JEE Main · 2023 · 13 Apr 2023 · Shift 1 · Matrices

The coefficient determinant is $\Delta =\left|\begin{array}{ccc}2& 4& 2a\\ 1& 2& 3\\ 2& -5& 2\end{array}\right|=18(3-a)$, so the system has a unique solution whenever $a\ne 3$, which confirms the unique-solution claims at $a=b=6$ and at $a=b=8$. When $a=3$, the first equation becomes $x+2y+3z=\frac{b}{2}$, consistent with the second equation $x+2y+3z=4$ only if $b=8$, in which case the system reduces to two independent equations and has infinitely many solutions. For $a=3,b=6$ the planes $x+2y+3z=3$ and $x+2y+3z=4$ are parallel and distinct, so the system is inconsistent and has no solution rather than infinitely many. The claim of infinitely many solutions for $a=3,b=6$ is the statement that is not correct.