JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Matrices

Factoring $a$ from the first row, the determinant of the coefficient matrix reduces to $a(15a^2+31a+37)$. The quadratic $15a^2+31a+37$ has discriminant $31^2-4\cdot 15\cdot 37=-1259<0$, so it never vanishes for real $a$. Hence the determinant is zero only at $a=0$, and for every $a\in \mathbb{R}-\{0\}$ the system has a unique solution. Therefore $S_1=\mathbb{R}-\{0\}$ and $S_2=\Phi$.