Question

Let $S_1$ and $S_2$ be respectively the sets of all $a \in R -\\{0\\}$ for which the system of linear equations
$a x+2 a y-3 a z=1$
$(2 a+1) x+(2 a+3) y+(a+1) z=2$
$(3 a+5) x+(a+5) y+(a+2) z=3$
has unique solution and infinitely many solutions. Then

• A. $S_1=\Phi$ and $S_2=R-\\{0\\}$
• B. $S_1$ is an infinite set and $n\left(S_2\right)=2$
• C. $S_1=R-\\{0\\}$ and $S_2=\Phi$
• D. $n \left( S _1\right)=2$ and $S _2$ is an infinite set

Answer 1

Answer: (C)

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

Answer 2

The correct answer is (C) : $S_1=R-\\{0\\}$ and $S_2=\Phi$
Δ=∣∣​a2a+13a+5​2a2a+3a+5​−3aa+1a+2​∣∣​
=a(15a2+31a+36)=0⇒a=0
Δ=0 for all a∈R−{0}
Hence S1​=R−{0}S2​=Φ