**Problem Analysis:**
To find the minimum time in which the motorcycle can cover a distance of 1.5 km, we need to consider the acceleration and retardation limits of the motorcycle's engine and brakes respectively. The motorcycle will accelerate until it reaches its maximum speed, and then it will decelerate using its brakes to come to a stop at the desired distance. We can break down the problem into two parts:

1. The time taken to accelerate to the maximum speed.
2. The time taken to decelerate from the maximum speed to a stop.

**Solution:**

**Part 1: Acceleration**
The motorcycle's engine can produce a maximum acceleration of 5 m/s^2. We need to find the time taken to reach the maximum speed.

Distance covered during acceleration, S = 1.5 km = 1500 m
Acceleration, a = 5 m/s^2

Using the formula: S = ut + (1/2)at^2
Where:
- S is the distance covered
- u is the initial velocity (0 in this case)
- a is the acceleration
- t is the time taken

1500 = 0 + (1/2) * 5 * t^2
3000 = 5t^2
t^2 = 600
t = √600
t ≈ 24.49 s

**Part 2: Deceleration**
The motorcycle's brakes can produce a maximum retardation of 10 m/s^2. We need to find the time taken to decelerate from the maximum speed to a stop.

Using the same formula as above, but with a negative acceleration (retardation) value:
1500 = 0 + (1/2) * (-10) * t^2
3000 = -5t^2
t^2 = -600
Since time cannot be negative, we can ignore this solution.

Therefore, the minimum time taken to cover a distance of 1.5 km is the sum of the acceleration and deceleration times:
Total time = 24.49 s + 24.49 s
Total time ≈ 48.98 s

**Conclusion:**
The minimum time in which the motorcycle can cover a distance of 1.5 km is approximately 48.98 seconds. This time includes the acceleration phase, where the motorcycle reaches its maximum speed, and the deceleration phase, where it comes to a stop using its brakes.