JEE Advanced 2013 Paper 1

On small sphere

4

3

πR3(ρ)g‌+kx=

4

3

πR3(2ρ)g‌‌ ........(i)
on second sphere (large)

4

3

πR3(3ρ)g‌=

4

3

πR3(2ρ)g+kx‌‌ ......(ii)
by equation (i) and (ii)
x=

4πR3ρg

3k

For equilibrium of the complete system, net force of buoyancy must be equal tothe total weight of the sphere which holds true in the given problem. So both the spheres are completely submerged