JEE Main 2024 (Online) 9th April Evening ShiftQuestion 1
Consider the matrices: $A=\begin{bmatrix}2&-5\\3&m\end{bmatrix}$, $B=\begin{bmatrix}20\\m\end{bmatrix}$ and $X=\begin{bmatrix}x\\y\end{bmatrix}$. 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 $8\int_{a}^{b}|A|\,dm$ is equal to $\_\_\_\_\_\_$ .
Determinant
Answer$450$
Solution
$|A|=2m+15$. Solving: $x=\dfrac{25m}{2m+15}$, $y=\dfrac{2(m-30)}{2m+15}$. If $2m+15>0$: need $m<0$ and $m<30$, giving $-\tfrac{15}2<m<0$. If $2m+15<0$: $x<0$ needs $m>0$, impossible. $(a,b)=(-\tfrac{15}2,0)$, $|A|=2m+15>0$ there. $8\int_{-15/2}^0(2m+15)\,dm=8[m^2+15m]_{-15/2}^0=8\cdot\tfrac{225}{4}=450$.