Consider the matrices : $A=\left[\begin{array}{ll}2 & -5 \ 3 & m\end{array} ight], B=\left[\begin{array}{l}20 \ m\end{array} ight]$ and $X=\left[\begin{array}{l}x \ y\end{array} ight]$. Let the set of all $m$, for which the system of equations $A X=B$ has a negative solution (i.e., $x

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Consider the matrices : $A=\left[\begin{array}{ll}2 & -5 \ 3 & m\end{array}ight], B=\left[\begin{array}{l}20 \ m\end{array}ight]$ and $X=\left[\begin{array}{l}x \ y\end{array}ight]$. Let the set of all $m$, for which the system of equations $A X=B$ has a negative solution (i.e., $x < 0$ and $y < 0$ ), be the interval $(a, b)$. Then $8 \int_a^b|A| d m$ is equal to_________

Aicharya · Accepted Answer

$\begin{aligned} & A=\left(\begin{array}{cc}2 & -5 \ 3 & m\end{array}ight), B=\left(\begin{array}{c}20 \ m\end{array}ight) \ & X=\left(\begin{array}{l}x \ y\end{array}ight) \ & 2 x-5 y=20 \ldots(1)\ & 3 x+m y=m \ldots(2)\ & \Rightarrow y=\frac{2 m-60}{2 m+15} \ & y Now, $\begin{aligned} & 8 \int_{\frac{-15}{2}}^0(2 \mathrm{~m}+15) \mathrm{dm}=8\left\{\mathrm{~m}^2+15 \mathrm{~m}ight\}_{\frac{-15}{2}}^0 \ & \Rightarrow 8\left\{-\left(\frac{225}{4}-\frac{225}{2}ight)ight\} \ & =8 	imes \frac{225}{4}=450 \end{aligned}$

Consider the matrices : $A=\left[\begin{array}{ll}2 & -5 \\ 3 & m\end{array}\right],…

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