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Kohli (111) and Ajinkya Rahane (45) were at the crease at close of play after putting on an unbeaten 122-run stand at Hyderabad's Uppal Stadium.
India wary of complacency against Bangladesh
But the highlight of the day was a 178-run second-wicket partnership between Vijay (108) and Cheteshwar Pujara (83) which served as a harsh introduction for Bangladesh in their first-ever Test in India.
Taskin Ahmed, Mehedi Hasan and Taijul Islam shared a wicket each for the visitors but the day belonged firmly to their Indian hosts who elected to bat after winning the toss.
Bangladesh started well as seamer Ahmed bowled Lokesh Rahul for two in the very first over of the match but soon the Indian batsmen started bossing the visitors around the park.
Vijay and Pujara laid a solid foundation for other batsmen to build on as they took the attack to the opposition.
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Pujara, who hit two centuries in the recent England series, surpassed former Indian batsman Chandu Borde's domestic record of 1,604 runs in a season during his innings.
He was eventually dismissed by teenager Hasan who otherwise remained fairly ineffective against the number Test team's top order.
Vijay, who survived a run-out chance on 35, smashed twelve fours and a six during his 160-ball stay before being bowled by Taijul early in the final session.
Kohli, who had walked into loud cheers from the crowd before tea, then piloted the innings with aplomb.
The skipper reached his half century in 70 deliveries and soon galloped to his 16th Test hundred as the Bangladesh bowlers had no answers to his effortless stroke-making.
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Kohli, who went past the 1,000-run mark in his 15th innings of the season, got runs all around the ground as he hit 12 boundaries.
Rahane, who was added to the starting line-up in place of Karun Nair — a triple centurion in India's last Test match against England — was happy to play second fiddle to his skipper.
India have won six of their eight encounters against Bangladesh, with two matches ending in draws.
All of the previous matches have been held in Bangladesh which was awarded full Test status 17 years ago.
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