Increasing the maximum range of a radar system requires increasing the peak power transmitted. In this case, the required increase in range is by a factor of two. Let's analyze the relationship between peak power and range to understand why the correct answer is option 'A' (increasing the peak power by a factor of 2).

1. Relationship between peak power and range:
- The range of a radar system is determined by the time it takes for the transmitted signal to travel to the target and return to the radar receiver.
- The time of flight is directly proportional to the distance traveled, which is the range.
- To increase the range, we need to increase the time of flight, which can be achieved by increasing the peak power of the transmitted signal.

2. Square law relationship:
- The received power at the radar receiver is proportional to the square of the transmitted power.
- Mathematically, P_r = P_t * G_t * G_r * (λ^2) * (A_e^2) / (4π)^3 * R^-4, where P_r is the received power, P_t is the transmitted power, G_t and G_r are the antenna gains, λ is the wavelength, A_e is the effective aperture area, and R is the range.
- From this equation, we can see that increasing the transmitted power will result in an increase in the received power, thus increasing the range.

3. Factor of two increase:
- If we increase the peak power transmitted by a factor of 2, the received power will increase by a factor of (2^2) = 4.
- This increase in received power will result in an increase in the range by a factor of the square root of 4, which is 2.
- Therefore, by increasing the peak power transmitted by a factor of 2, the maximum range of the radar system can be increased by a factor of 2.

In conclusion, the correct answer is option 'A' (increasing the peak power transmitted by a factor of 2) because it will result in a proportional increase in the received power and, consequently, the maximum range of the radar system.