In $\triangle ABC$,if $B=90^{\circ}$,then $2(r+R)=$

With usual notations in $\triangle ABC$,if $\angle B = \frac{\pi}{2}$,and $\tan \frac{A}{2}, \tan \frac{C}{2}$ are roots of the equation $px^2 + qx + r = 0$,$p \neq 0$,then:

The angles of depression of the top and the foot of a chimney as seen from the top of a second chimney,which is $150 \, m$ high and standing on the same level as the first,are $\theta$ and $\phi$ respectively. If $\tan \theta = \frac{4}{3}$ and $\tan \phi = \frac{5}{2}$,then the distance between their tops is.......$m$.

If $x, y$ and $z$ are the distances of the incentre from the vertices $A, B$ and $C$ of the triangle $ABC$ respectively,then $\frac{abc}{xyz}$ is equal to

In $\triangle ABC$,if $\angle C = \frac{\pi}{3}$,then $\frac{3}{a+b+c} - \frac{1}{a+c}$ equals

For $x \in (0, \pi)$,the equation $\sin x + 2 \sin 2x - \sin 3x = 3$ has