b> : Read the following information carefully and answer the questions that follow :
I. Seven students P, Q, R, S, T, U and V take a series of tests.
II. No two students get similar marks.
III. V always scored more than P.
IV. P always scored more than Q.
V. Each time either R scored the highest and T gets the least or alternatively S scores the highest and U or Q scores the least.

If V is ranked fifth, which of these must be true ?

Correct option is D

Information Given:

1. Seven students: P, Q, R, S, T, U, V.
2. No two students get the same marks.
   (All rankings are unique.)
3. V always scores more than P.
   $V>P$
4. P always scores more than Q.
   $P>Q$
5. Each time either:
   • R scores the highest, and T scores the least, or
   • S scores the highest, and U or Q scores the least.

Question:

If V is ranked fifth, which of the given options must be true?

Step-by-Step Analysis:

Step 1: Ranking V at fifth

If V is fifth, this means:

1. Four students scored higher than V.
2. Two students scored lower than V.

Step 2: Apply the conditions for the highest and lowest scorers:

1. Condition 1: If R is the highest, then T must be the least.
2. Condition 2: If S is the highest, then U or Q must be the least.

Step 3: Key Observations from the Given Conditions:

1. Since V is ranked fifth, the rankings above V must include the highest scorer and three other students.
2. V > P > Q: From this condition, $Q$ is definitely ranked below P and V.
3. If $Q$ or $U$ is the least, the second condition applies, and S is the highest.
4. If $T$ is the least, the first condition applies, and R is the highest.

Step 4: Evaluate the Options:

1. A. Q is ranked fourth:
   This is not necessarily true because $Q$ must be below both $P$ and $V$, so $Q$ cannot be ranked fourth.

2. B. T is ranked third:
   $T$ could be the least scorer if R is the highest, so $T$ cannot always be ranked third. This is not necessarily true.

3. C. S scores the highest:
   This is a possible scenario when $U$ or $Q$ is the least. However, it is not guaranteed because R could also score the highest in another scenario.

4. D. R is ranked second:
   This must always be true when $V$ is fifth because:

   • In Condition 1, R scores the highest, so R is first.
   • In Condition 2, S scores the highest, meaning R is still one of the top scorers. Thus, R always ranks above V and most others.

Final Answer:

D: R is ranked second.