In $\triangle ABC$,if $r=3$ and $R=5$,then $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} = $

Let $ABC$ be a triangle such that $\angle ACB = \frac{\pi}{6}$ and let $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively. The value$(s)$ of $x$ for which $a = x^2+x+1, b = x^2-1$ and $c = 2x+1$ is (are)

If $b+c=3a$,then $\cot \frac{B}{2} \cot \frac{C}{2}$ is equal to :

In $\triangle ABC$,if the line joining the circumcentre $(O)$ and the incentre $(I)$ is parallel to $BC$,then $\cos B + \cos C = $

In a triangle $ABC$,let $AB = \sqrt{23}$,$BC = 3$,and $CA = 4$. Then the value of $\frac{\cot A + \cot C}{\cot B}$ is:

If in a $\triangle ABC$, $r_1=2$, $r_2=3$ and $r_3=6$, then $a$ equals to