Question

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

• A. $abc>-11$
• B. $abc<-8$
• C. $abc>-8$
• D. None of these

Answer 1

The correct option is C $abc>-8$

$\Delta =\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|=abc-(a+b+c)+2$ As $\Delta >0\therefore abc+2>a+b+c$ ...(A)Using A. M. $>G.M.$

$\Rightarrow \frac{a+b+c}{3}>{\left(abc\right)}^{1/3}$or $a+b+c>3{\left(abc\right)}^{1/3}$ ...(B)By (A) and (B) we have$\therefore abc+2>a+b+c>3{\left(abc\right)}^{1/3}$

$\Rightarrow abc+2>3{\left(abc\right)}^{1/3}$

$\Rightarrow 2+x^3>3x(x={\left(abc\right)}^{1/3})$

$\Rightarrow x^3-3x+2>0$

$\Rightarrow {(x-1)}^2(x+2)>0\therefore x>-2$

${\left(abc\right)}^{1/3}>2$

$abc>-8$