A solid sphere of mass 5 kg and radius 10 cm is kept in contact...

For two solid spheres in contact with the tangent line at the contact point.
Sphere 1: $m_1 = 5$ kg, $r_1 = 0.1$ m
Sphere 2: $m_2 = 10$ kg, $r_2 = 0.2$ m
For a solid sphere, $I_{cm} = \frac{2}{5}mr^2$
The tangent at contact is perpendicular to the line joining centers. The perpendicular distance from each sphere's center to this tangent line equals its radius.
Using parallel axis theorem: $I_{total} = I_{cm} + md^2$
For sphere 1: $I_1 = \frac{2}{5}(5)(0.1)^2 + 5(0.1)^2 = 0.02 + 0.05 = 0.07$ kg·m²
For sphere 2: $I_2 = \frac{2}{5}(10)(0.2)^2 + 10(0.2)^2 = 0.16 + 0.4 = 0.56$ kg·m²
Total: $I = 0.07 + 0.56 = 0.63$ kg·m²