Cook, a tormentor of India in the past Test series especially in their last visit four years ago, made amends for his first innings failure to compile a well-crafted 130 before applying declaration when his side reached 260 for three, leaving the hosts with a target of 310 to get in 49 overs.
This was Cook's sixth Test hundred against India.
Chasing the improbable target, India made 49 for two at tea, needing 261 more runs to get in a minimum 31 overs in the last session of the Test, which was heading towards a tame draw.
India commenced their second innings poorly by losing Gautam Gambhir for a duck with the team score too on nil in the second over bowled by Chris Woakes.
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Gambhir, who made 29 in the first innings, fended awkwardly at a short and rising ball from the tall fast bowler and guided it straight to the second slip fielder.
This left first innings centurions Vijay and one-down Cheteshwar Pujara to bat out till the end of the session. They nearly did it, albeit with some luck against the sharply spinning ball, before Pujara fell for 18, leg before to leggie Adil Rashid when playing back, at the stroke of tea.
In the morning, leg-spinner Amit Mishra struck twice in the space of 11 balls after being brought on late into the attack, but England carried on merrily after this twin strike to reach 211 for 2 at lunch with Cook setting a new benchmark by becoming the first overseas batsman to notch up five hundreds on Indian soil.
Cook, playing in his 136th Test, was level with West Indians Everton Weekes and Clive Lloyd and South African Hashim Amla with four centuries in India when the series commenced and went past the trio today.
The duo hardly looked in trouble on a fifth day track, after commencing at the overnight 114 for no loss.
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