JEE MAIN - Mathematics (2023 - 29th January Morning Shift - No. 17)
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.
It has a solution for all $$\alpha\ne-1$$ and $$\beta=2$$
It has no solution if $$\alpha=-1$$ and $$\beta\ne2$$
It has no solution for $$\alpha=-1$$ and for all $$\beta \in \mathbb{R}$$
It has no solution for $$\alpha=3$$ and for all $$\beta\ne2$$
Explanation
$D=\left|\begin{array}{ccc}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & 2\end{array}\right|=0 \Rightarrow \alpha=-1,3$
$D_{x}=\left|\begin{array}{ccc}2 & 1 & 1 \\ 3 & 1 & 1 \\ \alpha & 2 & \beta\end{array}\right|=0 \Rightarrow \beta=2$
$D_{y}=\left|\begin{array}{ccc}\alpha & 1 & 1 \\ 2 \alpha & 1 & 1 \\ 3 & 2 & \beta\end{array}\right|=0$
$D_{z}=\left|\begin{array}{ccc}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & \beta\end{array}\right|=0$
$\beta=2, \alpha=-1$
$\alpha=-1, \beta=2$ Infinite solution
$D_{x}=\left|\begin{array}{ccc}2 & 1 & 1 \\ 3 & 1 & 1 \\ \alpha & 2 & \beta\end{array}\right|=0 \Rightarrow \beta=2$
$D_{y}=\left|\begin{array}{ccc}\alpha & 1 & 1 \\ 2 \alpha & 1 & 1 \\ 3 & 2 & \beta\end{array}\right|=0$
$D_{z}=\left|\begin{array}{ccc}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & \beta\end{array}\right|=0$
$\beta=2, \alpha=-1$
$\alpha=-1, \beta=2$ Infinite solution
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