IIMB 2001 (QA) PYQs

A room, with a length of 18m, a height of 15 m and a total surface area of 1332 m² is to be divided into 4 m wide small rooms, by inserting plywood sheets in the room. To minimize the expenditure on the purchase of these sheets, how many 18 m × 15 m sheets will be required to construct the maximum possible number of such small rooms, if the length and the height of the small rooms remain the same as that of the original room?

In triangle $\triangle A B C$, the internal angle bisector of $\angle B A C$ meets side $B C$ at point $D$ and, when extended, intersects the circumcircle of $\triangle A B C$ at point $E$. The incenter of $\triangle A B C$ is denoted as $I$. Given that $A I=12 \mathrm{~cm}$ and $B E=8 \mathrm{~cm}$, find the length of $D E$.