Statement (A): The line $2x-3y+1=0$  does not pass through$Q_4$.

Statement (B): The line $y=mx+c,$ where $m>0$ and $c>0$ does not pass through IV quadrant.

Statement-I:

Given line $2x-3y+1=0$

$2x-3y=-1$

$\frac{x}{\frac{-1}{2}}+\frac{y}{\frac{1}{3}}=1$

$x-\text{intercept}=\left(\frac{-1}{2},0\right)$

$y-\text{intercept}=\left(0,\frac{1}{3}\right)$

The line passing through these points in $Q_1,Q_2,Q_3$ but not $Q_4$

$\therefore$ Statement-I  is correct

Statement-II:

Given line $y=mx+c$ where $m>0,c>0$ then the line does not passing through the $Q_4{ }^{th}$ quadrant