = 0

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