JEE Main 2022 · 24 Jun · Shift 1, Mathematics question 6

Determinant.
$$\begin{vmatrix}1&1&1\\\alpha&2\alpha&3\\1&3\alpha&5\end{vmatrix}=(10\alpha-9\alpha)-(5\alpha-3)+(3\alpha^2-2\alpha)=3(\alpha-1)^2.$$
Only $\alpha=1$ can make the system inconsistent.

Check $\alpha=1$. The equations are $x+y+z=1$, $x+2y+3z=-1$, $x+3y+5z=4$. Subtracting in pairs gives $y+2z=-2$ and $y+2z=5$, a contradiction.

So exactly one value.

(B)