A body of mass m is raised to a height h above the surface of the earth of mass M and radius R until its gravitational potential energy increases by $\frac{1}{3}$mgR. The value of h is 

PE on the surface of earth =$-\frac{G M m}{R}$

PE at a height h above the surface of earth 

$=-\frac{G M m}{(R+h)}$

$\therefore$Increase in PE  $=-\frac{G M m}{(R+h)}-\left(-\frac{GMm}{R}\right)$

                        $=GMm\left(\frac{1}{R}-\frac{1}{R+h}\right)$

                      = $GMm\left[\frac{h}{R\left(R+h\right)}\right]$

               $=\frac{g Rmh}{(R+h) } (\therefore g=\frac{GM}{R^2})$

      $\therefore$ PE will increase by $\frac{1}{3}$ mgR at a value of h given by 

                    $\frac{g R m h}{(R+h)}=\frac{1}{3} mgR$

  or    $\frac{h}{R+h}=\frac{1}{3}$ or  h=$\frac{R}{2}$, 

which is choice· (2).