In a sphere of radius $r$, a right circular cone of height $h$ having maximum curved surface area is inscribed. The expression for the square of curved surface of cone is:

$2\pi^2r(2rh^2 - h^3)$

The correct answer is Option (3) → $2\pi^2r(2rh^2 - h^3)$ ##

Here, $\text{CSA of cone} = \pi Rl$

$\text{Radius of sphere} = r$

$\text{Height of cone} = h$

In $\triangle AOC$,

$AO^2 = AC^2 + OC^2$

$\Rightarrow r^2 = R^2 + (h - r)^2$

$\Rightarrow R^2 = 2hr - h^2$

$∴\text{Radius of cone, } R = \sqrt{2hr - h^2} \quad \dots(i)$

In $\triangle ABC$,

$AB^2 = AC^2 + BC^2$

$\Rightarrow l^2 = R^2 + h^2$

$\Rightarrow l^2 = 2hr - h^2 + h^2$

$∴\text{Slant height, } l = \sqrt{2hr} \quad \dots(ii)$

$\text{CSA of cone} = \pi Rl$

$= \pi \sqrt{2hr - h^2} \sqrt{2hr}$

$(\text{CSA of cone})^2 = \pi^2 (2hr - h^2)(2hr)$

$= 2\pi^2 hr(2hr - h^2)$

$= 2\pi^2 r(2rh^2 - h^3)$