Question

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

Answer 1

$x.x^2\left|\begin{array}{ccc}1& 1& 1+x^3\\ 2& 4& 1+8x^3\\ 3& 9& 1+27x^3\end{array}\right|=10\Rightarrow x^3\left|\begin{array}{ccc}1& 1& 1\\ 2& 4& 1\\ 3& 9& 1\end{array}\right|+x^6\left|\begin{array}{ccc}1& 1& 1\\ 2& 4& 8\\ 3& 9& 27\end{array}\right|=10\Rightarrow x^3\left|\begin{array}{ccc}0& 0& 1\\ 1& 3& 1\\ 2& 8& 1\end{array}\right|+x^6\left|\begin{array}{ccc}1& 0& 0\\ 2& 2& 6\\ 3& 6& 24\end{array}\right|=10\Rightarrow x^3(8-6)+x^6(48-36)=10\Rightarrow 6x^6+x^3-5=0\Rightarrow (6x^3-5)(x^3+1)=0\Rightarrow x^3=\frac{5}{6},x^3=-1\Rightarrow x=(\frac{5}{6}{)}^{\frac{1}{3}},x=-1\Rightarrow \text{Two distinct value}$