Matrices & Determinants
Matrix Equation — Integral over Parameter Range
nta_pyq_2024_apr
Grade 12
Question:
Consider the matrices $A=\begin{pmatrix}2&-5\\3&m\end{pmatrix}$, $B=\begin{pmatrix}20\\m\end{pmatrix}$ and $X=\begin{pmatrix}x\\y\end{pmatrix}$. Let the set of all $m$ for which $AX=B$ 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 ________.
Step-by-Step Solution
Key Concept: Solve $AX=B$: $x=25m/(2m+15)$, $y=(2m-60)/(2m+15)$. For $x<0$ and $y<0$: need $m\in(-15/2,0)$. $|A|=2m+15$.
$m\in(-15/2,0)$. $8\int_{-15/2}^0(2m+15)dm=450$.
Correct Answer: 450