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Since you have born in Rajasthan but all your studied life in Delhi, you may not get a direct reservation of being a Rajasthan resident to write the exam for state governments. Usually it requires belonging to any state either by continuing there, or you have educated yourself there, or received a domicile certificate from that state.
You would probably have to get a domicile certificate from the state of Rajasthan to become a resident of Rajasthan. A domicile certificate is issued when a person has properties or even his or her family has their property holdings in that state, or when one is residing in that state for a relatively longer period, or in such cases where a person's parents have some government service ties within that particular state. You may check the eligibility criteria for domicile in Rajasthan and apply there, or through local authorities or even online on the portals of the Rajasthan government if needed.
This certificate will serve to determine your residency status for state exams.
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