Directions : Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. What is the speed of the train? (M.B.A., 2002) I. The train crosses 300 metres long platform in 21 seconds. II. The train crosses another stationary train of equal length in $19\frac{1}{2}$ seconds. III. The train crosses a signal pole in $9\frac{3}{4}$ seconds.

### Concept & Logic We need to establish equations relating the train's speed ($v$) and length ($x$) based on the given crossing times. If two statements yield identical equations, they are redundant, and we only need one of them along with a completely independent statement. ### Step-by-Step Solution * Let the length of the train be $x$ and its speed be $v$. * **From Statement I:** Crossing a 300 m platform in 21 seconds. $v = \frac{x + 300}{21}$ * **From Statement II:** Crossing a stationary train of equal length in $19\frac{1}{2}$ seconds. $v = \frac{x + x}{19.5} \Rightarrow v = \frac{2x}{19.5} \Rightarrow v = \frac{x}{9.75}$ * **From Statement III:** Crossing a signal pole in $9\frac{3}{4}$ seconds. $v = \frac{x}{9.75}$ * Notice that Statements II and III yield the exact same mathematical relationship between $v$ and $x$. They are essentially the same clue. * To solve for $v$, we need the equation from Statement I, paired with the relationship from either Statement II or Statement III. ### Exam Strategy & Shortcut Recognize that crossing a stationary train of equal length takes exactly twice as much time as crossing a point object (like a pole). $19\frac{1}{2}$ is exactly double $9\frac{3}{4}$. Thus, statements II and III provide identical constraints. You need Statement I and just one of the other two. ### Common Pitfall Students often assume that all three statements provide unique data and might incorrectly choose "None of these" or overcomplicate the system by trying to use all three simultaneously. ### Final Answer Therefore, the correct answer is **I and either II or III only**.