Consider the matrices : A = [ 2 - 5 3 m ] , B = [ 20 m ] and X = [ x y ] . Let the set of all m , for which the system of equations A X = B has a negative solution (i.e., x

Question

Consider the matrices : $A = [\begin{matrix} 2 & - 5 \\ 3 & m \end{matrix}], B = [\begin{matrix} 20 \\ m \end{matrix}]$ and $X = [\begin{matrix} x \\ y \end{matrix}] .$ 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 ______ . [2024]

Smart Achievers · Accepted Answer

(450)

$A X = B$ has a negative solution

So, $x, y < 0$

Now, $2 x - 5 y = 20$ ...(i)

$X = [\begin{matrix} x \\ y \end{matrix}] .$ ...(ii)

On solving (i) and (ii), we get

$m,$

$\Rightarrow \frac{2 m - 60}{2 m + 15} < 0 \Rightarrow 2 m - 60 < 0 and 2 m + 15 > 0$

$\Rightarrow m < 30 and m > \frac{- 15}{2}$

$\Rightarrow m \in (\frac{- 15}{2}, 30)$

$Also, x = \frac{25 m}{2 m + 15} < 0$

$\Rightarrow m \in (\frac{- 15}{2}, 0)$

$∴ m \in (\frac{- 15}{2}, 0)$

$Now, 8 \int_{a}^{b} | A | d m = 8 \int_{- 15 / 2}^{0} (2 m + 15) d m$

$= 8 {[m^{2} + 15 m]}_{- 15 / 2}^{0} = 450$

Solution

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Solution

Q. Consider the matrices : A=[2-53m], B=[20m] and X=[xy]. Let the set of all m, for which the system of equations AX=B has a negative solution (i.e., x<0 and y<0), be the interval (a,b). Then 8∫ab|A| dm is equal to ______ . [2024]

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