Question
Download Solution PDFω100 + ω200 + ω300 किसके बराबर है, जहाँ ω इकाई (यूनिटी) का घनमूल है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
एकल का घनमूल 1, ω और ω2 हैं।
यहाँ, ω = \(\frac{{ - {\rm{\;}}1{\rm{\;}} + {\rm{\;\;i}}\sqrt 3 }}{2}\) और ω2 = \(\frac{{ - {\rm{\;}}1{\rm{\;}} - {\rm{\;\;i}}\sqrt 3 }}{2}\)
एकल के घनमूल का गुण
- ω3 = 1
- 1 + ω + ω2 = 0
- ω = 1 / ω 2 और ω2 = 1 / ω
- ω3n = 1
गणना:
हमें ω100 + ω200 + ω300 का मान ज्ञात करना है।
⇒ ω100 + ω200 + ω300
= ω99 ω + ω198 ω2 + (ω3)100
= (ω3)33 ω + (ω3)66 ω2 + 1 (∵ ω3 = 1)
= ω + ω2 + 1
= 0 (∵ 1 + ω + ω2 = 0)
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