Question

Let $\omega$ be the complex number representing the point $M \left(\frac{-1}{2}, \frac{\sqrt{3}}{2}\right)$ and $a , b , c , \alpha, \beta, \gamma$ be non-zero complex numbers such that
$a+b+c=\alpha $
$a+b \omega+c \omega^2=\beta$
$a+b \omega^2+c \omega=\gamma$
Number of distinct complex numbers $z$ satisfying the equation $ (z+1)\begin{vmatrix} z+\omega^2 & 1 \\ 1 & z+\omega \end{vmatrix}+\omega\begin{vmatrix} 1 & \omega \\ z+\omega & \omega^2 \end{vmatrix}+\omega^2\begin{vmatrix} \omega & z+\omega^2 \\ \omega^2 & 1 \end{vmatrix}=0$ is equal to

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (B)

$(z+1)\begin{vmatrix}z+\omega^2 & 1 \\ 1 & z+\omega\end{vmatrix}+\omega\begin{vmatrix}1 & \omega \\ z+\omega & \omega^2\end{vmatrix}+\omega^2\begin{vmatrix}{cc}\omega & z+\omega^2 \\ \omega^2 & 1\end{vmatrix}=0$
$\text { As } 1+\omega+\omega^2=0, \omega^4=\omega $
$\Rightarrow ( z +1)\left( z ^2+ z \left(\omega+\omega^2\right)+\omega^3-1\right)+\omega\left(\omega^2- z \omega-\omega^2\right)+\omega^2\left(\omega- z \omega^2-\omega^4\right)=0 $
$\Rightarrow ( z +1)\left( z ^2- z \right)+ z \left(-\omega^2-\omega\right)=0 $
$\Rightarrow z ^3- z ^2+ z ^2- z + z =0$
$\Rightarrow z ^3=0 $
$\Rightarrow z =0 \text { only } 1 \text { complex number. }$