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Q.
If $\Delta=\begin{vmatrix}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{vmatrix}$ and $\Delta_1=\begin{vmatrix}1 & 1 & 1 \\ y z & z x & x y \\ x & y & z\end{vmatrix}$, then
Determinants
Solution:
We have, $\Delta_1=\begin{vmatrix}1 & 1 & 1 \\ y z & z x & x y \\ x & y & z\end{vmatrix}$
By interchanging rows and columns, we get
$\Delta_1=\begin{vmatrix}1 & y z & x \\ 1 & z x & y \\ 1 & x y & z\end{vmatrix}=\frac{1}{x y z}\begin{vmatrix}x & x y z & x^2 \\ y & x y z & y^2 \\ z & x y z & z^2\end{vmatrix}=\frac{x y z}{x y z} \cdot\begin{vmatrix}x & 1 & x^2 \\ y & 1 & y^2 \\ z & 1 & z^2\end{vmatrix}$
$=(-1)\begin{vmatrix}1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2\end{vmatrix}=-\Delta$ (interchanging $C_1$ and $C_2)$
$\Rightarrow \Delta_1+\Delta=0$