Differential compound motor is a type of DC motor that combines the features of both the shunt-wound and series-wound motors. It has both a series field winding and a shunt field winding. The series field winding is connected in series with the armature, while the shunt field winding is connected in parallel with the armature.

The differential compound motor has the following characteristics:

1. Speed Regulation: The differential compound motor provides good speed regulation due to the presence of both the series and shunt field windings. The series field winding provides a high starting torque, while the shunt field winding provides a stable speed under varying load conditions.

2. Starting Torque: The differential compound motor has a high starting torque, making it suitable for applications that require high torque at startup.

3. Speed-Torque Characteristics: The speed-torque characteristics of a differential compound motor are non-linear. At low loads, the motor operates in the cumulative compound mode, where the series field assists the shunt field, providing high starting torque. At high loads, the motor operates in the differential compound mode, where the series field opposes the shunt field, providing a reduced speed and increased torque.

Now, let's discuss the applications where a differential compound motor is commonly used:

a) Hoist: A hoist is a device used for lifting or lowering heavy loads vertically. Hoists require high starting torque to lift heavy loads, and the differential compound motor provides the necessary torque for this application.

b) Cranes: Cranes are used for lifting and moving heavy objects horizontally. Similar to hoists, cranes require high starting torque and variable speed control, which can be achieved using a differential compound motor.

c) Drilling Machines: Drilling machines often require high torque at low speeds for drilling through tough materials. The differential compound motor can provide the required torque and speed control for drilling applications.

d) None of the mentioned: This option is incorrect because the differential compound motor is commonly used in applications that require high starting torque, variable speed control, and good speed regulation, such as hoists, cranes, and drilling machines.

In conclusion, the correct answer is option 'A' - Hoist. The differential compound motor is commonly used in hoists due to its high starting torque and variable speed control capabilities.