Let's break down the given information step by step:

1. Understanding the Alignment: Two identical straight rods are painted in five distinct colours. Because they are identical, the sequence of colours on both rods is exactly the same. When they are kept parallel side by side and their portions "match", it means they align perfectly and have the same colour. If they were shifted, none of the distinct colours would match. If they were placed in opposite directions, at most only the middle portion would match in colour. However, the problem gives us three matching pairs, which means the rods must be aligned perfectly in the same direction (Position 1 to 1, Position 2 to 2, etc.).

2. Analyzing the Conditions: Let the 5 positions on the rods be 1, 2, 3, 4, and 5. At each position, there is one $P$ label (from Rod 1) and one $Q$ label (from Rod 2). The given matching conditions tell us which labels share the same position:

• $P1$ and $Q3$ are at the same position.
• $P4$ and ($Q1$ or $Q2$) are at the same position.
• $Q4$ and ($P3$ or $P5$) are at the same position.
• $Q3$ and $Q5$ are adjacent (their position numbers differ by 1).

3. Evaluating the Statements (Checking for Possibility): The question asks which of the statements is/are possible. We just need to find if there is at least one valid arrangement that satisfies all conditions for each statement.

Let's try to construct an arrangement where $Q3$ is in the middle (Position 3) and $P2$ is at an extreme (Position 1 or 5):

• Position 1: $P2$ and $Q2$
• Position 2: $P3$ and $Q4$
• Position 3: $P1$ and $Q3$
• Position 4: $P5$ and $Q5$
• Position 5: $P4$ and $Q1$

Let's verify if this arrangement violates any rules:

• Does $P1$ match $Q3$? Yes, both are at Position 3.
• Does $P4$ match $Q1$ or $Q2$? Yes, $P4$ matches $Q1$ at Position 5.
• Does $Q4$ match $P3$ or $P5$? Yes, $Q4$ matches $P3$ at Position 2.
• Are $Q3$ and $Q5$ adjacent? Yes, $Q3$ is at Position 3 and $Q5$ is at Position 4.
• Are all labels $P1-P5$ and $Q1-Q5$ used exactly once? Yes.

4. Conclusion: In our perfectly valid arrangement:

• $Q3$ is at the middle portion (Position 3). Thus, Statement I is possible.
• $P2$ is at an extreme portion (Position 1). Thus, Statement II is possible.

Since both scenarios can occur without violating any given conditions, both I and II are possible.