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) $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{cc}x & 3 \\ 2 x & 5\end{array}\right|$
(iii) $\left|\begin{array}{ll}3 & x \\ x & 1\end{array}\right|=\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|$
(iv) If $\left|\begin{array}{cc}3 x & 7 \\ 2 & 4\end{array}\right|=10$, find the value of $x$
(v) $\left|\begin{array}{ll}x+1 & x-1 \\ x-3 & x+2\end{array}\right|=\left|\begin{array}{cc}4 & -1 \\ 1 & 3\end{array}\right|$
(vi) $\left|\begin{array}{cc}2 x & 5 \\ 8 & x\end{array}\right|=\left|\begin{array}{ll}6 & 5 \\ 8 & 3\end{array}\right|$
(i)
Given : $\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$
$\Rightarrow 2-20=2 x^{2}-24$
$\Rightarrow-18=2 x^{2}-24$
$\Rightarrow 2 x^{2}=6$
$\Rightarrow x^{2}=3$
$\Rightarrow x=\pm \sqrt{3}$
(ii)
Given: $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{cc}x & 3 \\ 2 x & 5\end{array}\right|$
$\Rightarrow 10-12=5 x-6 x$
$\Rightarrow-2=-x$
$\Rightarrow \quad x=2$
(iii)
Given: $\left|\begin{array}{ll}3 & x \\ x & 1\end{array}\right|=\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|$
$\Rightarrow 3-x^{2}=3-8$
$\Rightarrow-x^{2}=-8$
$\Rightarrow x^{2}=8$
$\Rightarrow x=\pm 2 \sqrt{2}$
(iv)
Given : $\left|\begin{array}{cc}3 x & 7 \\ 2 & 4\end{array}\right|=10$
$\Rightarrow 12 x-14=10$
$\Rightarrow 12 x=24$
$\Rightarrow x=2$
(v)
Given : $\left|\begin{array}{ll}x+1 & x-1 \\ x-3 & x+2\end{array}\right|=\left|\begin{array}{cc}4 & -1 \\ 1 & 3\end{array}\right|$
$\Rightarrow(x+1)(x+2)-(x-3)(x-1)=12+1$
$\Rightarrow x^{2}+3 x+2-x^{2}+4 x-3=13$
$\Rightarrow 7 x-1=13$
$\Rightarrow 7 x=14$
$\Rightarrow x=2$
(vi)
Given : $\left|\begin{array}{cc}2 x & 5 \\ 8 & x\end{array}\right|=\left|\begin{array}{cc}6 & 5 \\ 8 & 3\end{array}\right|$
$\Rightarrow 2 x^{2}-40=18-40$
$\Rightarrow 2 x^{2}=18$
$\Rightarrow x^{2}=9$
$\Rightarrow x=\pm 3$
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