CAT (QA) PYQs

ABCD is a rectangle with sides AB = 56 cm and BC = 45 cm, and E is the midpoint of side CD. Then, the length, in cm, of radius of in circle of $\triangle$ADE is

Let $ABC$ be a right-angled isosceles triangle with hypotenuse $BC$. Let $BQC$ be a semi-circle, away from $A$, with diameter $BC$. Let $BPC$ be an arc of a circle centred at $A$ and lying between $BC$ and $BQC$. If $AB$ has length $6$ cm ,then the area, in sq. cm, of the region enclosed by $BPC$ and $BQC$ is:

In a circle with center $O$ and radius $1 \mathrm{~cm}$, an arc $A B$ makes an angle $60$ degrees at $O$. Let $R$ be the region bounded by the radii $\mathrm{OA}, \mathrm{OB}$ and the arc $AB$ . If $C$ and $D$ are two points on $OA$ and $OB$ , respectively, such that $\mathrm{OC}=\mathrm{OD}$ and the area of triangle $OCD$ is half that of $R$ , then the length of $OC$ , in cm , is