In a $\Delta ABC$,${a^2}\sin 2C + {c^2}\sin 2A = $

The number of roots of the equation $(81)^{\sin ^{2} x} + (81)^{\cos ^{2} x} = 30$ in the interval $[0, \pi]$ is equal to

Consider the following statements.
$I$. In $\triangle ABC$,if $c=6$ and $\cos C=-\frac{11}{25}$,then $R=\frac{25}{2\sqrt{14}}$.
$II$. In $\triangle ABC$,if $a=3, b=4, c=6$,then $\triangle ABC$ is an acute-angled triangle.
Which of the above statements is/are true?

In $\triangle ABC$,if $a, b$ and $c$ are in arithmetic progression,then $\cos A + 2 \cos B + \cos C =$

In $\triangle ABC$, if $R = \frac{65}{8}$, $rr_1 = 42$ and $r_1 - r = 6.5$, then $s(s-a) = $

Let in a right-angled triangle,the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right-angled triangle,then $\sin \theta$ is equal to: