STD 12 - 3 and 4 . determinant and metrices — JEE STD 11 Science — Question

$\Rightarrow x^3 \times \operatorname{det}\left[\begin{array}{ccc}1 & 1 & 1+x^3 \\ 2 & 4 & 1+8 x^3 \\ 3 & 9 & 1+27 x^3\end{array}\right]=10$

$\Rightarrow x^3 \times\left[4+108 x^3-9-72 x^3-2-54 x^3+3+24 x^3+6+6 x^3\right]=10$

$\Rightarrow x^3\left(12 x^3+2\right)=10$

$\Rightarrow 6 x^6+x^3-5=0$

$\Rightarrow 6 x^6+6 x^3-5 x^3-5=0$

$\Rightarrow\left(x^3+1\right)\left(6 x^3-5\right)=0$

$\Rightarrow x^3=-1, \frac{5}{6}$, both of which have one real root respectively.

Hence, total number of distinct values of $x$ is $2$ .