JEE Math Practice Question

Consider the following system of equations $$\alpha x+2y+z=1$$ $$2\alpha x+3y+z=1$$ $$3x+\alpha y+2z=\beta$$ for some $$\alpha,\beta\in \mathbb{R}$$. Then which of the following is NOT correct.

1. A.It has a solution for all $$\alpha\ne-1$$ and $$\beta=2$$
2. B.It has no solution if $$\alpha=-1$$ and $$\beta\ne2$$
3. C.It has no solution for $$\alpha=-1$$ and for all $$\beta \in \mathbb{R}$$
4. D.It has no solution for $$\alpha=3$$ and for all $$\beta\ne2$$