The simple harmonic displacement equation is , x =Asinwt
Sub. x=A in the above equation , we get , wt = π/2 , and sub. w =2π/12 (T =12 s given) and solve for t . We get, t= 3s...

Explanation:
To find the minimum time taken by a particle to cover a distance equal to the amplitude (A) of its motion, we need to understand the concept of time period and harmonic motion.

Harmonic Motion:
Harmonic motion is a type of periodic motion where the restoring force acting on the particle is directly proportional to its displacement from the equilibrium position and is always directed towards the equilibrium position. The motion of a particle undergoing harmonic motion can be described by a sinusoidal function.

Time Period:
The time period of a periodic motion is the time taken for one complete cycle of the motion. In the case of harmonic motion, the time period is the time taken for the particle to complete one oscillation from its initial position, through its equilibrium position, to the opposite extreme position, and back to the equilibrium position.

Formula for Time Period:
The formula to calculate the time period (T) of a harmonic motion is given by:
T = 2π√(m/k)
where T is the time period, m is the mass of the particle, and k is the spring constant or the constant of proportionality between the force and displacement.

Minimum Time Taken:
In harmonic motion, the particle oscillates symmetrically around its equilibrium position. The amplitude (A) of the motion is the maximum distance the particle travels from its equilibrium position. To cover a distance equal to the amplitude, the particle needs to travel from its equilibrium position to one extreme position and then back to the equilibrium position in one complete cycle.

Since the time period (T) is the time taken for one complete cycle, the minimum time taken by the particle to cover a distance equal to the amplitude is equal to half the time period:
Minimum Time Taken = T/2 = (2π√(m/k))/2 = π√(m/k)

Conclusion:
The minimum time taken by a particle to cover a distance equal to the amplitude of its motion is given by the formula π√(m/k), where m is the mass of the particle and k is the spring constant. This formula is derived from the concept of time period in harmonic motion and represents the time taken for the particle to travel from its equilibrium position to one extreme position and back to the equilibrium position in one complete cycle.