Question

The total number of distinct $x \in R$ for which $\begin{vmatrix}x & x^{2} & 1+x^{3} \\ 2 x & 4 x^{2} & 1+8 x^{3} \\ 3 x & 9 x^{2} & 1+27 x^{3}\end{vmatrix}=10$ is

Answer 1

$\begin{vmatrix}x & x^{2} & 1+x^{3} \\ 2 x & 4 x^{2} & 1+8 x^{3} \\ 3 x & 9 x^{2} & 1+27 x^{3}\end{vmatrix}=10$
$\Rightarrow \begin{vmatrix}x & x^{2} & 1 \\ 2 x & 4 x^{2} & 1 \\ 3 x & 9 x^{2} & 1\end{vmatrix}+\begin{vmatrix}x & x^{2} & x^{3} \\ 2 x & 4 x^{2} & 8 x^{3} \\ 3 x & 9 x^{2} & 27 x^{3}\end{vmatrix}=10$
$\Rightarrow x ^{3}\begin{vmatrix}1 & 1 & 1 \\ 2 & 4 & 1 \\ 3 & 9 & 1\end{vmatrix}+ x ^{6}\begin{vmatrix}1 & 1 & 1 \\ 2 & 4 & 8 \\ 3 & 9 & 27\end{vmatrix}=10$
$\Rightarrow 2 x ^{3}+12 x ^{6}=10$
$\Rightarrow 6 x ^{6}+ x ^{3}-5=0$
$\Rightarrow \left(6 x ^{3}-5\right)\left( x ^{3}+1\right)=0$
$\Rightarrow x ^{3}=\frac{5}{6}, $
$x ^{3}=-1$
So two real roots