Kapur, the best Indian in the first round with a 70, gave away two double bogeys on 15th and 18th and in between had bogeys on 16th and 17th to finish at five-over.
The cut came at two-over.
Also making an early exit were Jeev Milkha Singh (72-75) and Gaganjeet Bhullar (73-76).
Meanwhile, at the top Shane Lowry closed with a brace of birdies to join Thomas Bjorn at the top of the leaderboard. Lowry responded to a double bogey on the ninth hole by completing the back nine in just 33 strokes.
The third place was shared between Spaniard Rafa Cabrera-Bello and two-time BMW PGA Champion Luke Donald, whose superb 67 was the lowest of the day.