Olympiad/HOTS corner

Observe the shapes $1, 2, 3$ and $4$ in the figures. Which of the following statements is not correct?

A circle with centre at $O$ and $5 \mathrm{cm}$ is given. Find the area of the shaded region in the given figure. (Take $\mathrm{\pi }=\frac{22}{7}$)

Study the given figure (not drawn to scale) carefully and select the correct option.

(i) Perimeter of the complete figure is $198 \mathrm{cm}$.

(ii) Total area of figure II and III is $4 {\mathrm{cm}}^2$ less than the total area of figure I and IV

A rectangular grassy land is $65 \mathrm{m}$ long and $50 \mathrm{m}$ broad. It is surrounded by $3 \mathrm{m}$ wide path. Find the cost of gravelling the path at $₹ 2.50 \mathrm{per} {\mathrm{m}}^2$.

Study the following statements carefully and select the correct option.

Statement-$\mathrm{I}$ : The moon is about $384000 \mathrm{km}$ from earth and its path around the earth is nearly circular. The length of path described by moon in one complete revolution is $2411520 \mathrm{km}$. (Take $\mathrm{\pi }=3.14$)

Statement-$\mathrm{II}$: If the difference between the circumference and diameter of a circle is $154 \mathrm{cm}$, then the radius of the circle is $30.93 \mathrm{cm}$.

$ABCD$ is a rectangle (not drawn to scale) having length $30 \mathrm{cm}$ and breadth $25 \mathrm{cm}$. $P, Q, R$ and $S$ are midpoints of $AB, BC, CD$ and $AD$ respectively. Find the area of the shaded region of the figure.

Inside a rectangular garden which is $20 \mathrm{m}$ long and $15 \mathrm{m}$ wide, a margin of $2 \mathrm{m}$ width has been left all around for planting marigold plant and grass has been planted in the rest of the area. If $4$ marigold plants can be planted in $2 {\mathrm{m}}^2$, then how many marigold plants can be planted in the garden$?$

The ratio of the outer and the inner circumference of a circular path as shown in the figure is $26:25$. If the path is $5 \mathrm{m}$ wide, then find:

(i) the area enclosed by the path.

(ii) the diameter of the inner circle. (Take $\mathrm{\pi }=\frac{22}{7}$)