جَاءَ حَضْرَمَوْتُ - Hadramawt is nominative with dammah.
رَأَيْتُ حَضْرَمَوْتَ - It is accusative with fathah.
مَرَرْتُ بِحَضْرَمَوْتَ - Without ال or idafah, it is genitive with fathah because it is ghayr munsarif.
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Worked Example: Hadramawt
Cause of Ghayr Munsarif: Compound Proper Names
جَاءَ حَضْرَمَوْتُ - Hadramawt is nominative with dammah.
رَأَيْتُ حَضْرَمَوْتَ - It is accusative with fathah.
مَرَرْتُ بِحَضْرَمَوْتَ - Without ال or idafah, it is genitive with fathah because it is ghayr munsarif.