Trichy to Hospet there are no direct options.
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One option i will suggest is take Mayiladuthurai-Mysore Express/16231 on 7th April from Trichy to Bangalore city & from there on 8th Morning you can take Mysore-Sainagar Shirdi Express/16217 from Bangalore City to Hospet.You have about 2 hrs of buffer time in between at Bangalore to change over the trains.You can use that time to refresh yourself.
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I you are a Foreign tourist there are 4 Sleeper class tickets available in Foreign tourist quota in 16231 you can book under that quota & for 16217 from Bangalore to Hospet plenty of tickets are available across classes Sleeper/3AC/2AC..