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Sharma led the fightback with a chancy unbeaten 113 and the 207 runs he shared with Abhishek Nayar (99) helped Mumbai survive Bhuvneshwar Kumar's five-wicket haul.
Uttar Pradesh captain Mohammad Kaif's decision to insert Mumbai on a greenish track paid rich dividends with pacer Bhuvneshwar Kumar dealing a triple strike, including the prized scalp of Sachin Tendulkar (0), to reduce the Wasim Jaffer-led side to 55 for four inside 20 overs.
Poor fielding soon came to hurt UP as Kaif, otherwise the safest pair of hands, twice dropped Rohit in the slips to allow Mumbai slacken the noose and regain much of the lost ground.
Rohit and Ajit Agarkar (21) were in the middle at stumps.
Rohit's unbeaten 206-ball knock, including 18 fours, may not count among his best but it came just when the team needed it most.
Nayar looked more solid of the duo and the left-hander hit 13 fours and a six en route to a 190-ball 99 at the Rajiv Gandhi International Stadium at Uppal near Hyderabad.
Bhuvneshwar, who added two more scalps in the afternoon to his morning tally, finished with a figure of five for 64.
Put into bat, Mumbai found their skipper back in the hut in the sixth over when Bhuvneshwar's inswinger trapped Jaffer (1) plumb in front of the stumps. More
Jaffer should have gone with a duck against his name when he was caught behind off Praveen Kumar but the umpires felt otherwise.
Looking in all sort of problems against the UP pacers, Vinayak Samant (26) perished in the same manner and 13 runs later, RP Singh trapped Ajinkya Rahane (23) just when the batsman seemed to have got his eyes in.
The biggest blow, however, was dealt in the next over when Bhuvneshwar brought one back into Tendulkar and a desperate Shivkant Shukla dived to pouch the ball which took the bat-pad route before lobbing up. It turned out to be Tendulkar's maiden duck in first class cricket.
Kaif soon ran out of luck and grassed two sitters that allowed Rohit prop up the Mumbai innings with adequate help from Nayar.
Batting on 16, Rohit edged Praveen and was relieved to see Kaif spill the regulation catch at first slip. Again on 39, Kaif dropped Rohit, this time off RP Singh.
The duo survived some torrid moments, and umpteen run out chances as well, before steadying the ship.
Nayar went after Piyush Chawla, coming down the track and hitting the leg-spinner over the long-on boundary and following it with a swept four that brought up Mumbai's 200.
Rohit also belted the chubby spinner, hitting him for three consecutive boundaries. Another four off Parvinder Singh took him to 99 and a single in the same over brought up Rohit's fourth first class century.
At the other end, Nayar hit Parvinder Singh for back-to-back fours but the century he so thoroughly deserved was to elude him.
Armed with the new ball, Bhuvneshwar dealt a double blow this time, trapping Nayar leg before and getting Sairaj Bahutule (0) caught behind to find himself on a hat-trick before next man in Ajit Agarkar thwarted his bid.