Consider the matrices : A = [ cc 2 & -5 3 & m ], B = [ l 20 m ] a… - Vidyadip

Question

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

✓

Answer

$A=(2 -5 \ 3 m ), B=\binom{20}{m}$
$X=\binom{x}{y}$
$2 x-5 y=20$
$3 x+my=m$
$\Rightarrow y=\frac{2 m-60}{2 m+15}$
$y<0 $
$\Rightarrow m \in\left(\frac{-15}{2}, 30\right)$
$x=\frac{25 m}{2 m+15}$
$x<0 $
$\Rightarrow m \in\left(\frac{-15}{2}, 0\right)$
$\Rightarrow m \in\left(\frac{-15}{2}, 0\right)$
$|A|=2 m+15$
Now,
$8 \int_{\frac{-15}{2}}^0(2 m+15) dm$
$=8\left\{m^2+15 m\right\}_{\frac{-15}{2}}^0$
$\Rightarrow 8\left\{-\left(\frac{225}{4}-\frac{225}{2}\right)\right\}$
$=8 \times \frac{225}{4}=450$

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