CAT (QA) PYQs

Let $ABC$ be a right-angled isosceles triangle with hypotenuse $BC$. Let $BQC$ be a semi-circle, away from $A$, with diameter $BC$. Let $BPC$ be an arc of a circle centred at $A$ and lying between $BC$ and $BQC$. If $AB$ has length $6$ cm ,then the area, in sq. cm, of the region enclosed by $BPC$ and $BQC$ is:

A triangle is drawn with its vertices on the circle C such that one of its sides is a diameter of C and the other two sides have their lengths in the ratio $a: b$. If the radius of the circle is $r$, then the area of the triangle is

In triangle ABC, altitudes AD and BE are drawn to the corresponding bases. If $\angle BAC = 45^\circ$ and $\angle ABC = \theta$, then $\frac{AD}{BE}$ equals