Question

If $\omega (\ne 1)$ is a cube root of unity and $(1+\omega {)}^7=A+B\omega$ Then, $(A,B)$ is equal to

• A. $(1,1)$
• B. $(1,0)$
• C. $(-1,1)$
• D. $(0,1)$

Answer 1

The correct option is A $(1,1)$

Given $\omega$ is a cube root of unity

$⟹1+\omega +{\omega }^2=0,{\omega }^3=1$

$(1+\omega {)}^7=A+B\omega$

$⟹(-{\omega }^2{)}^7=A+B\omega$

$⟹-{\omega }^{14}=A+B\omega$

$⟹\left({\omega }^3{)}^4\right(-{\omega }^2)=A+B\omega$

$⟹-{\omega }^2=A+B\omega$

$⟹A+B\omega =1+\omega$

$⟹A=1,B=1$

So $(A,B)=(1,1)$