CAT (QA) PYQs

On a triangle $A B C$, a circle with diameter $B C$ is drawn, intersecting $A B$ and $A C$ at points $P$ and $Q$, respectively. If the lengths of $A B, A C$, and $C P$ are $30 \mathrm{~cm}, 25 \mathrm{~cm}$, and $20 \mathrm{~cm}$ respectively, then the length of $BQ$ , in cm , is

Two circles, each of radius $4$ cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

In a circle, two parallel chords on the same side of a diameter have lengths $4$ cm and $6$ cm. If the distance between these chords is $1$ cm, then the radius of the circle, in cm, is