To understand the answer, let's break down the given equilibria and analyze each one individually:

1. AB2(g) ⇌ 2AB(g)
This equilibrium represents the dissociation of AB2 into two molecules of AB. The equilibrium constant for this reaction is denoted as K1.

2. B2C(g) ⇌ BC(g) + C(g)
This equilibrium represents the dissociation of B2C into BC and C. The equilibrium constant for this reaction is denoted as K2.

3. AB3(g) ⇌ AB(g) + B2C(g)
This equilibrium represents the dissociation of AB3 into AB and B2C. The equilibrium constant for this reaction is denoted as K3.

4. BC(g) ⇌ B2C(g)
This equilibrium represents the conversion of BC into B2C. The equilibrium constant for this reaction is denoted as K4.

5. B2C(g) ⇌ B3C2(g)
This equilibrium represents the conversion of B2C into B3C2. The equilibrium constant for this reaction is denoted as K5.

Now, let's consider the reaction starting only with AB2 and B2C. Initially, there is no AB3, BC, or B3C2 present.

At equilibrium, the concentrations of all species will remain constant. Based on this information and the given equilibria, we can analyze the options:

a) [AB3]eq = [BC]eq
This option suggests that the equilibrium concentrations of AB3 and BC are equal. However, there is no direct relationship between these two species in any of the equilibria given. Therefore, option a) is not necessarily true.

b) [A]eq = [B2C]eq
This option suggests that the equilibrium concentrations of A and B2C are equal. However, the equilibrium concentrations of A and B2C are not necessarily equal based on the given equilibria. Therefore, option b) is not necessarily true.

c) [AB3]eq = [B3C2]eq
This option suggests that the equilibrium concentrations of AB3 and B3C2 are equal. From the given equilibria, we can see that AB3 is a reactant in the equilibrium AB3 ⇌ AB + B2C, and B3C2 is a product in the equilibrium B2C ⇌ B3C2. Therefore, at equilibrium, the concentrations of AB3 and B3C2 can be expected to be equal. Hence, option c) is necessarily true.

d) [AB]aq = [BC]eq
This option suggests that the initial concentration of AB is equal to the equilibrium concentration of BC. However, there is no direct relationship between these two species in any of the equilibria given. Therefore, option d) is not necessarily true.

In summary, options c) and d) are necessarily true at equilibrium. Option c) is true because AB3 and B3C2 are related in the given equilibria, and option d) is not true because there is no direct relationship between AB and BC in the given equilibria.