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However, a decline in IT stocks ahead of the earnings season and a weak rupee against the greenback restricted the market rally, traders said.
The 30-share BSE benchmark Sensex climbed 234.12 points or 0.30 per cent to settle at 78,199.11. During the day, it jumped 487.75 points or 0.62 per cent to 78,452.74.
The NSE Nifty gained 91.85 points or 0.39 per cent to 23,707.90.
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In contrast, Zomato, HCL Tech, Tata Consultancy Services, Tech Mahindra, Kotak Mahindra Bank and Infosys were among the losers.
In Asian markets, Seoul, Tokyo, and Shanghai settled in positive territory while Hong Kong ended lower.
European markets were trading with gains in mid-session deals. US markets ended mostly higher on Monday.
The rupee fell 5 paise to settle at 85.73 (provisional) against the US dollar on Tuesday.
Foreign Institutional Investors (FIIs) offloaded equities worth Rs 2,575.06 crore on Monday, according to exchange data.
Global oil benchmark Brent crude dipped 0.12 per cent to USD 76.21 a barrel.
In the previous session, the BSE benchmark tanked 1,258.12 points or 1.59 per cent to close below the 78,000 level at 77,964.99. The Nifty slumped 388.70 points or 1.62 per cent to 23,616.05.