JEE Main 2023 (Online) 6th April Morning ShiftQuestion 1
If the system of equations
$x+y+az=b$
$2x+5y+2z=6$
$x+2y+3z=3$
has infinitely many solutions, then $2a+3b$ is equal to
Determinant
Solution
$\det\begin{pmatrix}1&1&a\\2&5&2\\1&2&3\end{pmatrix}=7-a=0\Rightarrow a=7$. For consistency, replace the $z$-column by the constants: $\begin{vmatrix}1&1&b\\2&5&6\\1&2&3\end{vmatrix}=0\Rightarrow b=3$. $2a+3b=14+9=23$. Option (B).