The syllabus of RA exams conducted in IIT Bombay for ECE is very much similar to that of GATE. The primary purpose of this written test is to ensure you have a holistic understanding of different subjects to work with and manage BTech students in labs of different branches.
There are 5 branches of ECE at IIT Bombay; Communication Engineering, Electronic Systems, Control, and Computing, MicroElectronics, and Power Electronics. Some core points related to the RA exam are
Pro tip: You need to have sound knowledge of the research work in the department. For instance, Wadhwani Electronics Lab(WEL) is one of the main labs in EE dept, and most of the basic electronics lab courses are conducted in this lab. So as an RA, you should be skilled in using CROs, DSOs, working breadboards, and so on. Additionally, your exposure to Micro Controllers or FPGA could increase your chances of selection.