JEE Main 2022 (Online) 28th June Evening ShiftQuestion 1
If the system of linear equations
$2x-3y=\gamma+5$,
$\alpha x+5y=\beta+1$, where $\alpha,\beta,\gamma\in\mathbf{R}$ has infinitely many solutions, then the value of $|9\alpha+3\beta+5\gamma|$ is equal to $\_\_\_\_\_\_$ .
Determinant
Answer$58$
Solution
Infinitely many solutions need $\frac{\alpha}{2}=\frac{5}{-3}=\frac{\beta+1}{\gamma+5}$. $\alpha=-\tfrac{10}3$, and $3(\beta+1)=-5(\gamma+5)\Rightarrow3\beta+5\gamma=-28$. $9\alpha+3\beta+5\gamma=-30-28=-58$, so absolute value $58$.