At the break, Cook was unbeaten on 56 with in-form Gary Ballance undefeated on 40 in an unbroken 82-run stand for the second wicket after Varun Aaron had removed opener Sam Robson (37) early in the day.
While Cook had six fours in 140 balls, fellow left-hander Ballance was more attacking having eight hits to the fence in 66 balls.
The Indian bowlers toiled hard but the conditions had changed quite a bit from the first day.
The day started with England at 62/0 overnight, trailing only by 86 runs. India needed early breakthroughs to force the hosts onto the backfoot with Kumar starting proceedings.
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And they did get a bright spark when in the second over of the day - 21st of the innings - Aaron bowled Sam Robson with quick straight delivery that beat the batsman.
It brought Ballance to the crease but he was quick to get off the mark and got a nice start first up to dash any hopes the Indians had of triggering a sudden collapse.