Let's examine each expression step by step:

Let's examine each expression step by step:

Here, is the angular momentum. However, torque is defined as the time derivative of angular momentum:

not as the cross product of the position vector with the angular momentum. In fact, if you write

you don't obtain the standard expression for torque. Thus, this expression is not correct.

For a particle, the angular momentum is defined as

Taking the time derivative gives

Since and , the term

is zero. This simplifies to

which is a standard expression for torque. So, this expression is correct.

This is the standard definition of torque, as is the net force . Hence, we can also write

This expression is correct.

This relation applies to rigid bodies rotating . about a fixed axis (where the moment of inertia is constant and can be treated as a scalar). It is a common form used in rotational dynamics, although one must be cautious since it is a special case. In the context of this problem, it is acceptable as a correct expression.

This is the fundamental definition of torque in physics. It directly relates the force applied to a particle and its lever arm. This expression is clearly correct.

To summarize:

Expression A is not a standard or generally valid expression for torque.

Expressions B, C, D, and E are acceptable under the usual assumptions in mechanics.