In a cyclic quadrilateral $ABCD$,diagonals $AC$ and $BD$ intersect at $P$. If $\angle BAC = 52^{\circ}$ and $\angle ADB = 78^{\circ}$,then find $\angle ABC$. (in $^{\circ}$)

In a circle with centre $P$,$AB$ is a chord and point $C$ is a point other than $A$ and $B$ on the major arc $AB$. If $\angle ACB = 47^{\circ},$ then $\angle APB = $ .......... (in $^{\circ}$)

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

Write True or False and justify your answer in each of the following:
If $A, B, C, D$ are four points such that $\angle BAC = 30^{\circ}$ and $\angle BDC = 60^{\circ}$, then $D$ is the centre of the circle through $A, B$ and $C$.

$AB$ and $CD$ are two parallel chords of a circle with centre $P$. Centre $P$ is not between the two chords $AB$ and $CD$. If $AB = 40\,cm$,$CD = 30\,cm$ and the radius of the circle is $25\,cm$,find the distance between $AB$ and $CD$. (in $,cm$)

The longest chord of a circle is a .......... of the circle.