Question

If $x, \, y$ and $z$ are the roots of the equation $2t^{3}-\left(tan \left[x + y + z\right] \pi \right)t^{2}-111t+2020=0,$ then $\begin{vmatrix} x & y & z \\ y & z & x \\ z & x & y \end{vmatrix}$ is equal to (where, $\left[x\right]$ denotes the greatest integral value less than or equal to $x$ )

• A. $20$
• B. $-10$
• C. $0$
• D. $1$

Answer 1

Answer: (C)

$\left[x + y + z\right]$ is an integer
$\Rightarrow tan \left(\left[x + y + z\right] \pi \right) = 0$
Hence, the sum of roots $=x+y+z=0$
$\Rightarrow \begin{vmatrix} x & y & z \\ y & z & x \\ z & x & y \end{vmatrix}=-\left(x + y + z\right)\left(x^{2} + y^{2} + z^{2} - x y - y z - z x\right)=0$