CAT (QA) PYQs

Two tangents drawn from a point $P$ touch a circle with center $O$ at points $Q$ and $R$. Points $A$ and $B$ lie on $PQ$ and $PR$, respectively, such that $AB$ is also a tangent to the same circle. If $\angle AOB = 50^\circ$, then $\angle APB$, in degrees, equals

In a triangle $ABC, AB = AC = 8$cm. $A$ circle drawn with $BC$ as diameter passes through $A$. Another circle drawn with center at $A$ passes through $B$ and $C$. Then the area, in sq. cm, of the overlapping region between the two circles is

$AB$ is a diameter of a circle of radius $5 \mathrm{~cm}$. Let $P$ and $Q$ be two points on the circle so that the length of $PB$ is $6 \mathrm{~cm}$, and the length of $AP$ is twice that of $AQ$ . Then the length, in cm , of $QB$ is nearest to