P, Q, R, S and T are ranked 1 to 5 (not necessarily in that order). The rank of P is 4, the rank of Q is not 5, the rank of R is 1, the rank of S is not 2, the rank of T is not 3. Then which of the following is/are correct?

The correct option is D - Neither I nor II.

Step-by-step Trace

First, let us list the fixed ranks and the constraints given in the question:
1. P = 4 (Fixed)
2. R = 1 (Fixed)
3. Q ≠ 5
4. S ≠ 2
5. T ≠ 3
The available ranks to be filled by Q, S, and T are 2, 3, and 5.

Evaluating Statement I: "If the rank of S is 3, then that of T is 2."
- If S = 3, the remaining ranks are 2 and 5.
- The remaining people are Q and T.
- Constraint check: Q cannot be 5 (Q ≠ 5). Therefore, Q must be 2.
- This leaves rank 5 for T.
- Result: If S = 3, then T = 5 and Q = 2.
- Conclusion: Statement I says T is 2, which is incorrect. Statement I is False.

Evaluating Statement II: "If the rank of Q is 3, then that of T is 5."
- If Q = 3, the remaining ranks are 2 and 5.
- The remaining people are S and T.
- Constraint check: S cannot be 2 (S ≠ 2). Therefore, S must be 5.
- This leaves rank 2 for T.
- Result: If Q = 3, then T = 2 and S = 5.
- Conclusion: Statement II says T is 5, which is incorrect. Statement II is False.

Why the other options are incorrect

• Option (a) - I only: This is incorrect because Statement I leads to T being 5, not 2, as shown in the logical trace above.
• Option (b) - II only: This is incorrect because Statement II leads to T being 2, not 5, due to the constraint that S cannot occupy rank 2.
• Option (c) - Both I and II: This is incorrect because both statements fail the logical test when the specific constraints (Q ≠ 5, S ≠ 2, T ≠ 3) are applied to the remaining slots.

Key Concept

This is a logical arrangement problem based on the process of elimination and satisfying multiple negative constraints simultaneously.