Question

Let $ω$ be the complex number representing the point $M\left(\frac{−1}{2},\frac{\sqrt{3}}{2}\right)$ and $a,b,c,Î\pm ,β,γ$ be non-zero complex numbers such that
$a+b+c=Î\pm$
$a+bω+c{ω}^2=β$
$a+b{ω}^2+cω=γ$
Number of distinct complex numbers $z$ satisfying the equation $(z+1)â\begin{array}{cc}z+{ω}^2& 1\\ 1& z+ω\end{array}â+ωâ\begin{array}{cc}1& ω\\ z+ω& {ω}^2\end{array}â+{ω}^2â\begin{array}{cc}ω& z+{ω}^2\\ {ω}^2& 1\end{array}â=0$ is equal to

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (B)

$(z+1)â\begin{array}{cc}z+{ω}^2& 1\\ 1& z+ω\end{array}â+ωâ\begin{array}{cc}1& ω\\ z+ω& {ω}^2\end{array}â+{ω}^2â\begin{array}{cc}ccω& z+{ω}^2\\ {ω}^2& 1\end{array}â=0$
$\text{ As }1+ω+{ω}^2=0,{ω}^4=ω$
$⇒(z+1)\left(z^2+z\left(ω+{ω}^2\right)+{ω}^3−1\right)+ω\left({ω}^2−zω−{ω}^2\right)+{ω}^2\left(ω−z{ω}^2−{ω}^4\right)=0$
$⇒(z+1)\left(z^2−z\right)+z\left(−{ω}^2−ω\right)=0$
$⇒z^3−z^2+z^2−z+z=0$
$⇒z^3=0$
$⇒z=0\text{ only }1\text{ complex number. }$