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:

If in $\Delta ABC$,$2b^2 = a^2 + c^2$,then $\frac{\sin 3B}{\sin B} = $

If the reciprocals of the lengths of the sides of a $\triangle ABC$ are in harmonic progression,then its ex-radii $r_1, r_2, r_3$ are in

In a triangle $ABC$,if $\cos A \cos B + \sin A \sin B \sin C = 1$,then $\sin A + \sin B + \sin C = $

In a $\triangle ABC$,the value of $\Sigma(b+c) \tan \frac{A}{2} \tan \left(\frac{B-C}{2}\right)$ is equal to

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):
$(A) 16$
$(B) 18$
$(C) 24$
$(D) 22$