Why are indian time zones on the half hour?

Answer 1

It is complicated but has to do, among several factors, the distance between Calcutta and Bombay. (Read the link pages for the whole explanation.

Calcutta time was 5 hours, 30 minutes, and 21 seconds in advance of GMT, while Bombay Time was 4 hours and 51 minutes ahead of GMT.

Many railway companies, however, standardized on using Madras time as being in between Bombay and Calcutta times, and often this, rather than Bombay time, was used in Indian timetables from the late 1880s onward.