This is a matching-type question where we need to identify the properties of standing waves in four different systems. Let's analyze each system based on fundamental physics principles.

Column II Definitions:

• (P) Longitudinal waves: Oscillations are parallel to the direction of wave propagation (e.g., sound waves in air).
• (Q) Transverse waves: Oscillations are perpendicular to the direction of wave propagation (e.g., waves on a string).
• (R) Fundamental wavelength is L ($\\\lambda_f = L$)
• (S) Fundamental wavelength is 2L ($\\\lambda_f = 2L$)
• (T) Fundamental wavelength is 4L ($\\\lambda_f = 4L$)

Step-by-step Analysis of Each System in Column I:

1. System (A): A string of length L fixed at both ends.

• Nature of Wave: Waves on a string are transverse waves. So, (A) matches with (Q).
• Fundamental Wavelength: For a string fixed at both ends, the ends must be nodes. The fundamental mode (lowest frequency) has the simplest standing wave pattern, which consists of a single antinode between two nodes at the ends. The distance between two consecutive nodes is half a wavelength ($\\\lambda/2$). $L = \frac{\lambda_f}{2}$ Therefore, the fundamental wavelength is $\\\lambda_f = 2L$. So, (A) matches with (S).
• Correct Match for (A): (Q), (S)

2. System (B): A string of length L fixed at one end and free at the other.

• Nature of Wave: Waves on a string are transverse waves. So, (B) matches with (Q).
• Fundamental Wavelength: The fixed end must be a node, and the free end must be an antinode. The fundamental mode has the simplest pattern, which is a single node at one end and a single antinode at the other. The distance between a node and an adjacent antinode is a quarter of a wavelength ($\\\lambda/4$). $L = \frac{\lambda_f}{4}$ Therefore, the fundamental wavelength is $\\\lambda_f = 4L$. So, (B) matches with (T).
• Correct Match for (B): (Q), (T)

3. System (C): An open pipe of length L (open at both ends).

• Nature of Wave: Sound waves in a pipe are longitudinal waves. So, (C) matches with (P).
• Fundamental Wavelength: In a pipe open at both ends, displacement antinodes are formed at the open ends. The fundamental mode has a node in the middle and antinodes at both ends. The distance between two consecutive antinodes is half a wavelength ($\\\lambda/2$). $L = \frac{\lambda_f}{2}$ Therefore, the fundamental wavelength is $\\\lambda_f = 2L$. So, (C) matches with (S).
• Correct Match for (C): (P), (S)

4. System (D): A closed pipe of length L (closed at one end, open at the other).

• Nature of Wave: Sound waves in a pipe are longitudinal waves. So, (D) matches with (P).
• Fundamental Wavelength: The closed end must be a displacement node, and the open end must be a displacement antinode. This case is analogous to the string fixed at one end. The distance between the node and the antinode is a quarter of a wavelength ($\\\lambda/4$). $L = \frac{\lambda_f}{4}$ Therefore, the fundamental wavelength is $\\\lambda_f = 4L$. So, (D) matches with (T).
• Correct Match for (D): (P), (T)

Conclusion and Comparison with Options

Based on our analysis, the correct matching is:

• (A) $\\to$ (Q), (S)
• (B) $\\to$ (Q), (T)
• (C) $\\to$ (P), (S)
• (D) $\\to$ (P), (T)

Now, let's examine the given options: A: (A) $\\to$(T); (B) $\\to$(P), (S); (C) $\\to$(Q), (S); (D) $\\to$(Q) B: (A) $\\to$(P), (T); (B) $\\to$(P); (C) $\\to$(Q), (S); (D) $\\to$(Q) C: (A) $\\to$(P); (B) $\\to$(P), (S); (C) $\\to$(Q); (D) $\\to$(Q), (R) D: (A) $\\to$(P), (T); (B) $\\to$(P), (S); (C) $\\to$(Q), (S); (D) $\\to$(Q), (R)

None of the provided options match our derived correct set of pairings. For instance, option D, the stored answer, states:

• (A) $\\to$ (P), (T): Incorrect. Should be (Q), (S).
• (B) $\\to$ (P), (S): Incorrect. Should be (Q), (T).
• (C) $\\to$ (Q), (S): Incorrect wave type (Q for P), but correct wavelength (S).
• (D) $\\to$ (Q), (R): Incorrect wave type (Q for P) and incorrect wavelength (R for T).

Since no option is correct, the question is flawed. This was a known issue with this question from the JEE Main 2018 exam. As an AI tutor, my derived answer is that no option is physically correct.