Question

Let $\alpha, \beta$ and $\gamma$ be real numbers. consider the following system of linear equations
$$\begin{aligned}
& x+2 y+z=7 \
& x+\alpha z=11 \
& 2 x-3 y+\beta z=\gamma
\end{aligned}$$
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 $\gamma 
eq 28$, then the system has
(R) If $\beta 
eq \frac{1}{2}(7 \alpha-3)$ where $\alpha=1$ and $\gamma 
eq 28$, then the system has
(S) If $\beta 
eq \frac{1}{2}(7 \alpha-3)$ 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 :

Accepted Answer

(A) $(\mathrm{P}) \rightarrow(3)(\mathrm{Q}) \rightarrow(2)(\mathrm{R}) \rightarrow(1)(\mathrm{S}) \rightarrow(4)$

Answer

(B) $(\mathrm{P}) \rightarrow(3)(\mathrm{Q}) \rightarrow(2)(\mathrm{R}) \rightarrow(5)(\mathrm{S}) \rightarrow(4)$

Answer

(C) $(\mathrm{P}) \rightarrow(2)(\mathrm{Q}) \rightarrow(1)(\mathrm{R}) \rightarrow(4)(\mathrm{S}) \rightarrow(5)$

Answer

(D) $(\mathrm{P}) \rightarrow(2)(\mathrm{Q}) \rightarrow(1)(\mathrm{R}) \rightarrow(1)(\mathrm{S}) \rightarrow(3)$

JEE Advanced · 2023 · 04 Jun 2023 · Paper 1 · Matrices

Solution

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