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

Let $D$ and $E$ be points on sides $A B$ and $A C$, respectively, of a triangle $A B C$, such that $A D: B D=2: 1$ and $A E: C E=2: 3$. If the area of the triangle $A D E$ is $8 \mathrm{sq} \mathrm{cm}$, then the area of the triangle $ABC$ , in sq 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