Question

If $\omega$ is fifth root of unity, then ${\log }_2\mid 1+\omega +{\omega }^2+{\omega }^3-{\omega }^{-1}\mid$ is equal to

• A. $1$
• B. $0$
• C. $-1$
• D. $2$

Answer 1

The correct option is A $1$

$1+w+w^2+...w^3-\frac{1}{w}$
$=\frac{w+w^2+w^3+...w^4-1}{w}$

$=\frac{1}{w}[\frac{w(1-w^4)}{1-w}-1]$

$=\frac{1}{w}[\frac{w-w^5}{1-w}-1]$

$=\frac{1}{w}[\frac{w-1}{1-w}-1]$

$=\frac{1}{w}[-2]$

Now
$|\frac{-2}{w}|$

$=\frac{|-2|}{|w|}$

$=\frac{2}{1}$

$=2$

$=lo{g}_2\left(2\right)$

$=1$
Hence, option 'A' is correct.