JEE Math Practice Question

Let $\alpha, \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations $x+2 y+z=7$ $x+\alpha z=11$ $2 x-3 y+\beta z=\gamma$ Match each entry in List-I to the correct entries in List-II. List-I (P) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma=28$, then the system has (Q) If $\beta=\frac{1}{2}(7 \alpha-3)$ and $\gamma \neq 28$, then the system has (R) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma \neq 28$, then the system has (S) If $\beta \neq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma = 28$, then the system has List-II (1) a unique solution (2) no solution (3) infinitely many solutions (4) $x=11, y=-2$ and $z=0$ as a solution (5) $x=-15, y=4$ and $z=0$ as a solution The correct option is:

1. A.$(P) \rightarrow(3) \quad(Q) \rightarrow(2) \quad(R) \rightarrow(1) \quad(S) \rightarrow(4)$
2. B.$(P) \rightarrow(3) \quad(Q) \rightarrow(2) \quad(R) \rightarrow(5) \quad(S) \rightarrow(4)$
3. C.$(P) \rightarrow(2) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(4) \quad(S) \rightarrow(5)$
4. D.$(P) \rightarrow(2) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(1) \quad$ (S) $\rightarrow$ (3)