Practicing Success
If the value of the determinant $\begin{vmatrix}a&1&1\\1 &b& 1\\1&1&c\end{vmatrix}$ is positive, then |
$abc >1$ $abc>-8$ $abc <-8$ $abc >-2$ |
$abc>-8$ |
We have, $Δ=\begin{vmatrix}a&1&1\\1 &b& 1\\1&1&c\end{vmatrix}=abc - (a+b+c) +2$ $∴Δ>0$ $⇒abc +2> a+b+c$ $⇒abc+2>3(abc)^{1/3}$ $\left[∵A.M.>G.M.⇒\frac{a+b+c}{3}>(abc)^{1/3}\right]$ $x^3+2> 3x$, where $x =(abc)^{1/3}$ $⇒x^3-3x+2>0$ $⇒(x-1)^2 (x+2) > 0$ $⇒ x+2>0⇒ x>-2⇒ (abc)^{1/3} >-2 ⇒ abc >-8$ |