Is the standard time in India decided by the local time at 82.5° E longitude?

Yes, the statement is true. Indian Standard Time is calculated from the clock tower in Mirzapur nearly exactly on the reference longitude of IST at 82°30'E, within 4 angular minutes. In 1905, the meridian passing east of Allahabad was declared as an IST and was declared as an IST in 1947 for the dominion of India. The longitude of 82°5'E passing through Naini near Allahabad was chosen as the standard meridian for the whole country because there is a time lag of more than an hour between western India and northeastern India, hence approximately synchronizing with UTC+05:30 in central India. Currently, the Council of Scientific and Industrial Research—National Physical Laboratory (CSIR-NPL) maintains the Indian Standard Time with the help of the Allahabad Observatory.