In cyclic quadrilateral $PQRS$,if $5 \angle Q = 7 \angle S$,then find the value of $\angle Q$. (in $^{\circ}$)

In the figure,$AB$ and $CD$ are two chords of a circle intersecting each other at point $E$. Prove that $\angle AEC = \frac{1}{2}$ (angle subtended by arc $CXA$ at the centre $+$ angle subtended by arc $DYB$ at the centre).

In a circle with centre $P$,the diameter of the circle is $70 \, cm$. The chord $AB$ is at a distance of $21 \, cm$ from the centre of the circle,then $AB = \dots \, cm$.

$ABCD$ is a cyclic quadrilateral in which $\angle A = 80^{\circ}$ and $\angle B = 75^{\circ},$ then $\angle D = \dots$ (in $^{\circ}$)

If $ABC$ is an equilateral triangle inscribed in a circle and $P$ is any point on the minor arc $BC$ which does not coincide with $B$ or $C$,prove that $PA$ is the angle bisector of $\angle BPC$.

In the figure,$AOC$ is a diameter of the circle and $\operatorname{arc} AXB = \frac{1}{2} \operatorname{arc} BYC$. Find $\angle BOC$.