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Q.
The number of positive integral solutions of the equation $\begin{vmatrix}x^3+1 & x^2 y & x^2 z \\ x y^2 & y^3+1 & y^2 z \\ x z^2 & y z^2 & z^3+1\end{vmatrix}=11$ is
Determinants
Solution:
Multiply $R_1$ by $x ; R_2$ by $y$ and $R_3$ by $z$ and divide the determinant by $x y z$
$\frac{1}{x y z}\begin{vmatrix}x^4+x & x^3 y & x^3 z \\ x y^3 & y^4+y & y^3 z \\ x z^3 & y z^3 & z^4+z\end{vmatrix}=11$
$=\frac{x y z}{x y z}\begin{vmatrix}x^3+1 & x^3 & x^3 \\ y^3 & y^3+1 & y^3 \\ z^3 & z^3 & z^3+1\end{vmatrix}=11$
use $R_1 \rightarrow R_1+R_2+R_3$
$D=\left(x^3+y^3+z^3+1\right)\begin{vmatrix}1 & 1 & 1 \\ y^3 & y^3+1 & y^3 \\ z^3 & z^3 & z^3+1\end{vmatrix}=11$
hence $x^3+y^3+z^3=10 $ (as the det. has the value 1)
$(2,1,1),(1,2,1),(1,1,2) \Rightarrow(B)$