CAT (QA) PYQs

The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length $1$ cm, $2$ cm and $4$ cm, then the total number of possible lengths of the fourth side is

In a parallelogram $ABCD$ of area $72 \mathrm{sq} \mathrm{cm}$, the sides $CD$ and $AD$ have lengths $9 \mathrm{~cm}$ and $16 \mathrm{~cm}$, respectively. Let P be a point on $C D$ such that $A P$ is perpendicular to $C D$. Then the area, in sq cm, of triangle $A P D$ is

In a rectangle ABCD, AB = 9 cm and BC = 6 cm. P and Q are two points on BC such that the areas of the figures ABP, APQ, and AQCD are in geometric progression. If the area of the figure AQCD is four times the area of triangle ABP, then BP : PQ : QC is