Let a,b, c be positive and not all equal, the value of the determinan…

The correct answer is option (2) : -ive

Let $Δ=c$. Then,

$Δ=\begin{bmatrix}a+b+c & b & c\\b+c+a & c & a\\c+a+b & a & b\end{bmatrix}$    [Applying $C_1→C_1+C_2+C_3]$

$⇒Δ= (a+b+c) \begin{bmatrix}1 & b & c\\1 & c & a\\1 & a & b\end{bmatrix}$       [Taking (a+b+c) common $C_1$]

$⇒Δ= (a+b+c) \begin{bmatrix}1 & b & c\\0 & c-b & a-c\\0 & a-b & b-c\end{bmatrix}$    $\begin{matrix} Applying \, R_2→R_2-R_1\\R_3→R_3-R_1\end{matrix}$

$⇒Δ= (a+b+c)1\begin{bmatrix}c-b & a- c\\a-b & b-c\end{bmatrix}$ [Expanding along $C_1$]

$⇒Δ= (a+b+c) (-(b-c)^2 - (a-c) (a-b))$

$⇒Δ= - (a + b + c) [(b=c)^2 + (a-c) (a-b)]$

$⇒Δ=-\frac{1}{2}(a+b+c) \begin{Bmatrix}(a-b)^2 + (b-c)^2 +(c-a)^2\end{Bmatrix}< 0 $   [∵a≠b≠c and a, b, c > 0]