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? (S.B.I.P.O., 2002) I. The train crosses a tree in 13 seconds. II. The train crosses a platform of length 250 metres in 27 seconds. III. The train crosses another train running in the same direction in 32 seconds.

### Concept & Logic To find the speed of the train, we need to relate its length and speed to the time taken to cross objects. When a train crosses a stationary point object (like a tree), the distance covered is its own length. When it crosses a platform, the distance covered is the sum of its length and the platform's length. $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$ ### Step-by-Step Solution * Let the length of the train be $x$ meters and its speed be $v$ m/s. * **From Statement I:** The train crosses a tree in 13 seconds. $v = \frac{x}{13} \Rightarrow x = 13v$ * **From Statement II:** The train crosses a 250 m platform in 27 seconds. $v = \frac{x + 250}{27}$ * By substituting $x = 13v$ into the second equation, we get $v = \frac{13v + 250}{27}$, which allows us to solve for $v$. Thus, statements I and II together are sufficient. * **From Statement III:** The train crosses another train in 32 seconds. This introduces two new variables (the length and speed of the other train), making it impossible to solve without more information. ### Exam Strategy & Shortcut In Data Sufficiency, you don't need to calculate the final value. Simply check if the equations provided are independent and equal in number to the unknown variables. Statements I and II give two independent linear equations with two variables ($x$ and $v$), which is perfectly sufficient. ### Common Pitfall A common mistake is thinking Statement III provides useful relative speed information. Without the length or speed of the second train, Statement III is virtually useless on its own or combined with just one other statement. ### Final Answer Therefore, the correct answer is **I and II only**.