With usual notations, if in $\mathrm{\Delta }ABC$, $s$ is semi-perimeter and $\left(s-a\right)\left(s-b\right)=s\left(s-c\right)$, then $\mathrm{\Delta }ABC$ is

In a $\Delta ABC$, points $X$ and $Y$ are on $AB$ and $AC$, respectively, such that $XY$ is parallel to $BC$. Which of the two following equalities always hold? (Here, $\left[PQR\right]$ denotes the area of $\Delta PQR$).

$\mathrm{I}$. $\left[BCX\right]=\left[BCY\right]$

$\mathrm{II}$. $\left[ACX\right]·\left[ABY\right]=\left[AXY\right]·\left[ABC\right]$

In the figure given below, if the areas of the two regions are equal then which of the following is true?

Suppose we have two circles of radius $2$ each in the plane such that the distance between their centres is $2\sqrt{3}$. The area of the region common to both circles lies between

In the figure given below, $ABCDEF$ is a regular hexagon of side length $1$ unit, $\mathrm{A}\mathrm{F}\mathrm{P}\mathrm{S}$ and $ABQR$ are squares. Then the ratio $\frac{\text{area of} △APQ}{\text{area of} △SRP}$ equals