If two adjacent sides of a cyclic quadrilateral are $2$ and $5$ and the angle between them is $60^{\circ}$. If the third side is $3$,then the remaining fourth side is :-

Let $\Delta$ denote the area of a $\triangle ABC$. If $\alpha, \beta, \gamma$ are the lengths of the altitudes of the $\triangle ABC$,then $\alpha^{-2}+\beta^{-2}+\gamma^{-2}=$

Let $f, g, h$ be the lengths of the perpendiculars from the circumcentre of the $\Delta ABC$ on the sides $a, b$ and $c$ respectively. If $\frac{a}{f} + \frac{b}{g} + \frac{c}{h} = \lambda \frac{abc}{fgh}$,then the value of $\lambda$ is:

In $\triangle ABC, (r_2 + r_3) \cot \left(\frac{B+C}{2}\right) = $

With usual notations,the perimeter of a triangle $ABC$ is $6$ times the arithmetic mean of the sines of its angles. If $a=1$,then $\angle A=$

With usual notations,the perimeter of a triangle $ABC$ is $6$ times the arithmetic mean of the sines of its angles. If $a=1$,then the measure of angle $A$ is: