Let S$_1$ and S$_2$ be respectively the sets of all $a \in \mathbb{R} - \{ 0\} $ for which the system of linear equations $ax + 2ay - 3az = 1$ $(2a + 1)x + (2a + 3)y + (a + 1)z = 2$ $(3a + 5)x + (a + 5)y + (a + 2)z = 3$ has unique solution and infinitely many solutions. Then

Question

Let S$_1$ and S$_2$ be respectively the sets of all $a \in \mathbb{R} - \{ 0\} $ for which the system of linear equations

$ax + 2ay - 3az = 1$

$(2a + 1)x + (2a + 3)y + (a + 1)z = 2$

$(3a + 5)x + (a + 5)y + (a + 2)z = 3$

has unique solution and infinitely many solutions. Then

$\mathrm{n({S_1}) = 2}$ and S$_2$ is an infinite set

$\mathrm{{S_1} = \Phi} $ and $\mathrm{{S_2} = \mathbb{R} - \{ 0\}}$

$\mathrm{{S_1} = \mathbb{R} - \{ 0\}}$ and $\mathrm{{S_2} = \Phi} $

S$_1$ is an infinite set and n(S$_2$) = 2

Please login to view the detailed solution steps...