In a triangle $ABC$,if $a < b < c$ and $\frac{a^3+b^3+c^3}{\sin^3 A+\sin^3 B+\sin^3 C}=8$,then the maximum value of $c$ is

In $\triangle ABC$,with usual notations,$m \angle C = \frac{\pi}{2}$. If $\tan \left(\frac{A}{2}\right)$ and $\tan \left(\frac{B}{2}\right)$ are the roots of the equation $a_1 x^2 + b_1 x + c_1 = 0$ $(a_1 \neq 0)$,then:

$ABC$ is a triangular park with $AB = AC = 100 \, m$. $A$ clock tower is situated at the mid-point $D$ of $BC$. The angles of elevation of the top of the tower at $A$ and $B$ are $\cot^{-1} 3.2$ and $\csc^{-1} 2.6$ respectively. The height of the tower is .... $m$.

In a triangle $ABC$,if $(r_1-r_3)(r_1-r_2)-2r_2r_3=0$,then $a^2-b^2=$

If $7 \cos \theta - \sin \theta = 5$ and $\tan \theta > 0$,then $\tan \theta =$

If $\alpha, \beta$ and $\gamma$ are the lengths of the altitudes of a $\triangle ABC$ with area $\Delta$,then $\frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)$ is equal to