JEE Main 2021 (Online) 27th July Morning ShiftQuestion 1
For real numbers $\alpha$ and $\beta$, consider the following system of linear equations : $x+y-z=2$, $x+2y+\alpha z=1$, $2x-y+z=\beta$. If the system has infinite solutions, then $\alpha+\beta$ is equal to ________.
Determinant
Answer$5$
Solution
Infinite solutions need $\Delta=0$: $\begin{vmatrix}1&1&-1\\1&2&\alpha\\2&-1&1\end{vmatrix}=3\alpha+6=0\Rightarrow\alpha=-2$. Also $\begin{vmatrix}1&1&2\\1&2&1\\2&-1&\beta\end{vmatrix}=0\Rightarrow 3\beta-21=0\Rightarrow\beta=7$. $\alpha+\beta=5$.