Consider the matrices : A = [ begin matrix 2 & -5 \\3 & \!\! \!\! m \\end matrix ], B = [ begin array 35 l 20 \\ \!\! \!\! m \\end array ] and X= [ begin array 35 l x \\ y \\end array ] . 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 A dm is equal to .

Mathematics · Definite Integration

JEE Main 2024 — 9 April, Shift 2 — Question 30

Consider the matrices : $\text{A}=\left[ \begin{matrix}2 & -5 \\3 & \text{ }\!\!~\!\!\text{ m} \\\end{matrix} \right],\text{B}=\left[ \begin{array}{*{35}{l}}20 \\ \text{ }\!\!~\!\!\text{ m} \\\end{array} \right]$

and $X=\left[ \begin{array}{*{35}{l}}x \\ y \\\end{array}\right]$ . Let the set of all $m$ , for which the system of equations $\mathrm{AX}=\mathrm{B}$ has a negative solution

(i.e., $\mathrm{x}<0$ and $\mathrm{y}<0$ ), be the interval (a, b). Then $8 \int_{a}^{b}|\mathrm{~A}| \mathrm{dm}$ is equal to $\qquad$ .

Answer: 450

Numerical answer — enter this value.

Step-by-step solution

$\text{A}=\left[ \begin{matrix}2 & -5 \\3 & \text{ }\!\!~\!\!\text{ m} \\\end{matrix} \right],\text{B}=\left[ \begin{array}{*{35}{l}}20 \\ \text{ }\!\!~\!\!\text{ m} \\\end{array} \right]$ and $X=\left[ \begin{array}{*{35}{l}}x \\ y \\\end{array}\right]$ $2 x-5 y=20$

$3 x+m y=m$

$\Rightarrow \mathrm{y}=\frac{2 \mathrm{~m}-60}{2 \mathrm{~m}+15}$

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

$x=\frac{25 m}{2 m+15}$

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

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

$|\mathrm{A}|=2 \mathrm{~m}+15$ Now, $8 \int_{\frac{-15}{2}}^{0}(2 m+15) \mathrm{dm}=8\left\{\mathrm{~m}^{2}+15 m\right\}_{\frac{-15}{2}}^{0}$

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

$=8 \times \frac{225}{4}=450$

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Exam: JEE Main 2024
Paper: 9 April, Shift 2
Subject: Mathematics
Chapter: Definite Integration
Topic: Evaluation of Definite Integrals

Answer: 450

Step-by-step solution

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