CAT (QA) PYQs

A triangle $ABC$ has area $32$ sq units and its side $BC$ , of length $8$ units, lies on the line $\mathrm{x}=4$. Then the shortest possible distance between $A$ and the point $(0,0)$ 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

In a $\triangle ABC$, points $D$ and $E$ are on the sides $BC$ and $AC$, respectively. $BE$ and $AD$ intersect at point $T$ such that $AD:AT = 4:3$, and $BE:BT = 5:4$. Point $F$ lies on $AC$ such that $DF$ is parallel to $BE$. Then, $BD:CD$ is