A solid sphere of radius R has a charge Q distributed in its volume with a charge density $$\rho = K{r^a}$$, where K and a are constants and r is the distance from its centre. If the electric field at $$r = R/2$$ is 1/8 times than at $$r = R$$, find the value of $$a$$.

A metal rod AB of length 10x has its one end A in ice at 0$$^\circ$$C and the other end B in water at 100$$^\circ$$C. If a point P on the rod is maintained at 400$$^\circ$$C, then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is 540 cal/g and latent heat of melting of ice is 80 cal/g. If the point P is at a distance of $$\lambda x$$ from the ice end A, find the value of $$\lambda$$. (Neglect any heat loss to the surrounding.)

Three objects A, B and C are kept in a straight line on a frictionless horizontal surface. These have masses m, 2m and m, respectively. The object A moves towards B with a speed 9 m/s and makes an elastic collision with it. Thereafter, B makes completely inelastic collision with C. All motions occur on the same straight line. Find the final speed (in m/s) of the object C.