What is the speed of the train ?
(i) the train crosses 300 metres long platform in 21 seconds
(ii) the train crosses another stationary train of equal length in `19(1)/(2)` seconds.
(iii) the train crosses a signal pole in `9(3)/(4)` seconds.

Options

A. (i) & (ii) only
B. (i) and either (ii) or (iii) only
C. (iii) & either (i) or (ii) only
D. All (i), (ii) and (iii)

Answer

To determine the speed of the train, we will analyze each statement provided in the question step by step.

### Step 1: Analyze Statement (i)
The first statement tells us that the train crosses a 300-meter long platform in 21 seconds.

- **Distance covered by the train** = Length of the platform = 300 meters
- **Time taken** = 21 seconds

Using the formula for speed:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
Substituting the values:
\[ \text{Speed} = \frac{300 \text{ meters}}{21 \text{ seconds}} \approx 14.29 \text{ m/s} \]

### Step 2: Analyze Statement (ii)
The second statement states that the train crosses another stationary train of equal length in \(19 \frac{1}{2}\) seconds.

- Let the length of the train be \(x\) meters. Since both trains are of equal length, the total distance covered when crossing each other is \(x + x = 2x\).
- **Time taken** = \(19.5\) seconds

Using the speed formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{2x}{19.5} \]
However, we do not have the value of \(x\), so we cannot determine the speed from this statement alone.

### Step 3: Analyze Statement (iii)
The third statement indicates that the train crosses a signal pole in \(9 \frac{3}{4}\) seconds.

- When crossing a signal pole, the distance covered is equal to the length of the train, which is \(x\) meters.
- **Time taken** = \(9.75\) seconds

Using the speed formula:
\[ \text{Speed} = \frac{x}{9.75} \]
Again, without knowing \(x\), we cannot determine the speed from this statement alone.

### Conclusion
From the analysis:
- **Statement (i)** provides enough information to calculate the speed of the train.
- **Statements (ii)** and **(iii)** do not provide enough information to determine the speed without knowing the length of the train.

Thus, we conclude that the speed of the train can be determined solely from **Statement (i)**.

### Final Answer
The speed of the train is approximately \(14.29 \text{ m/s}\) based on Statement (i).

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