CAT (QA) PYQs

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

In a right-angled triangle $\triangle A B C$, the altitude $A B$ is $5 \mathrm{~cm}$, and the base $B C$ is $12 \mathrm{~cm}$. $P$ and $Q$ are two points on $B C$ such that the areas of $\triangle \mathrm{ABP}, \triangle \mathrm{ABQ}$ and $\triangle \mathrm{ABC}$ are in arithmetic progression. If the area of $\triangle \mathrm{ABC}$ is $1.5$ times the area of $\triangle \mathrm{ABP}$, the length of PQ in cm , is

Let $P$ be an interior point of a right-angled isosceles triangle $ABC$ with hypotenuse $AB$ . If the perpendicular distance of P from each of $A B, B C$, and $C A$ is $4(\sqrt{2}-1) \mathrm{cm}$, then the area, in sq cm, of the triangle $A B C$ is