In a circle with centre $P$,the length of chord $AB$ is $40 \, cm$. $AB$ lies at a distance of $21 \, cm$ from the centre $P$. Find the diameter of the circle. (in $, cm$)

Two circles of radii $8 \ cm$ with centers $P$ and $Q$ are given. The chord $AB$ of the circle with center $P$ and the chord $CD$ of the circle with center $Q$ are equal. If $\angle PAB = 40^{\circ}$,then find $\angle CQD$. (in $^{\circ}$)

$AB$ and $AC$ are two chords of a circle with centre $P$. If the bisector of $\angle BAC$ passes through the centre $P$,prove that $AB = AC$.

$A$ quadrilateral $ABCD$ is inscribed in a circle such that $AB$ is a diameter and $\angle ADC = 130^{\circ}$. Find $\angle BAC$.

In the given figure,if $\angle BCD = 40^{\circ}$ and $\angle BAE = 65^{\circ},$ then find the values of $a, b,$ $c$ and $d$.

State whether the following statement is True or False and justify your answer: $ABCD$ is a cyclic quadrilateral such that $\angle A = 90^{\circ}, \angle B = 70^{\circ}, \angle C = 95^{\circ}$ and $\angle D = 105^{\circ}$.