Question

${\left[\frac{-1+i\sqrt{3}}{2}\right]}^6+{\left[\frac{-1-i\sqrt{3}}{2}\right]}^6+{\left[\frac{-1+i\sqrt{3}}{2}\right]}^5+{\left[\frac{-1-i\sqrt{3}}{2}\right]}^5=$

• A. -1
• B. 1
• C. 2
• D. none of these

Answer 1

Answer: (B)

1

${\left[\frac{-1+i\sqrt{3}}{2}\right]}^6+{\left[\frac{-1-i\sqrt{3}}{2}\right]}^6+{\left[\frac{-1+i\sqrt{3}}{2}\right]}^5+{\left[\frac{-1-i\sqrt{3}}{2}\right]}^5$
$={\omega }^6+{\left({\omega }^2\right)}^6+{\omega }^5+{\left({\omega }^2\right)}^5$----(1)
Where $\omega$ is cube root of unity.
i.e ${\omega }^3=1$ and $1+\omega +{\omega }^2=0$
from (1)
${\omega }^6+{\left({\omega }^2\right)}^6+{\omega }^5+{\left({\omega }^2\right)}^5=1+1+{\omega }^2+\omega =1$
Hence, option B.