JEE Main 2025 — 2 April, Morning Shift — Question 31

$3 x+y+\beta z=3$

$2 x+\alpha y-z=-3$

$x+2 y+z=4$

has infinite solution

$\Rightarrow \Delta=0, \Delta_{1}=\Delta_{2}=\Delta_{3}$

$\Delta=0 \Rightarrow\left|\begin{array}{ccc}3 & 1 & \beta\\ 2 & \alpha & -1\\ 1 & 2 & 1\end{array}\right|=0$

$\Delta_{2}=0 \Rightarrow\left|\begin{array}{ccc}3 & 3 & \beta\\ 2 & -3 & -1\\ 1 & 4 & 1\end{array}\right|=0$

$\Rightarrow 3(-3+4)-3(2+1)+\beta(8+3)=0$

$\Rightarrow 3-9+11 \beta=0$

$\Rightarrow \quad \beta=\frac{6}{11}$ $\Delta_{3}=0 \Rightarrow\left|\begin{array}{ccc}3 & 1 & 3\\ 2 & \alpha & -3\\ 1 & 2 & 4\end{array}\right|=0$

$\Rightarrow 3(4 \alpha+6)-1(8+3)+3(4-\alpha)=0$

$12 \alpha+18-11+12-3 \alpha=0$

$9 \alpha=-19$

$\alpha=\frac{-19}{9}$

$\therefore \quad 22 \beta-9 \alpha=31$