If $\cos \alpha =\frac{-12}{13},\cot \beta =\frac{24}{7},90°<\alpha <180°$ and $180°<\beta <270°$ then the quadrant in which $\alpha +\beta$ lies.

$\Rightarrow 90°<\alpha <180°$                    $\Rightarrow 180<\beta <270$

$\therefore$ $270<\alpha +\beta <450$

$\Rightarrow \alpha +\beta \in Q_4\left(or\right)Q_1$

$\Rightarrow \sin \left(\alpha +\beta \right)<0$

$\Rightarrow \alpha +\beta \in Q_4$