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

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

The length of each side of an equilateral triangle ABC is $3 \mathrm{~cm}$. Let D be a point on BC such that the area of triangle ADC is half the area of triangle $A B D$. Then the length of $A D$, in cm , is