Question

If the value of the determinant $\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|$ is positive then $\left(a,b,c>0\right)$

• A. $abc>1$
• B. $abc>-8$
• C. $abc<-8$
• D. $abc>-2$

Answer 1

Answer: (B)

$\Delta =\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|=abc-\left(a+b+c\right)+2$
$\therefore \Delta >0\Rightarrow abc+2>a+b+c$
$\Rightarrow abc+2>3{\left(abc\right)}^{1/3}$
$\left[\because A.M.>G.M\Rightarrow \frac{a+b+c}{3}>{\left(abc\right)}^{1/3}\right]$
$\Rightarrow x^3+2>3x,$ where $x={\left(abc\right)}^{1/3}$
$\Rightarrow x^3-3x+2>0$
$\Rightarrow {\left(x-1\right)}^2\left(x+2\right)>0$
$\Rightarrow x+2>0$
$\Rightarrow x>-2$
$\Rightarrow {\left(abc\right)}^{1/3}>-2$
$\Rightarrow abc>-8$