Question

Find the values of x, if

(i) $\left|\begin{array}{cc}2& 4\\ 5& 1\end{array}\right|=\left|\begin{array}{cc}2x& 4\\ 6& x\end{array}\right|$

(ii) $\left|\begin{array}{cc}2& 3\\ 4& 5\end{array}\right|=\left|\begin{array}{cc}x& 3\\ 2x& 5\end{array}\right|$

(iii) $\left|\begin{array}{cc}3& x\\ x& 1\end{array}\right|=\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$

(iv) If $\left|\begin{array}{cc}3x& 7\\ 2& 4\end{array}\right|=10$, find the value of x.

(v) $\left|\begin{array}{cc}x+1& x-1\\ x-3& x+2\end{array}\right|=\left|\begin{array}{cc}4& -1\\ 1& 3\end{array}\right|$

(vi) $\left|\begin{array}{cc}2x& 5\\ 8& x\end{array}\right|=\left|\begin{array}{cc}6& 5\\ 8& 3\end{array}\right|$

Answer 1

(i)
$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}\left|\begin{array}{cc}2& 4\\ 5& 1\end{array}\right|=\left|\begin{array}{cc}2x& 4\\ 6& x\end{array}\right| \\ \Rightarrow 2-20=2x^2-24 \\ \Rightarrow -18=2x^2-24 \\ \Rightarrow 2x^2=6 \\ \Rightarrow x^2=3 \\ \Rightarrow x=\pm \sqrt{3} \end{gathered}$$

(ii)
$$\begin{gathered} \mathrm{Given}:\left|\begin{array}{cc}2& 3\\ 4& 5\end{array}\right|=\left|\begin{array}{cc}x& 3\\ 2x& 5\end{array}\right| \\ \Rightarrow 10-12=5x-6x \\ \Rightarrow -2=-x \\ \Rightarrow x=2 \end{gathered}$$

(iii)
$$\begin{gathered} \mathrm{Given}:\left|\begin{array}{cc}3& x\\ x& 1\end{array}\right|=\left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right| \\ \Rightarrow 3-x^2=3-8 \\ \Rightarrow -x^2=-8 \\ \Rightarrow x^2=8 \\ \Rightarrow x=\pm 2\sqrt{2} \end{gathered}$$

(iv)
$$\begin{gathered} \mathrm{Given}:\left|\begin{array}{cc}3x& 7\\ 2& 4\end{array}\right|=10 \\ \Rightarrow 12x-14=10 \\ \Rightarrow 12x=24 \\ \Rightarrow x=2 \end{gathered}$$

(v)
$$\begin{gathered} \mathrm{Given}:\left|\begin{array}{cc}x+1& x-1\\ x-3& x+2\end{array}\right|=\left|\begin{array}{cc}4& -1\\ 1& 3\end{array}\right| \\ \Rightarrow \left(x+1\right)\left(x+2\right)-\left(x-3\right)\left(x-1\right)=12+1 \\ \Rightarrow x^2+3x+2-x^2+4x-3=13 \\ \Rightarrow 7x-1=13 \\ \Rightarrow 7x=14 \\ \Rightarrow x=2 \end{gathered}$$

(vi)
$$\begin{gathered} \mathrm{Given}:\left|\begin{array}{cc}2x& 5\\ 8& x\end{array}\right|=\left|\begin{array}{cc}6& 5\\ 8& 3\end{array}\right| \\ \Rightarrow 2x^2-40=18-40 \\ \Rightarrow 2x^2=18 \\ \Rightarrow x^2=9 \\ \Rightarrow x=\pm 3 \end{gathered}$$