Consider the system of linear equations

$-x+y+2z=0$

$3x-ay+5z=1$

$2x-2y-az=7$

Let $S_1$ be the set of all $a\in R$ for which the system is inconsistent and $S_2$ be the set of all $a\in R$ for which the system has infinitely many solutions. If $\mathrm{n}\left({\mathrm{S}}_1\right)$ and $\mathrm{n}\left({\mathrm{S}}_2\right)$ denote the number of elements in ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ respectively, then

• Correct answer
  A

  $\mathrm{n}\left({\mathrm{S}}_1\right)=2, \mathrm{n}\left({\mathrm{S}}_2\right)=0$
• B

  $\mathrm{n}\left({\mathrm{S}}_1\right)=2,\mathrm{n}\left({\mathrm{S}}_2\right)=2$
• C

  $\mathrm{n}\left({\mathrm{S}}_1\right)=0, \mathrm{n}\left({\mathrm{S}}_2\right)=2$
• D

  $\mathrm{n}\left({\mathrm{S}}_1\right)=1,\mathrm{n}\left({\mathrm{S}}_2\right)=0$

Hence, the correct answer is option: 'A'.