Question

The total number of distinct $x\in R$ for which $\left[\begin{array}{c}x\\ 2x\\ 3x\end{array}\begin{array}{c}{x}^2\\ 4x^2\\ 9x^2\end{array}\begin{array}{c}1+x^3\\ 1+8x^3\\ 1+27x^3\end{array}\right]=10$ is

Answer 1

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

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

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

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

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

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

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

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

respectively.

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