Q1: 2^x − x − 1 = 0

∴ 2^x = x + 1

This equation indicates the intersection of an exponential curve, lying in the I and II quadrants, and a straight line. ∴ It can have two intersection points at the most. In this case, we can find two intersection points by trial and error as x = 0 and x = 1

Q2:

Ratio of areas of two spheres = s1 : s2 = 4 : 1

∴ Ratio of their radii = r1 : r2 = 2 : 1

∴ Ratio of their volumes = v1 : v2 = 8 : 1

Volume of A is 12.5% (1/8th) of the volume of B.

But, volume of A is k% less than B.

∴ k = (100 − 12.5) = 87.5% Hence, option 4.