Option A is the correct answer: A unitless quantity never has a nonzero dimension.

Explanation:
A unitless quantity is a quantity that does not have any associated units. In other words, it is a pure numerical value without any physical dimensions. For example, the number 5 is a unitless quantity because it does not have any units attached to it.

In physics, quantities are often described using dimensions. Dimensions are the physical properties or characteristics of a quantity, such as length, time, mass, temperature, etc. These dimensions are represented by units, such as meters for length, seconds for time, kilograms for mass, etc.

When we say that a quantity has a nonzero dimension, it means that it has a physical property associated with it and can be measured using a specific unit. For example, if we say that the length of an object is 5 meters, we are stating that the object has a dimension of length and it can be measured in meters.

On the other hand, a unitless quantity does not have any physical property or dimension associated with it. It is simply a numerical value. For example, if we say that the ratio of two lengths is 5, we are not specifying any particular unit for the ratio. It is just a pure number.

Therefore, a unitless quantity never has a nonzero dimension because it does not have any physical property or characteristic that can be measured with a specific unit. It is important to note that a unitless quantity can still have a value and can be used for calculations, but it does not have any associated units or dimensions.

In conclusion, a unitless quantity never has a nonzero dimension because it does not have any physical property or characteristic that can be measured with a specific unit. It is simply a pure numerical value.