A) Purely real
B) Purely imaginary
C) Complex
D) 0
Correct Answer: B
Solution :
[b] We have \[\overline{\Delta }=\left| \begin{matrix} 0 & -\overline{y} & -\overline{z} \\ y & 0 & -\overline{x} \\ z & x & 0 \\ \end{matrix} \right|=\left| \begin{matrix} 0 & y & z \\ -\overline{y} & 0 & x \\ -\overline{z} & -\overline{x} & 0 \\ \end{matrix} \right|\] [Interchanging rows and columns] \[={{(-1)}^{3}}\left| \begin{matrix} 0 & -y & -z \\ \overline{y} & 0 & -x \\ \overline{z} & \overline{x} & 0 \\ \end{matrix} \right|\] [Taking -1 common from each row] \[=-\Delta \] \[\therefore \overline{\Delta }+\Delta =0\Rightarrow 2\operatorname{Re}(\Delta )=0\] \[\therefore \Delta \] is purely imaginary.
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