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 $(a,b,c>0)$

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

Answer 1

Answer: (B)

$abc>-8$

we have

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

$\therefore \Delta >0\Rightarrow abc+2>a+b+c\Rightarrow abc+2>3{\left(abc\right)}^{1/3}[\because A.M>G.M\Rightarrow \frac{a+b+c}{3}>{\left(abc\right)}^{1/3}]$

$\Rightarrow x^3+2>3x[x={\left(abc\right)}^{1/3}]\Rightarrow x^3-3x+2>0\Rightarrow (x-1{)}^2(x+2)>0\Rightarrow xϵ(-2,1)\cup (1,\infty )\Rightarrow x>-2\Rightarrow {\left(abc\right)}^{1/3}>-2\Rightarrow abc>-8$