Let S 1 and S 2 be respectively the sets of all a ∈ R - 0 for which the system of linear equations a x…

Given,

$S_1$ and $S_2$ be respectively the sets of all $a\in R-\left\{0\right\}$ for which the system of linear equations

$ax+2ay-3az=1 .........\left(1\right)$

$\left(2a+1\right) x+\left(2a+3\right) y+\left(a+1\right)z=2 ....\left(2\right)$

$\left(3a+5\right) x+\left(a+5\right) y+\left(a+2\right) z=3 .....\left(3\right)$

Now from above equations finding $\Delta =\left|\begin{array}{lll}a& 2a& -3a\\ 2a+1& 2a+3& a+1\\ 3a+5& a+5& a+2\end{array}\right|$

$=a\left(15a^2+31a+36\right)=0\Rightarrow a=0$

As $15a^2+31a+36$ cannot be zero as ${31}^2-4\times 15\times 36<0$

So, $\Delta \ne 0$ for all $a\in R-\left\{0\right\}$

Hence, $S_1=R-\left\{0\right\} \& S_2=\varphi$