QJEEM22Ma994f: JEE Main 2022 Mathematics - Class 12 JEEM22Ma994f Solution: Let the system of linear equations , , be inconsistent. Then is …

$x + 2y + z = 2$

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

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

$$\mathrm{\Delta }=\left|\begin{array}{ccc}1& 2& 1\\ \alpha & 3& -1\\ -\alpha & 1& 2\end{array}\right|=1(6+1)-2(2\alpha -\alpha )+1(\alpha +3\alpha )$$

$= 7 + 2\alpha$

$\Delta = 0 \Rightarrow \alpha = - {7 \over 2}$

$${\mathrm{\Delta }}_1=\left|\begin{array}{ccc}2& 2& 1\\ \alpha & 3& -1\\ -\alpha & 1& 2\end{array}\right|=14+2\alpha \ne 0$$ for $\alpha = - {7 \over 2}$

$\therefore$ For no solution $\alpha = - {7 \over 2}$