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

Question

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

Then \(8 \int\limits_{a}^{b}|\mathrm{~A}| \mathrm{dm}\) is equal to ______.

Answer 1

Correct answer is : 450

\(\mathrm A=\left(\begin{array}{cc}2 & -5 \\ 3 & \mathrm{~m}\end{array}\right), \mathrm{B}=\left(\begin{array}{cc}20 \\ \mathrm{m}\end{array}\right)\)

\(\mathrm{X}=\left(\begin{array}{cc}\mathrm x\\ \mathrm{y}\end{array}\right)\)

\(2 x-5 y=20\)   .......(1)

\(3 x+m y=m\)  ........(2)

\(\Rightarrow \mathrm{y}=\frac{2 \mathrm{m}-60}{2 \mathrm{m}+15}\)

\(\mathrm{y}<0 \Rightarrow \mathrm{m} \in\left(\frac{-15}{2}, 30\right)\)

\(\mathrm{x}=\frac{25 \mathrm{m}}{2 \mathrm{m}+15}\)

\(\mathrm{x}<0 \Rightarrow \mathrm{m} \in\left(\frac{-15}{2}, 0\right)\)

\(\Rightarrow \mathrm{m} \in\left(\frac{-15}{2}, 0\right)\)

\(|\mathrm{A}|=2 \mathrm{m}+15\)

Now,

\(8 \int\limits_{\frac{-15}{2}}^{0}(2 m+15) d m=8\left\{m^{2}+15 m\right\}_{\frac{-15}{2}}^{0}\)

\(\Rightarrow 8\left\{-\left(\frac{225}{4}-\frac{225}{2}\right)\right\}\)

\(=8 \times \frac{225}{4}=450\)