**Explanation:**

To understand why the degree of dissociation of the weak electrolyte at 0.1 M is 0.01, we need to look at the concept of equivalent conductivity and the behavior of weak acids.

**Equivalent Conductivity:**

Equivalent conductivity is a measure of the ability of an electrolyte to conduct electricity. It depends on the concentration of the electrolyte and is defined as the ratio of the conductivity at a specific concentration to the concentration itself.

**Dissociation of Weak Acids:**

Weak acids are substances that only partially dissociate into ions in aqueous solution. They have a low degree of dissociation compared to strong acids. The degree of dissociation (α) is defined as the fraction of the initial concentration that dissociates into ions.

**Relationship between Equivalent Conductivity and Degree of Dissociation:**

The equivalent conductivity (Λ) of a weak acid at a certain concentration is related to the equivalent conductivity at infinite dilution (Λ°) by the following equation:

Λ = Λ° × α

In this case, it is given that the equivalent conductivity at 0.1 M is 100 times less than that at infinite dilution. Mathematically, this can be expressed as:

Λ(0.1 M) = Λ° × α = Λ°/100

**Calculation:**

We are asked to find the degree of dissociation (α) at 0.1 M. To do this, we rearrange the equation:

α = Λ(0.1 M) / Λ°

Since it is given that Λ(0.1 M) = Λ°/100, we can substitute this value:

α = (Λ°/100) / Λ° = 1/100

Therefore, the degree of dissociation (α) is 0.01, which corresponds to option C.

**Conclusion:**

The degree of dissociation of the weak electrolyte at 0.1 M is 0.01. This means that only 0.01 or 1% of the weak acid molecules dissociate into ions in the solution. The rest of the molecules remain undissociated. This behavior is characteristic of weak acids, which only partially ionize in water.