Question

If $\omega$ is the cubic root of unity, then value of the ${(1+\omega -{\omega }^2)}^2+{(1-\omega +{\omega }^2)}^2+1$ is

• A. $-1$
• B. $7$
• C. $1$
• D. $-3$

Answer 1

The correct option is D $-3$

${(1+\omega -{\omega }^2)}^2+{(1-\omega +{\omega }^2)}^2+1$
$={(-2{\omega }^2)}^2+{(-2\omega )}^2+1$
$=4({\omega }^4+{\omega }^2)+1$ $[\because 1+\omega +{\omega }^2=0,{\omega }^3=1]$
$=4(\omega \cdot {\omega }^3+{\omega }^2)+1$
$=4(\omega +{\omega }^2)+1$
$=4(-1)+1=-4+1=-3$.