JEE Main 2024 — 6 April, Shift 1 — Question 21

$\alpha \beta \gamma=45, \alpha, \beta, \gamma \in R$

$x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)$

$x, y, z \in R, x y z \neq 0$,

$\alpha x+y+2 z=0$

$x+\beta y+3 z=0$

$2 x+2 y+\gamma z=0$

$\mathrm{xyz} \neq 0 \Rightarrow$ non-trivial

\left| \begin{array}{*{35}{l}}\alpha & 1 & 2 \\1 & \beta & 3 \\2 & 2 & \gamma \\\end{array} \right|=0

$\Rightarrow \alpha(\beta \gamma-6)-1(\gamma-6)+2(2-2 \beta)=0$

$\Rightarrow \alpha \beta \gamma-6 \alpha-\gamma+6+4-4 \beta=0$

$\Rightarrow 6 \alpha+4 \beta+\gamma=55$