CSIR UGC NET Physical Science June 2019 Solution 1D potential well

Question goes like this:
The Infinite square well potential of a particle in a box of size a is modified as shown in the figure below. The energy of the ground state compared to the ground state energy before the perturbation was added

The second problem goes something like this

A Quantum particle of mass M in One dimension confined to a rigid box as shown in the figure is in its ground state. And infinitesimally thin wall is very well slowly raised to infinity at the centre of the box in such a way that the system remains in its ground state at all times as you mean that no energy is lost in raising the wall the work done on the system when the wall is fully taste eventually separating the original box into two Compartment

Solution is below

The Slides can be found here

Csr ugc net phy june19 physics q(8 9 of 55) solution from Bibhatsu Bibhatsu