IIMB 2001 (QA) PYQs

Let $\triangle A B C$ be an isosceles triangle such that $A B$ and $A C$ are of equal length. $A D$ is the altitude from $A$ on $B C$ and $B E$ is the altitude from $B$ on $AC$. If $AD$ and $BE$ intersect at $0$ such that $\angle A O B=105^{\circ}$, then $\frac{AD}{BE}$ equals

Given here is a right-angled triangle ABC, if the shaded region represents half of the triangle's area, what is the length of the line segment AD?

In $\triangle A B C, \angle A=66^{\circ} .$ $AB$ and $A C$ are produced to points $D$ and $E$ , respectively. If the bisectors of angle $CBD$ and angle $B C E$ meet at the point $O$, then $\angle B O C$ is equal to: