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

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$S_{1} = Φ and S_{2} = R - {0}$

S₁ is an infinite set and $n (S_{2}) = 2$

$S_{1} = R - {0} and S_{2} = Φ$

$n (S_{1}) = 2$ and S₂ is an infinite set

answer is C.

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Detailed Solution

$Δ = |\begin{matrix} a & 2 a & - 3 a \\ 2 a + 1 & 2 a + 3 & a + 1 \\ 3 a + 5 & a + 5 & a + 2 \end{matrix}| = 0 \Rightarrow a = 0$

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