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 $, cm$)

Write True or False and justify your answer in each of the following:
Two chords $AB$ and $AC$ of a circle with centre $O$ are on the opposite sides of $OA$. Then $\angle OAB = \angle OAC$.

In a circle with centre $O$,the lengths of two chords $AB$ and $CD$ are $12 \, cm$ and $16 \, cm$ respectively. If the chord $AB$ is at a distance $8 \, cm$ from the centre,what is the distance of the chord $CD$ from the centre (in $, cm$)?

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

$AB$ is a chord of a circle with centre $P$. Point $C$ is a point other than $A$ and $B$ on the major arc $AB$. If $\angle ACB + \angle APB = 135^{\circ}$,then find $\angle ACB$ and $\angle APB$.