In a right-angled triangle,if the difference between the two acute angles is $60^{\circ}$,then the ratio of the length of the hypotenuse to the length of the perpendicular drawn to the hypotenuse from its opposite vertex is: (in $: 1$)

In $\triangle ABC$,if $A = 60^{\circ}$ and $B = 105^{\circ}$,then find the value of $\frac{2R^2(b-c) \sin A \sin B \sin C}{(b+c)(s-a \cos C - c \cos A)(s-a \cos B - b \cos A)}$.

In $\triangle ABC$,if $A, B, C$ are in arithmetic progression,then $\frac{c}{a} \sin 2A + \frac{a}{c} \sin 2C =$

Statement-$I$: In the interval $[0, 2\pi]$,the number of common solutions of the equations $2 \sin^2 \theta - \cos 2\theta = 0$ and $2 \cos^2 \theta - 3 \sin \theta = 0$ is two.
Statement-$II$: The number of solutions of $2 \cos^2 \theta - 3 \sin \theta = 0$ in $[0, \pi]$ is two.

With usual notations,in $\triangle ABC$,if $a=2, b=3, c=5$ and $\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{k+7}{30}$,then $k=$

If $x, y$ and $z$ are the distances of the incentre from the vertices $A, B$ and $C$ of the triangle $ABC$ respectively,then $\frac{abc}{xyz}$ is equal to