If x - y = 2 and y - z = 3 then value of, […] =

x - y = 2 and y - z = 3 , z-x = -5

\(\begin{vmatrix}1 & x & x^2\\1 & y & y^2\\1 & z & z^2\end{vmatrix}\)

$R_2 → R_2 - R_1$

$R_3 → R_3 - R_1$

\(=\begin{vmatrix}1 & x & x^2\\0 & y-x & (y-x)(y+x)\\0 & z-x & (z-x)(z+x)\end{vmatrix}\)

\((y-x)(z-x)\begin{vmatrix}1 & x & x^2\\0 & 1 & y+x\\0 & 1 & z+x\end{vmatrix}\)

$=(y-x)(z-x)[1.(1.(z+x)-1.(y+x)]$

$=(y-x)(z-x)[z+x-y-x]$

$=(x-y)(y-z)(z-x)$

$=(2)(3)(z-x)=6\times -5 = -30$