An object may have

Options

A. varying speed without having varying velocity.
B. varying velocity without having varying speed.
C. non-zero acceleration without having varying velocity.
D. non-zero acceleration without having varying speed.

Explanation

To solve the question regarding the conditions under which an object may have varying speed, varying velocity, non-zero acceleration, and their interrelations, let's analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Speed and Velocity**:
- Speed is a scalar quantity that refers to how fast an object is moving, while velocity is a vector quantity that includes both speed and direction. Therefore, velocity can change if either the speed or the direction changes.

2. **Analyzing Option 1: Varying Speed without Varying Velocity**:
- If an object's speed is varying, it means that the magnitude of its velocity is changing. Since velocity includes speed, it is impossible for speed to vary without velocity also varying.
- **Conclusion**: This option is **incorrect**.

3. **Analyzing Option 2: Varying Velocity without Varying Speed**:
- An object can have a constant speed while changing direction. For example, an object moving in a circular path at a constant speed has a varying velocity due to the change in direction.
- **Conclusion**: This option is **correct**.

4. **Analyzing Option 3: Non-Zero Acceleration without Varying Velocity**:
- Acceleration is defined as the rate of change of velocity. If the velocity is not changing (i.e., it is constant), then the acceleration must be zero. Therefore, it is not possible to have non-zero acceleration without varying velocity.
- **Conclusion**: This option is **incorrect**.

5. **Analyzing Option 4: Non-Zero Acceleration without Varying Speed**:
- An object can have a constant speed but still experience acceleration if its direction is changing. For example, a car turning around a curve at a constant speed is accelerating due to the change in direction.
- **Conclusion**: This option is **correct**.

### Final Summary:
- The correct options are:
- **Option 2**: Varying velocity without varying speed (Correct)
- **Option 4**: Non-zero acceleration without varying speed (Correct)
- The incorrect options are:
- **Option 1**: Varying speed without varying velocity (Incorrect)
- **Option 3**: Non-zero acceleration without varying velocity (Incorrect)