Introduction:
In this scenario, we have two conducting concentric hollow spheres, A and B, with radii a and b respectively, where A is inside B. The common potential between the spheres is V. We are given that sphere A is given some charge such that its potential becomes zero. We need to determine the new potential of sphere B after this charge redistribution.

Analysis:
To solve this problem, we will consider the charge distribution on both spheres A and B. Let's analyze the scenario step by step:

Step 1: Charge distribution on Sphere A:
When sphere A is given some charge, it redistributes itself to reach a state of equilibrium. As a result, the charge will accumulate on the outer surface of sphere A, since the inner surface is connected to the common potential V and thus neutralizes any charge on it. Therefore, the charge on the outer surface of sphere A will be such that it creates a potential of zero inside A.

Step 2: Potential of Sphere B:
Now let's consider the potential of sphere B. Since sphere B is larger than sphere A, it will have a larger surface area. The potential at any point outside a charged conducting sphere depends only on the charge enclosed within it and the radius of the sphere. In this case, the charge enclosed within sphere B is the charge on the outer surface of sphere A.

Step 3: Applying the Potential Formula:
According to the formula for the potential of a conducting sphere, the potential at any point outside the sphere is given by V = kQ/r, where V is the potential, k is the Coulomb's constant, Q is the charge enclosed within the sphere, and r is the radius of the sphere.

Step 4: Calculating the Potential of B:
In this case, the potential of B is given by V = kQ_B/r_B, where Q_B is the charge on the outer surface of sphere A and r_B is the radius of sphere B. Since the charge on the outer surface of A creates a potential of zero inside A, Q_B = 0. Therefore, the potential of B becomes V = k(0)/r_B = 0.

Conclusion:
From the above analysis, we can conclude that the new potential of sphere B after the charge redistribution on sphere A will be 0. Therefore, the correct answer is (a) 0.