For a particle moving in straight line with increasing speed the appropriate sign of acceleration a and velocity v can be:

Options

A. `agt0` and `vgt0`
B. `alt0` and `vlt0`
C. `agt0` and `vlt0`
D. `alt0` and `vgt0`

Explanation

To determine the appropriate signs of acceleration \( a \) and velocity \( v \) for a particle moving in a straight line with increasing speed, we can analyze the relationship between velocity and acceleration.

### Step-by-Step Solution:

1. **Understanding Velocity and Acceleration**:
- Velocity (\( v \)) indicates the speed and direction of the particle.
- Acceleration (\( a \)) indicates the rate of change of velocity. It can be positive (speeding up) or negative (slowing down).

2. **Case 1: Both Velocity and Acceleration are Positive**:
- If \( v > 0 \) and \( a > 0 \), the particle is moving in the positive direction and speeding up.
- This scenario leads to an increase in speed.
- **Conclusion**: This case is valid.

3. **Case 2: Both Velocity and Acceleration are Negative**:
- If \( v < 0 \) and \( a < 0 \), the particle is moving in the negative direction and speeding up (in the negative direction).
- This also results in an increase in speed (though the speed is considered negative).
- **Conclusion**: This case is also valid.

4. **Case 3: Positive Acceleration and Negative Velocity**:
- If \( v < 0 \) and \( a > 0 \), the particle is moving in the negative direction but experiencing positive acceleration.
- This means the particle is slowing down (decelerating) in the negative direction, which results in a decrease in speed.
- **Conclusion**: This case is invalid.

5. **Case 4: Negative Acceleration and Positive Velocity**:
- If \( v > 0 \) and \( a < 0 \), the particle is moving in the positive direction but experiencing negative acceleration.
- This means the particle is slowing down, which results in a decrease in speed.
- **Conclusion**: This case is also invalid.

### Final Conclusion:
The valid cases where the particle is moving in a straight line with increasing speed are:
- Both \( a \) and \( v \) are positive.
- Both \( a \) and \( v \) are negative.

Thus, the correct options are:
- Option A: \( a > 0 \), \( v > 0 \)
- Option B: \( a < 0 \), \( v < 0 \)