Embedded OS is the correct answer.

Embedded operating systems are designed to be self-contained in a device and to reside in ROM (Read-Only Memory). They are specifically developed for embedded systems, which are computing devices that are part of a larger system or product. These devices have dedicated functions and are often embedded in other machines or appliances. Examples include smartphones, tablets, smart TVs, automotive systems, industrial control systems, and medical devices.

Here is a detailed explanation of why the other options are incorrect:

a) Multiprocessing OS: Multiprocessing operating systems are designed to support multiple processors or cores in a system. They are not specifically designed to be self-contained in a device and resident in ROM.

b) Batch Processing OS: Batch processing operating systems are designed to process large amounts of data in batches. They are not specifically designed to be self-contained in a device and resident in ROM.

c) Multi-threading OS: Multi-threading operating systems are designed to support multiple threads of execution within a single process. They are not specifically designed to be self-contained in a device and resident in ROM.

d) Embedded OS: As mentioned earlier, embedded operating systems are designed to be self-contained in a device and to reside in ROM. They are tailored to the specific requirements of embedded systems and often have real-time capabilities.

e) Real-Time OS: Real-time operating systems are designed to provide timely and deterministic responses to events. While some embedded operating systems may have real-time capabilities, not all real-time operating systems are self-contained in a device and resident in ROM.

In conclusion, an embedded operating system is defined as self-contained in a device and resident in ROM. It is specifically designed for embedded systems and is not to be confused with other types of operating systems such as multiprocessing, batch processing, multi-threading, or real-time operating systems.