Question

If $\omega$ is a complex cube root of unity, then the value of $(a+b{)}^2+{(a\omega +b{\omega }^2)}^2+{(a{\omega }^2+b\omega )}^2$ is

• A. $6ab$
• B. $3ab$
• C. $12ab$
• D. $ab$

Answer 1

The correct option is A $6ab$

The given expression is
$(a+b{)}^2+{(a\omega +b{\omega }^2)}^2+{(a{\omega }^2+b\omega )}^2$
$=(a^2+b^2+2ab)+(a^2{\omega }^2+b^2{\omega }^4+2ab{\omega }^3)+(a^2{\omega }^4+b^2{\omega }^2+2ab{\omega }^3)$

$=(a^2+b^2+2ab)+(a^2{\omega }^2+b^2\omega +2ab)+(a^2\omega +b^2{\omega }^2+2ab)$
$[\because {\omega }^3=1]$

$=a^2(1+\omega +{\omega }^2)+b^2(1+\omega +{\omega }^2)+6ab$
$=6ab(\because 1+\omega +{\omega }^2=0)$