Let a, b, c be distinct real numbers and D be the determinant given by $D=\begin{vmatrix}a&1&1\\1&b&1\\1&1&c\end{vmatrix}$

Statement-1: If $D >0$, then $abc >-8$

Statement-2: A.M.>G.M.

Statement-1 is True, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

We have,

$D=\begin{vmatrix}a&1&1\\1&b&1\\1&1&c\end{vmatrix}=abc-(a+b+c) + 2$

$∴D >0$

$⇒abc - (a+b+c)+2>0$

$⇒abc +2> a+b+c$  ...(i)

But, A.M. >G.M.

$⇒\frac{a+b+c}{3}>(abc)^{1/3}$

$⇒a+b+c>(abc)^{1/3}$  ...(ii)

From (i) and (ii), we get

$abc+2> 3 (abc)^{1/3}$

$⇒ x^3+2>3x$, where $x = (abc)^{1/3}$

$⇒ x^3-3x+2>0 (x-1)^2 (x+2) > 0$

$⇒ x+2>0⇒ (abc)^{1/3}+2>0$

$⇒ (abc)^{1/3} >-2⇒ abc >-8$