CAT (QA) PYQs

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

The midpoints of sides $AB$, $BC$, and $AC$ in $\triangle ABC$ are $M$, $N$, and $P$, respectively. The medians drawn from $A$, $B$, and $C$ intersect the line segments $MP$, $MN$ and $NP$ at $X$, $Y$, and $Z$, respectively. If the area of $\triangle ABC$ is $1440$ sq cm, then the area, in sq cm, of $\triangle XYZ$ is

Let $ABC$ be a right-angled triangle with hypotenuse $BC$ of length $20$ cm. If $AP$ is perpendicular on $BC$, then the maximum possible length of $AP$, in cm, is