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)

$ \Delta=\begin{vmatrix}a & 2 a & -3 a \\ 2 a+1 & 2 a+3 & a+1 \\ 3 a+5 & a+5 & a+2\end{vmatrix} $
$ =a\left(15 a^2+31 a+36\right)=0 \Rightarrow a=0 $
$ \Delta \neq 0 \text { for all } a \in R-\{0\} $
Hence $ S_1=R-\{0\} S_2=\Phi$