Meanwhile, Anirban Lahiri (71-68) was hanging by a thin thread after two rounds and Jeev Milkha Singh missed out with rounds of 74-69 as the cut was expected to fall either at two -under or one-under, in which case Lahiri will play on the weekend.
Kapur, who last made a cut in April in Volvo China at Shenzhen, survived the cut despite a double bogey on par-3 third and bogeys on fifth and 10th as he had four other birdies.
Lahiri, winner of Hero Indian Open and Malaysian Open this season, was going great in first round at three-under after five and then finished at one-over 71. In the second round he had four birdies against two bogeys on seventh and 12th.
Jeev after a 74 in first round faced a tough task and even a 68 was not enough with three birdies and two bogeys.
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Daniel Brooks (64-65) was 11-under and three shots ahead of Johan Carlsson (65-67), Shane Lowry 66-66) and Justin Rose (66-66). Also at eight-under was Richard McEvoy but he had 10 more holes to play.
Rose, the former US Open Champion recovered from a bogey on the first with five birdies in the next seven holes, but could only manage nine straight pars on the back nine.
Playing partner Phil Mickelson, who won the 2013 Open a mile down the road at Muirfield, was in danger of missing the halfway cut when he bogeyed the 11th and 12th, but responded in style with a hat-trick of birdies from the 14th to finish three under.