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The rupee is trading at 60.45/46 versus its Thursday's close of 60.29/30 as dollar demand for gas and oil imports aids.
Three traders cite dollar bidding from GAIL India for around $50 million in late afternoon trade.
Traders say state-run banks seen buying the greenback on behalf of state-run oil companies but good selling by a few foreign banks helps limit a much sharper rise in the pair.
The BSE Sensex and Nifty hit record highs as lenders such as Axis Bank rallied ahead of their financial results starting later this week, while strong gains in debt markets reduced concerns about potential losses in the sector's bond portfolios.
Most emerging Asian currencies eased on Monday as the dollar stayed firm after upbeat U.S. economic data and on caution over tensions in Ukraine, while trading in the region was subdued with some global markets still closed for the Easter holiday.