There is a square field of side $500 \mathrm{m}$ length each. It has a compound wall along its perimeter. At one of its corners, a triangular area of the field is to be cordoned off by erecting a straight-line fence. The compound wall and the fence will form its borders. If the length of the fence is $100 \mathrm{m},$ what is the maximum area that can be cordoned off?

Consider a circle with unit radius. There are seven adjacent sectors, $\mathrm{S}1, \mathrm{S}2, \mathrm{S}3,..., \mathrm{S}7,$ in the circle such that their total area is $1/8$ of the area of the circle. Further, the area of the jth sector is twice that of the $(\mathrm{j}-1) \mathrm{th}$ sector, for $\mathrm{j}=2, \dots , 7.$ What is the angle, in radians, subtended by the arc of $\mathrm{S}1$ at the centre of the circle?

In the figure, $\mathrm{AB}=\mathrm{BC}=\mathrm{CD}=\mathrm{DE}=\mathrm{EF}=\mathrm{FG}=\mathrm{GA}.$ Then $\angle \mathrm{DAE}$ is approximately

Two sides of a plot measure $32 \mathrm{m}$ and $24 \mathrm{m}$ and the angle between them is a perfect right angle. The other two sides measure $25 \mathrm{m}$ each and the other three angles are not right angles.

What is the area of the plot?

In $∆\mathrm{DEF}$ shown below, points $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are taken on $\mathrm{DE}, \mathrm{DF}$ and $\mathrm{EF}$ respectively such that $\mathrm{EC}=\mathrm{AC}$ and $\mathrm{CF}=\mathrm{BC}.$ If $\angle \mathrm{D}=40°,$ then $\angle \mathrm{ACB}=$