CAT (QA) PYQs

Two ships are approaching a port along straight routes at constant speeds. Initially, the two ships and the port formed an equilateral triangle with sides of length $24$ km. When the slower ship travelled $8$ km, the triangle formed by the new positions of the two ships and the port became right-angled. When the faster ship reaches the port, the distance, in km, between the other ship and the port will be

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 sum of the perimeters of an equlateral triangle and a rectangle is $90 \mathrm{~cm}$ the area, $T$ , of the triangle and the area, $R$ , of the rectangle, both in sq cm, satisfy the relationship $R=T^{2}$. If the sides of the rectangle are in the ratio $1: 3$, then the length, in cm , of the longer side of the rectangle, is