The total number of distinct x∈R for which \begin{vmatrix} x^2 & x^3 & x \ 2 & 4 & 1 \ 29 & 21 & 3 \end{vmatrix} = 10

\begin{vmatrix} x^2 & x^3 & x \ 2 & 4 & 1 \ 29 & 21 & 3 \end{vmatrix} = 10 \implies x^3 \times det\begin{vmatrix} 1 & 1 & 1 \ 2 & 4 & 1 \ 29 & 21 & 3 \end{vmatrix} = 10 \implies x^3 \times [4(3-1)-1(6-29)+1(42-116)]=10 \implies x^3 \times [8+23-74]=10 \implies x^3(-43)=10 \implies x^3 = -\frac{10}{43} \implies x = \sqrt[3]{-\frac{10}{43}} which has one real root. \begin{vmatrix} x^2 & x^3 & x \ 2 & 4 & 1 \ 29 & 21 & 3 \end{vmatrix} = 10 \implies x^3 \times det\begin{vmatrix} 1 & 1 & 1 \ 2 & 4 & 1 \ 29 & 21 & 3 \end{vmatrix} = 10 \implies x^3 \times [1(12-21)-1(6-29)+1(42-116)] = 10 \implies x^3 \times [-9+23-74] = 10 \implies x^3(-60) = 10 \implies x^3 = -\frac{10}{60} = -\frac{1}{6} \implies x = \sqrt[3]{-\frac{1}{6}} which has one real root. Hence, total number of distinct values of x is 1.