NOC is generally valid for 3 months (check). You will have to get the car re-registered in Karnataka within the validity period by paying life-time tax as applicable to old vehicles on depreciated value assessed by RTO (Karnataka).
Thereafter, once the re-registration comes through, you may seek pro-rata refund of life-time tax paid earlier in UP (cumbersome job it is).
Honesty is not a Spare Wheel that you pull out when in trouble. It's a Steering Wheel that keeps you on the right path throughout the life's journey.
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