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The location approved for setting up the India Based Neutrino Observatory is in the district of Theni.
Option 1, Emakulam, is incorrect. This is because the India Based Neutrino Observatory is not planned to be set up in the district of Emakulam.
Option 2, Guntur, is incorrect. The India Based Neutrino Observatory is not planned to be set up in the district of Guntur.
Option 3, Theni, is the correct answer. The India Based Neutrino Observatory is planned to be set up in the district of Theni. This location was chosen for its suitable geological features and minimal human interference.
Option 4, Udupi, is incorrect. The India Based Neutrino Observatory is not planned to be set up in the district of Udupi.
Therefore, option 3 - Theni, is the correct answer.