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The total number of distinct $x \in \mathbb{R}$ for which $\left|\begin{array}{ccc}x & x^2 & 1+x^3 \\ 2 x & 4 x^2 & 1+8 x^3 \\ 3 x & 9 x^2 & 1+27 x^3\end{array}\right|=10$ is
$2$
$3$
$5$
$4$
Solution
$\left[\begin{array}{ccc} x & x ^2 & 1+ x ^3 \\ 2 x & 4 x ^2 & 1+8 x ^3 \\ 3 x & 9 x ^2 & 1+27 x ^3\end{array}\right]=10$
$\Rightarrow x^3 \times \operatorname{det}\left[\begin{array}{ccc}1 & 1 & 1+x^3 \\ 2 & 4 & 1+8 x^3 \\ 3 & 9 & 1+27 x^3\end{array}\right]=10$
$\Rightarrow x^3 \times\left[4+108 x^3-9-72 x^3-2-54 x^3+3+24 x^3+6+6 x^3\right]=10$
$\Rightarrow x^3\left(12 x^3+2\right)=10$
$\Rightarrow 6 x^6+x^3-5=0$
$\Rightarrow 6 x^6+6 x^3-5 x^3-5=0$
$\Rightarrow\left(x^3+1\right)\left(6 x^3-5\right)=0$
$\Rightarrow x^3=-1, \frac{5}{6}$, both of which have one real root respectively.
Hence, total number of distinct values of $x$ is $2$ .
