An object may have
( a) varying speed without having varying velocity
(b) varying velocity without having varying speed
(c) nonzero acceleration without having varying velocity
(d) nonzero acceleration without having varying speed.

Options

A. a, b & c are correct
B. b & d are correct
C. a, b & d are correct
D. a & d are correct

Answer

To solve the question regarding the relationships between speed, velocity, and acceleration, 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, regardless of its direction.
- Velocity is a vector quantity that includes both the speed of an object and the direction of its motion.

2. **Analyzing Option (a)**: Varying speed without varying velocity.
- This is **not possible** because if the speed of an object is changing, the magnitude of the velocity is also changing. Therefore, this option is incorrect.

3. **Analyzing Option (b)**: Varying velocity without having varying speed.
- This is **possible** if the direction of the velocity changes while the speed remains constant. An example is an object moving in a circular path at a constant speed. The velocity changes due to the change in direction, but the speed remains constant. Therefore, this option is correct.

4. **Analyzing Option (c)**: Nonzero acceleration without having varying velocity.
- This is **not possible**. Acceleration is defined as the rate of change of velocity. If there is nonzero acceleration, it implies that the velocity must be changing. Therefore, this option is incorrect.

5. **Analyzing Option (d)**: Nonzero acceleration without varying speed.
- This is **possible**. An example is an object moving in a circular path at a constant speed. The object experiences centripetal acceleration directed towards the center of the circle, even though its speed (magnitude of velocity) remains constant. Therefore, this option is correct.

### Conclusion:
Based on the analysis:
- The correct options are **(b)** varying velocity without varying speed, and **(d)** nonzero acceleration without varying speed.