If $\begin{vmatrix}x-2 & 1 & 2\\1 & 2x & 1\\ -1 & 1 & 1\end{vmatrix} =3 $ then the values of x are :

$1, -\frac{1}{2}$

The correct answer is Option (3) → $1, -\frac{1}{2}$

$\begin{vmatrix}x-2 & 1 & 2\\1 & 2x & 1\\ -1 & 1 & 1\end{vmatrix} =3$

$C_2→C_2+C_1,C_3→C_3+C_1$

$\begin{vmatrix}x-2 & x-1 & x\\1 & 2x+1 & 2\\ -1 & 0 & 0\end{vmatrix} =3$

$⇒2x^2+x-2x+2=3⇒2x^2-x-1=0$

$⇒2x^2-2x+x-1=0$

$2x(x-1)+1(x-1)=0$

$x=-\frac{1}{2},1$