CAT (QA) PYQs

From a triangle $ABC$ with sides of lengths $40$ ft, $25$ ft and $35$ ft, a triangular portion $GBC$ is cut off where $G$ is the centroid of $ABC$. The area, in sq ft, of the remaining portion of triangle $ABC$ is

Suppose the medians $BD$ and $CE$ of a triangle $ABC$ intersect at a point $O$. If area of triangle $ABC$ is $108$ sq. cm., then, the area of the triangle $EOD$, in sq. cm., 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