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 < 0 and y < 0 ), be the interval ( a , b ) . Then 8 a b | A | ? d

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Solution

Q.
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]

Ans.
(450)

$A X = B$ has a negative solution

So, $x, y < 0$

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

$3 x + m y = m$ ...(ii)

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

$y = \frac{2 m - 60}{2 m + 15}, Since y < 0$

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

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

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

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

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

$∴ m \in (\frac{- 15}{2}, 0)$

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

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

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