In a cyclic quadrilateral $ABCD$,$\angle A = 2x - 10^{\circ}$ and $\angle C = 3x - 35^{\circ}$,then $\angle A =$ .......... (in $^{\circ}$)

In the same plane,two circles can have a maximum of ............ common chords.

In the figure,two congruent circles have centers $O$ and $O'$. Arc $AXB$ subtends an angle of $75^{\circ}$ at the center $O$ and arc $A'YB'$ subtends an angle of $25^{\circ}$ at the center $O'$. Then the ratio of the lengths of arcs $AXB$ and $A'YB'$ is:

In a cyclic quadrilateral,$\angle A = 3x + 15^{\circ}$ and $\angle C = 2x + 15^{\circ}$. Find the value of $x$.

If $BM$ and $CN$ are the perpendiculars drawn on the sides $AC$ and $AB$ of the triangle $ABC$,prove that the points $B, C, M$ and $N$ are concyclic.

Write True or False and justify your answer in each of the following:
$A$ circle of radius $3 \, cm$ can be drawn through two points $A$ and $B$ such that $AB = 6 \, cm$.