Question

Consider the matrices: \[ A = \begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}, \quad B = \begin{bmatrix} 20 \\ m \end{bmatrix}, \quad \text{and} \quad X = \begin{bmatrix} x \\ y \end{bmatrix}. \] 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 \[ 8 \int_a^b |\det(A)| \, dm \] is equal to _____ .

Answer 1

Correct Answer: 450

Given:

\[A = \begin{pmatrix} 2 & -5 \\ 3 & m \end{pmatrix}, \quad B = \begin{pmatrix} 20 \\ m \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}\]

From the equations:

\(2x - 5y = 20\)       (1)

\(3x + my = m\)      (2)

We get:

\[y = \frac{2m - 60}{2m + 15}\]

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

Similarly:

\[x = \frac{25m}{2m + 15}\]

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

Thus, combining conditions:

\[m \in \left(-\frac{15}{2}, 0\right)\]

The determinant of matrix \(A\) is:

\[|A| = 2m + 15\]

Now:

\[8 \int_{-\frac{15}{2}}^{0} (2m + 15) \, dm = 8 \left[ m^2 + 15m \right]_{-\frac{15}{2}}^{0}\]

\[= 8 \left\{ \frac{225}{4} - \frac{225}{2} \right\}\]

\[= 8 \times \frac{225}{4} = 450\]

Final Answer: 450