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The India skipper said Cook's determination reflected in his batting as he scored 147, his 33rd century, in his last Test for England.
Cook, who hit a 71 and 147 in the two innings, described James Anderson, who became Test cricket's highest wicket-taking fast bowler, as England's best player of all time.
Once the 204-run sixth wicket stand between Rahul and Pant ended, India tried to hang on for a respectable draw but it was of no avail.
By the time Alastair Cook was done, he had also brought up that century 'is 33rd in Test cricket' before ending on 147.
The Essex left-handed opener, who is retiring from international cricket after his match, was the first batsman since Mohammad Azharuddin in 2000 to score hundreds in both their first and last Tests.
The veteran did not disappoint the 19,000 spectators at the south London ground, reaching his 33rd century in 161 Tests in bizarre fashion, courtesy of four overthrows.
Batting at 96, Cook drove Ravindra Jadeja for a single in the cover region. Bumrah aimed hard at stumps and it resulted in giving five runs to the home opener.
Following a disappointing year, Cook's late flurry has left him with an impressive career average of 45.35.
The day certainly belonged to Cook, who scored his 33rd hundred his last appearance to join a select band of players having scored a hundred on debut and final Test.
Cook said that he had "nothing left in the tank" but what he didn't say was that there was a last ounce present and it was meant for the 'Three Lions'.
Cook's 181-run third-wicket partnership with his successor as captain, Joe Root, who is unbeaten on 92, has taken England's lead to 283.
Alastair Cook, in his final innings before international retirement, was 46 not out and Joe Root, his successor as England captain, 29 not out.
Strauss, now the England and Wales Cricket Board director, addressed the England team before handing Cook a cap numbered 161 to mark the final international match of a 12-year international career.
Former England skipper Alastair Cook top-scored for the hosts with a patient 71 off 190 balls before he was removed by Jasprit Bumrah.
Cook, who will retire from international cricket after the fifth test against India, said that as captain, he was involved in the decision to sack Pietersen but the final decision was collective.