The National Standard Examination in Chemistry is an entrance exam that precedes the International Chemistry Olympiad.

In general, it's determined based on:

School Enrollment: In most cases, the state in which your school is registered and where you are a bonafide student is your home state for academic examinations-including NSEC.
Domicile Certificate: Sometimes, you might be required to submit a domicile certificate issued by a state as proof of residence.
Considering your case:

School in Rajasthan: Your school falls in the place of Rajasthan. That is why you would automatically satisfy one of the conditions eligible to apply under the quota of the state, Rajasthan.
Home in Madhya Pradesh: If your home falls in Madhya Pradesh, they might not consider your residential address for home state purposes in NSEC unless there are specific rules that make provisions for domicile certificate.
Why School Enrollment Usually Prevalent:

Consistency: Using the criterion of school enrollment, consistency is ensured across all students and makes it easy.
Academic Affiliation: The school is affiliated with a particular state's education board, and their student records are maintained by the respective state's education department.
Exceptions and Clarifications:

Special Cases: In some rare cases, students continuously resident in a state for some number of years may be eligible for the quota of that state, irrespective of where their schools are located.
Organizing Committee Rules: The awarding of eligibility for state quotas is on the discretion of the organizing committee of the NSEC. It is better to refer to the official notification and guidelines for the particular year to which you are making an application.
Recommendations:

Refer the Official Notification: Always refer to the official notification and guidelines for the NSEC to gather proper and current information regarding the eligibility criteria.
Contact the Organizing Committee: All confusion or specific queries should better be addressed to the organizing committee directly.
Gather Requirements: Ensure that one has all the required documents in hand, like a school leaving certificate, mark sheets, etc., as applicable and specified according to the requirement.
Hence, based on general knowledge of the home state quotas for acedemic exams, it is probable that your home state is going to be Rajasthan because your school is based there, but you should check the final communication and contact the organizing committee if required. https://school.careers360.com/articles/iapt-registration