Find the values of $x$,if: $(i)$ $\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$ (ii) $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{ll}x & 3 \\ 2 x & 5\end{array}\right|$

(A) $(i)$ Given $\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$
Expanding the determinants:
$(2 \times 1) - (5 \times 4) = (2x \times x) - (6 \times 4)$
$2 - 20 = 2x^2 - 24$
$-18 = 2x^2 - 24$
$2x^2 = 6$
$x^2 = 3$
$x = \pm \sqrt{3}$
(ii) Given $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{ll}x & 3 \\ 2 x & 5\end{array}\right|$
Expanding the determinants:
$(2 \times 5) - (3 \times 4) = (x \times 5) - (3 \times 2x)$
$10 - 12 = 5x - 6x$
$-2 = -x$
$x = 2$