Two identical straight rods are painted in five distinct colours so that each of them gets divided into five equal parts along the length. In one of them, the portions are marked P1, P2, P3, P4 and P5 (not necessarily in that order) whereas in the other, they are marked Q1, Q2, Q3, Q4 and Q5 (not necessarily in that order). When the rods are kept parallel to each other side by side, P1 and Q3 match, P4 matches Q1 or Q2, and Q4 matches P3 or P5. If Q3 and Q5 are adjacent, which of the following is/are possible?

The correct option is C - Both I and II.

Solution

Let the five positions on the rods be 1, 2, 3, 4, and 5. Since the rods are identical and kept parallel, the positions for P and Q correspond directly (e.g., if P1 is at position 1, it matches with whatever Q is at position 1).

Given Constraints:
1. P1 matches Q3. (P1 and Q3 are at the same position).
2. P4 matches Q1 or Q2.
3. Q4 matches P3 or P5.
4. Q3 and Q5 are adjacent.
5. There are 5 distinct colors/parts: {P1, P2, P3, P4, P5} and {Q1, Q2, Q3, Q4, Q5}.

Testing Statement I: Q3 is at the middle (Position 3).
If Q3 is at position 3, then P1 is also at position 3.
Since Q3 and Q5 are adjacent, Q5 must be at position 2 or 4. Let's assume Q5 is at position 4.
Now we need to place Q4. Q4 matches P3 or P5. Let's place Q4 at position 1.
Now we check P4. P4 matches Q1 or Q2. If we place Q1 at position 5, P4 can be at position 5.
This leaves Q2 at position 2.
Arrangement:
Position: 1 - 2 - 3 - 4 - 5
Rod Q: Q4 - Q2 - Q3 - Q5 - Q1
Rod P: P3 - P2 - P1 - P5 - P4
Check: P1 matches Q3 (Yes), P4 matches Q1 (Yes), Q4 matches P3 (Yes), Q3/Q5 adjacent (Yes).
Conclusion: Statement I is possible.

Testing Statement II: P2 is at an extreme (Position 1 or 5).
Let's try placing P2 at position 1.
If P2 is at position 1, then Q must have a value there (say Q4).
If Q4 is at position 1, it must match P3 or P5. Let's say P3 is at position 1. But we just placed P2 there. This is a contradiction.
Wait, let's try another configuration:
Position: 1 - 2 - 3 - 4 - 5
Rod P: P2 - P4 - P1 - P3 - P5
Rod Q: Q4 - Q1 - Q3 - Q5 - Q2
Check:
1. P1 matches Q3? Yes (Position 3).
2. P4 matches Q1 or Q2? Yes (P4 matches Q1 at Position 2).
3. Q4 matches P3 or P5? Yes (Q4 matches P2... wait, let's adjust).
Let's try:
Rod P: P2 - P3 - P1 - P4 - P5
Rod Q: Q2 - Q4 - Q3 - Q1 - Q5
Check:
1. P1 matches Q3 (Pos 3).
2. P4 matches Q1 (Pos 4).
3. Q4 matches P3 (Pos 2).
4. Q3 and Q5 are adjacent? No (Q3 is at 3, Q5 is at 5).
Let's adjust: Rod Q: Q2 - Q5 - Q3 - Q4 - Q1.
Rod P: P2 - P5 - P1 - P3 - P4.
Check: P1=Q3 (Pos 3), P4=Q1 (Pos 5), Q4=P3 (Pos 4), Q3/Q5 adjacent (Pos 3 & 2).
In this valid case, P2 is at Position 1 (Extreme).
Conclusion: Statement II is possible.

Why the other options are incorrect

• Option (a) - I only: This is incorrect because it ignores the possibility of Statement II. As shown in the step-by-step trace, a valid configuration exists where P2 occupies an extreme position while satisfying all given constraints.
• Option (b) - II only: This is incorrect because it ignores the possibility of Statement I. A configuration where Q3 is in the middle (Position 3) is logically consistent with the adjacency of Q5 and the matching rules for P1, P4, and Q4.
• Option (d) - Neither I nor II: This is incorrect because both scenarios are "possible." In UPSC CSAT logic, "possible" means there exists at least one valid distribution of P and Q labels that satisfies the conditions. Since both can be satisfied, this option is false.

Key Concept

Logical consistency and existence of permutations under multiple constraints.