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

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

Answer 1

The correct option is B $abc>-8$

$\Delta =\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|=abc-(a+b+c)+2$

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

$abc+2>a+b+c>3{\left(abc\right)}^{1/3}$
$\Rightarrow abc+2>3{\left(abc\right)}^{1/3}$
$\Rightarrow 2+x^3>3x(x={\left(abc\right)}^{1/3})$
$\Rightarrow x^3-3x+2>0$
$\Rightarrow {(x-1)}^2(x+2)>0$

$\therefore x>-2$
$\Rightarrow {\left(abc\right)}^{1/3}>-2$
$\Rightarrow abc>-8$