Manjeet wants to donate a rectangular plot of land for a school in his village. When he has asked to give dimensions of the plot, he told that if its length is decreased by 50 m and breadth is increased by 50 m, then its area will remain same, but if its length is decreased by 10 m and breadth is decreased by 20 m, then its area will decrease by 5300 m². If length and breadth of the plot are x and y respectively, then based on above information answer the question.

The length of the plot is:

200 m  

$∴ AX = B ⇒ X = A^{-1}B$   ....(iii)

$\begin{vmatrix}1&-1\\2&1\end{vmatrix}=1-(-2)=1+2=3$

$Adj.A=\begin{bmatrix}1&-2\\1&1\end{bmatrix}=\begin{bmatrix}1&1\\-2&1\end{bmatrix}$

$A^{-1}=\frac{1}{3}\begin{bmatrix}1&1\\-2&1\end{bmatrix}$

From (iii)

$\begin{bmatrix}x\\y\end{bmatrix}=\frac{1}{3}\begin{bmatrix}1&1\\-2&1\end{bmatrix}\begin{bmatrix}50\\550\end{bmatrix}$

$\begin{bmatrix}x\\y\end{bmatrix}=\frac{1}{3}\begin{bmatrix}600\\450\end{bmatrix}=\begin{bmatrix}200\\150\end{bmatrix}$ ...(iv)

x = length = 200 m

option 3 is correct.