Pick the correct statements:

Options

A. Average speed of a particle in a given time is never less than the magnitude of the avrage velocity
B. It is possible to have a situation in which `|(vec(dv))/(dt)|!=0` but `d/(dt)|vecv|=0`.
C. the average velocity of a particle is zero inn a time interval. It is possible that eh instantaneous velocity is never zero in the interval.
D. The average velocity of a particle moving on a straight line is zero in a time interval. It is possible that the instantaneous velocity is never zero in the interval. (Infinite accelerations are not allowed).

Answer

To solve the question, we need to analyze each of the given statements about average speed, average velocity, and instantaneous velocity.

### Step-by-Step Solution:

1. **Understanding Average Speed and Average Velocity**:
- Average speed is defined as the total distance traveled divided by the total time taken.
- Average velocity is defined as the total displacement divided by the total time taken.
- It is known that displacement can never be greater than the distance traveled. Therefore, the average speed is always greater than or equal to the magnitude of average velocity.

**Conclusion**: **Option 1 is correct**.

2. **Analyzing the Second Statement**:
- The statement suggests that it is possible for one expression to be non-zero while another is zero.
- In uniform circular motion, the angular velocity (ω) is constant, which means the centripetal acceleration is non-zero (ω²r is non-zero). However, the linear displacement over one complete revolution is zero (since the starting and ending points are the same).

**Conclusion**: **Option 2 is correct**.

3. **Evaluating the Third Statement**:
- The statement claims that the average velocity can be zero while the instantaneous velocity is never zero.
- In uniform circular motion, the average velocity is zero because the particle returns to its starting point (displacement is zero). However, at any point in time during the motion, the instantaneous velocity is not zero.

**Conclusion**: **Option 3 is correct**.

4. **Examining the Fourth Statement**:
- This statement posits that if the average velocity of a particle moving in a straight line is zero, then it is possible for the instantaneous velocity to be never zero.
- For a particle to have an average velocity of zero in a straight line, it must return to its original position. This means that at some point, the particle must have an instantaneous velocity of zero (when it changes direction).

**Conclusion**: **Option 4 is incorrect**.

### Final Answers:
- **Correct Statements**: Options 1, 2, and 3 are correct.
- **Incorrect Statement**: Option 4 is incorrect.