In $\Delta ABC$,$a \cot A + b \cot B + c \cot C = . . . $ (where $r$ is inradius and $R$ is circumradius.)

If the median of a triangle $ABC$ through $A$ is perpendicular to $AB$,then the value of $\frac{\tan A}{\tan B}$ is equal to

In a $\triangle ABC$,$(b^2-c^2) \cot A + (c^2-a^2) \cot B =$

In a triangle $ABC$,angle $A$ is greater than angle $B$. If the measures of angles $A$ and $B$ satisfy the equation $3\sin x - 4\sin^3 x - k = 0$ for $0 < k < 1$,then the measure of angle $C$ is:

In a triangle $PQR$,$P$ is the largest angle and $\cos P = \frac{1}{3}$. Further,the incircle of the triangle touches the sides $PQ, QR$ and $RP$ at $N, L$ and $M$ respectively,such that the lengths of $PN, QL$ and $RM$ are consecutive even integers. Then the possible length$(s)$ of the side$(s)$ of the triangle is (are):
$(A) 16$
$(B) 18$
$(C) 24$
$(D) 22$

In a triangle $PQR$ with usual notations,$\angle R = \frac{\pi}{2}$. If $\tan \frac{P}{2}$ and $\tan \frac{Q}{2}$ are the roots of the equation $ax^2 + bx + c = 0$ $(a \neq 0)$,then: