Question : Two identical circles each of radius $2\;\mathrm{cm}$ intersect each other such that the circumference of each one passes through the centre of the other. What is the area (in $\mathrm{cm^2}$) of the intersecting region?

Option 1: $\frac{8\pi }{3}-2\sqrt{3}$

Option 2: $\frac{8\pi }{3}-\sqrt{3}$

Option 3: $\frac{4\pi }{3}-\sqrt{3}$

Option 4: $\frac{4\pi }{3}-2\sqrt{3}$

Question : Three circles of equal radius '$a$' cm touch each other. The area of the shaded region is:

Option 1: $\left ( \frac{\sqrt{3}+\pi}{2} \right )a^{2}$ sq. cm

Option 2: $\left ( \frac{6\sqrt{3}-\pi }{2} \right )a^{2}$ sq. cm

Option 3: $\left ( \sqrt{3}-\pi \right )a^{2}$ sq. cm

Option 4: $\left ( \frac{2\sqrt{3}- \pi}{2} \right )a^{2}$ sq. cm

Question : If the area of an equilateral triangle is $a$ and height $b$, then the value of $\frac{b^2}{a}$ is:

Option 1: $3$

Option 2: $\frac{1}{3}$

Option 3: $\sqrt3$

Option 4: $\frac{1}{\sqrt3}$

Question : If $\frac{1}{x^{2}}+x^{2}$ represents the radius of circle $P$ and $\frac{1}{x}+x=17$, which of the following best approximates the circumference of circle $P$? 

Option 1: $287\pi$

Option 2: $547\pi$

Option 3: $574\pi$

Option 4: $278\pi$

Question : The height of a cylinder is $\frac{2}{3}$rd of its diameter. Its volume is equal to the volume of a sphere whose radius is 4 cm. What is the curved surface area (in cm²) of the cylinder?

Option 1: $\frac{112}{3} \pi$

Option 2: $32 \pi$

Option 3: $\frac{128}{3} \pi$

Option 4: $40 \pi$