Answer:

Hydraulic turbines are used to convert the potential energy of water into mechanical energy. The head available for a hydraulic turbine, denoted by H, refers to the height difference between the water level above the turbine and the tailwater level.

The power, speed, and discharge of a hydraulic turbine are all influenced by the head available. Let's examine each parameter separately:

Power:
The power output of a hydraulic turbine is proportional to the head available raised to the power of 3/2 (H^3/2). This relationship is derived from the fundamental equation of power:

Power = (density of water) * (acceleration due to gravity) * (discharge) * (head available)

Since the density of water and acceleration due to gravity are constants, we can see that the power is directly proportional to the discharge and head available. Therefore, the power is proportional to H^3/2.

Speed:
The speed of a hydraulic turbine is proportional to the square root of the head available (H^1/2). This relationship is due to the conservation of energy principle. As the head available increases, the potential energy of the water also increases. This increase in potential energy is converted into kinetic energy, resulting in higher turbine speeds.

Discharge:
The discharge of a hydraulic turbine refers to the volume of water passing through the turbine per unit time. It is directly proportional to the head available (H^1/2). The higher the head available, the greater the pressure difference across the turbine, leading to a higher discharge.

Conclusion:
Considering the relationships discussed above, we can conclude that the power, speed, and discharge of a hydraulic turbine are proportional to H^3/2, H^1/2, and H^1/2, respectively. Therefore, the correct answer is option 'B' - H^3/2, H^1/2, H^1/2.