Introduction:
A particle performing Simple Harmonic Motion (SHM) moves back and forth along a straight line about a mean position. The motion of the particle is periodic, meaning that it repeats itself at regular intervals. This question asks about the time taken by the particle to appear at a particular point on its path, and which point it corresponds to.

Explanation:
To answer this question, we need to understand the nature of SHM and the positions of the particle during its motion. SHM is characterized by the following key points:

1. Mean Position: The mean position is the equilibrium position of the particle where there is no net force acting on it. It is the central position between the extreme points of the motion. In SHM, the particle oscillates back and forth around this mean position.

2. Extreme Positions: The extreme positions are the points in the motion where the displacement of the particle from the mean position is maximum. These points represent the farthest positions reached by the particle in both directions.

Now, let's consider the different options provided in the question:

A. Extreme Position:
If the particle appears at a particular point A in consecutive appearances, it must be an extreme position. This is because the particle will take the same amount of time to reach the same extreme position during each oscillation.

B. Mean Position:
If the particle appears at a particular point A in consecutive appearances, it cannot be the mean position. This is because the particle does not stay at the mean position for a significant amount of time during its motion. Instead, it passes through the mean position quickly and spends more time near the extreme positions.

C. Between Positive Extreme and Mean Position:
If the particle appears at a particular point A in consecutive appearances, it cannot be between the positive extreme and mean position. This is because the particle takes different amounts of time to travel from the positive extreme to the mean position and vice versa. It spends more time near the mean position compared to the extreme positions.

D. Between Negative Extreme and Mean Position:
If the particle appears at a particular point A in consecutive appearances, it cannot be between the negative extreme and mean position. This is because the particle takes different amounts of time to travel from the negative extreme to the mean position and vice versa. It spends more time near the mean position compared to the extreme positions.

Conclusion:
Based on the explanations above, if a particle performing SHM takes time T in consecutive appearances at a particular point A, it corresponds to an extreme position as the particle takes the same amount of time to reach the same extreme position during each oscillation.

(A) Extreme Position...