Question

A train running at a certain speed crosses a platform in $30$ seconds. What is the speed of the train?
I. Length of the train is $240$ m.
II. The train crosses a man who is on the platform in $12$ seconds. 

• A. If I alone is sufficient but II alone is not sufficient.
• B. If II alone is sufficient but I alone is not sufficient.
• C. If either I alone or II alone is sufficient.
• D. If even I + II together are not sufficient.
• E. If I + II together are necessary to answer the question.

Hint:

When a train "crosses a platform", use $(L+P)/v$; when it “crosses a man”, use $L/v$. Often, length + one more time measurement together fix the speed.

Answer 1

Answer: (E)

Let train length $L$, platform length $P$, speed $v$. From the stem, \[ \frac{L+P}{v}=30. \tag{1} \] I alone: $L=240$ gives two unknowns ($P,v$) in (1) $\Rightarrow$ not sufficient.
II alone: time to cross a man $=\dfrac{L}{v}=12 \Rightarrow v=\dfrac{L}{12}$, but $L$ unknown $\Rightarrow$ not sufficient.
I + II: from II, $v=L/12$; with $L=240$, $v=20$ m/s. So speed is determined. Hence (e). 

Answer 2

Set up the two unknowns clearly: the train's length \( L \) and the platform's length \( P \), related by \( \dfrac{L+P}{30}=v \). Test each statement, and their combination, by counting how many independent unknowns remain.

1. Option A (I alone sufficient, II not): Statement I only fixes \( L=240 \). The stem's single equation \( \dfrac{L+P}{30}=v \) still has two unknowns, \( P \) and \( v \); one equation with two unknowns cannot be solved. So I alone is not sufficient, and this option is rejected.
2. Option B (II alone sufficient, I not): Statement II gives \( \dfrac{L}{12}=v \), introducing a second unknown \( L \) alongside \( v \), with no numerical value for \( L \); it cannot pin down \( v \) alone. So II alone is also not sufficient, and this option is rejected.
3. Option C (either alone sufficient): Since neither I alone nor II alone works, as shown above, this option is rejected.
4. Option D (even together not sufficient): Combining I and II gives \( L=240 \) from I, substituted into \( v=L/12 \) from II, yielding \( v=240/12=20 \) m/s, a single definite value. Since the two together do determine \( v \), this option, which claims even the combination fails, is rejected.
5. Option E (I + II together necessary): As shown, neither statement alone fixes \( v \), but together they give exactly one value, \( v=20 \) m/s. This matches.

Counting unknowns against equations for each case confirms both statements are required together.

Hence, the correct answer is If I + II together are necessary to answer the question.