In a triangle $ABC$,$\angle B < \angle C$ and the values of $B$ and $C$ satisfy the equation $2 \tan x - k (1 + \tan^2 x) = 0$,where $0 < k < 1$. Then the measure of angle $A$ is:

In any triangle $ABC$,the value of $a(b^2 + c^2)\cos A + b(c^2 + a^2)\cos B + c(a^2 + b^2)\cos C$ is

If $a \cos 2 \theta + b \sin 2 \theta = c$ has $\alpha$ and $\beta$ as its roots,then the value of $\tan \alpha + \tan \beta$ is

In $\triangle PQR$,$\angle R = \frac{\pi}{4}$. If $\tan \left(\frac{P}{3}\right)$ and $\tan \left(\frac{Q}{3}\right)$ are the roots of the equation $ax^2 + bx + c = 0$,then:

In a triangle $ABC$,if $A = \frac{\pi}{4}$ and $\tan B \tan C = K$,then $K$ must satisfy:

If the incircle of the $\Delta ABC$ touches its sides at $L, M$ and $N$ respectively,and if $x, y, z$ are the circumradii of the triangles $\Delta MIN, \Delta NIL$ and $\Delta LIM$ respectively,where $I$ is the incentre,then the product $xyz$ is equal to: