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Q.
If $a, b, c$ are sides of a scalene triangle, then the value of $\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}$ is always
Determinants
Solution:
$\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}=3 abc - a ^3- b ^3- c ^3$
$=-(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right) $
$=-(a+b+c) \times \frac{1}{2} \times\left[(a-b)^2+(b-c)^2+(c-a)^2\right]$
$=$ always negative as $a , b , c$ are sides of the triangle.