સુરેખ સમીકરણ સંહતિ
$2 x-y+3 z=5$
$3 x+2 y-z=7$
$4 x+5 y+\alpha z=\beta$
માટે નીચેના માથી ક્યૂ સાચું નથી?
JEE MAIN 2023, Difficult
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$\Delta=\left|\begin{array}{ccc}2 & -1 & 3 \\ 3 & 2 & -1 \\ 4 & 5 & \alpha\end{array}\right|=7(\alpha+5)$
$\Delta_1=\left|\begin{array}{ccc}5 & -1 & 3 \\ 7 & 2 & -1 \\ \beta & 5 & \alpha\end{array}\right|=17 \alpha-5 \beta+130$
$\Delta_2=\left|\begin{array}{ccc}2 & 5 & 3 \\ 3 & 7 & -1 \\ 4 & \beta & \alpha\end{array}\right|=-11 \beta+\alpha+104$
$\Delta_3=\left|\begin{array}{ccc}2 & -1 & 5 \\ 3 & 2 & 7 \\ 4 & 5 & \beta\end{array}\right|=7(\beta-9)$
For infinitely many solutions
$\Delta=\Delta_1=\Delta_2=\Delta_3=0$
For $\alpha=-5$ and $\beta=9$
Hence option $(3)$ is incorrect
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