A hemisphere and a cone have equal bases. If their heights are also equal, then ratio of their curved surface will be

Take the common base radius as r. A hemisphere of radius r has height r, so the cone has height h = r as well. Its slant height is l = √(r² + r²) = r√2.

Curved surface of hemisphere = 2πr². Curved surface of cone = πrl = πr × r√2 = √2πr². So hemisphere : cone = 2 : √2 = √2 : 1. Option C is correct.

Why the others fail: A (2:1) would need the cone's curved surface to be πr², which requires slant height r, impossible when the height is also r. B and D are the inverse of the correct ratio, a common slip when numerator and denominator get swapped.

Core concept: curved surface areas of solids of revolution, with 2πr² for a hemisphere (curved part only, base excluded) and πrl for a cone, plus Pythagoras to get the slant height. This sits in the Mensuration block of quantitative aptitude and recurs in dome and funnel design problems.

Trap: Using total surface area for the hemisphere (adding πr²) when the question asks only for curved surface. That changes the ratio and leads to a wrong option.