Thank you for reporting, we will resolve it shortly
Q.
If the value of the determinant $\left|\begin{matrix}a&1&1\\ 1&b&1\\ 1&1&c\end{matrix}\right|$ is positive then $\left(a, b, c >0\right)$
Determinants
Solution:
$\Delta=\left|\begin{matrix}a&1&1\\ 1&b&1\\ 1&1&c\end{matrix}\right|=a b c-\left(a +b+ c\right)+2$
$\therefore \Delta>0\Rightarrow a b c+2>a + b +c$
$\Rightarrow a b c +2>3\left(a b c\right)^{1/3}$
$\left[\because A.M. >G.M\Rightarrow \frac{a +b+ c}{3}>\left(a bc\right)^{1/3}\right]$
$\Rightarrow x^{3}+2>3x,$ where $x=\left(a b c\right)^{1/3}$
$\Rightarrow x^{3}-3x +2>0$
$\Rightarrow \left(x-1\right)^{2} \left(x+2\right)>0$
$\Rightarrow x+2>0$
$\Rightarrow x > -2$
$\Rightarrow \left(a b c\right)^{1/3} > -2$
$ \Rightarrow a b c > -8$