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, where $a\ne b\ne c$, then the value of $abc$.

• A. Cannot be less than 1
• B. Is greater than -8
• C. Is less than -8
• D. Must be greater than 8

Answer 1

Answer: (B)

Is greater than -8

Solution:

Let $△=\left[\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right]=a(bc-1)-1(c-1)+1(1-b)$

$=abc-a-b-c+2$

$\because △>0$

$\therefore abc-a-b-c+2>0$

$⟹abc+2>a+b+c$

$\because a\ne b\ne c$

$\therefore AM$ of $a,b,c>GM$ of $a,b,c$

$⟹\frac{a+b+c}{3}>(abc{)}^{\frac{1}{3}}$

$⟹abc+2>3(abc{)}^{\frac{1}{3}}$

Now, let $x=(abc{)}^{\frac{1}{3}}$, we have

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

$⟹(x-1{)}^2(x+2)>0$

$⟹x+2>0$ ..... $[\because (x-1{)}^2>0]$

$⟹x>-2⟹ab{c}^{\frac{1}{3}}>-2$

$⟹abc>-8$

Hence, B is the correct option.