The differential equation of all circles having their centres on the line $y=5$ and touching ( X -axis) is $\qquad$

$$ \int_0^{\frac{\pi}{4}} \frac{\cos ^2 x \sin ^2 x}{\cos ^3 x+\sin ^3 x} d x= $$

If $\{(\mathrm{p} \wedge \sim \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r})\} \rightarrow \sim \mathrm{p} \vee \mathrm{q}$ has truth value false then truth values of the statements $p, q, r$ are respectively

In a triangle ABC , with usual notations, $\tan \left(\frac{\mathrm{A}}{2}\right)=\frac{5}{6}, \tan \left(\frac{\mathrm{C}}{2}\right)=\frac{2}{5}$, then