JEE Main 2026 — 21 January, Evening Shift — Question 18

for no solution $\Delta=0$

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

$\Rightarrow 3(\mathrm{a}+2)+1(-1-2)+4(4-\alpha)=0$

$\Rightarrow 19-\alpha=0 \Rightarrow \alpha=19$ & for $\alpha=19$

$\Delta_{\mathrm{x}}=\left|\begin{array}{ccc}3 & 1 & 4 \\-3 & 19 & -1 \\4 & 2 & 1\end{array}\right|=3(21)+1(-1)+4(-82)$

$\neq 0$ ∴ no solution for $\alpha=19$