Each of the following questions consists of a question followed by information in three statements. You have to study the question and the statements and decide which of the statements is/are sufficient to answer the question.
What is the speed of train A?
I Train A crosses a signal post in 24 seconds.
II Train A crosses another train B of equal length in 120 seconds, When running in same direction and in 30 seconds when running in opposite directions on parallel tracks.
III Length of each train is 600 m.

Options

A. Only I and II
B. I and either II or III
C. II and either I or III
D. III and either I or II

Answer

To determine the speed of train A, we will analyze the given statements one by one and see how they can help us find the answer.

### Step-by-Step Solution:

1. **Understanding the Question**:
We need to find the speed of train A.

2. **Analyzing Statement I**:
- Statement I states that "Train A crosses a signal post in 24 seconds."
- This means that the time taken to cross the signal post is 24 seconds.
- However, we do not know the length of train A from this statement alone, so we cannot calculate the speed (Speed = Distance/Time).
- **Conclusion**: Statement I alone is insufficient.

3. **Analyzing Statement II**:
- Statement II states that "Train A crosses another train B of equal length in 120 seconds when running in the same direction and in 30 seconds when running in opposite directions."
- Let the length of each train be L.
- When running in the same direction, the relative speed is (Speed of A - Speed of B), and the distance covered is 2L (both trains).
- Therefore, we have the equation:
\[ \frac{2L}{(V_A - V_B)} = 120 \quad \text{(1)} \]
- When running in opposite directions, the relative speed is (Speed of A + Speed of B), and we have:
\[ \frac{2L}{(V_A + V_B)} = 30 \quad \text{(2)} \]
- We have two equations but three unknowns (Speed of A, Speed of B, and Length L), so we cannot find the speed of train A from this statement alone.
- **Conclusion**: Statement II alone is insufficient.

4. **Analyzing Statement III**:
- Statement III states that "The length of each train is 600 m."
- Knowing the length of train A allows us to substitute L = 600 m in the equations derived from Statement II.
- Substituting L = 600 m into equation (1):
\[ \frac{1200}{(V_A - V_B)} = 120 \implies V_A - V_B = 10 \quad \text{(3)} \]
- Substituting L = 600 m into equation (2):
\[ \frac{1200}{(V_A + V_B)} = 30 \implies V_A + V_B = 40 \quad \text{(4)} \]
- Now we have two equations (3) and (4) with two unknowns (Speed of A and Speed of B):
- From (3): \(V_A - V_B = 10\)
- From (4): \(V_A + V_B = 40\)
- Adding these two equations:
\[ 2V_A = 50 \implies V_A = 25 \text{ m/s} \]
- **Conclusion**: Statement III is sufficient to find the speed of train A when combined with Statement II.

5. **Combining Statements**:
- We found that Statement III is essential to find the speed of train A when combined with Statement II.
- Therefore, we can conclude that either:
- Statement I and Statement III together are sufficient.
- Statement II and Statement III together are sufficient.

### Final Conclusion:
The answer is that the speed of train A can be determined using Statements II and III together or Statements I and III together. Thus, the answer is:

**Option 4: Statements II and III are sufficient to answer the question.**