Yes, India does make cryogenic engines. An example is the CE-20 cryogenic engines that are used in GSLV. You can read about the GSLV Mk III – A Comprehensive Overview of Heavy Lift Launch Vehicle in the given link.
Further readings:
- Indian Space Research Organisation [ISRO Notes for UPSC]
- Mars Orbiter Mission (MOM) – A Complete Overview of Mangalyaan Mission
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