1. $m\sin x-(2m+1)\cos x=m-1$ có nghiệm:
$\Leftrightarrow m^2+(2m-1)^2\ge (m-1)^2$
$\Leftrightarrow m^2+4m^2-4m+1\ge m^2-2m+1$
$\Leftrightarrow 4m^2-2m\ge 0$
$\Leftrightarrow m\in (-\infty;0]\cup \Big[\dfrac{1}{2};+\infty)$
2. $3\sin x-m\cos x=2m-1$ có nghiệm:
$\Leftrightarrow 3^2+m^2\ge (2m-1)^2$
$\Leftrightarrow m^2+9\ge 4m^2-4m+1$
$\Leftrightarrow 3m^2-4m-8\le 0$
$\Leftrightarrow m\in \Big[\dfrac{2-2\sqrt7}{3}; \dfrac{2+2\sqrt7}{3}\Big]$